Skip to main content

Full text of "A complete physics written for London medical students and general use"

See other formats 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/completephysicswOOwhituoft 


A  COMPLETE  PHYSICS 


WRITTEN    AT    MY    PLEASURE     FOR    YOURS 


BRINGING    ME    ONLY    THE    ONE    THING    THAT    MATTERS 


THE    CERTAINTY    THAT    IT    CAN     SERVE     YOU     WELL 


THIS    BOOK    IS 


DEDICATED 


TO    YOU    ALL 


A  COMPLETE 

PHYSICS 

WRITTEN  FOR 

LONDON  MEDICAL  STUDENTS 

AND  GENERAL  USE 


\    BY 

W.  H;  WHITE 

SENIOR  OF  THE  LECTURERS  AND  EXAMINERS  IN 
PHYSICS  IN  THE  MEDICAL  SCHOOLS  OF  LONDON 


Containing  420  Diagrams 

and  1000  Exam  Questions 

with  Brief  Solutions 


PRINTED   AND   BOUND   BY 

RICHARD  CLAY  &  SONS,  LTD. 

BLACKFRIARS   HOUSE,    E.C.4,    AND    BUNGAY,    SUFFOLK 

SOLD   AT   THE   USUAL   MEDICAL   SCHOOL   BOOKSTORES 

A2^D   OBTAINABLE   FROM  ALL   BOOKSELLERS 

OB  POST  FBEE  FROM  THE   PRINTERS 

Price   15/- 


"  Behold,  there  were  very  many  .  .  . 
.  .  .  and  lo,  they  were  very  dry. 

.  .  .  Can  these  bones  live  ?  " 


COMPLETE   PHYSICS.-Please  make  these  alterations  :- 


Page  34 
161 
178 
188 
261 
268 
272 
279 
280 
284 
286 
306 
321 
341 
451 
459 
479 
491 
496 
541 
557 
566 
577 
581 
604 
609 
638 

686 
688 
689 
731 
744 
770 
775 

779 
786 
789 
792 
802 
808 
811 
138 
252 
340 


1  o wer  formula  alter  M  /m 

top 

half-way  down 


line  5 

half-way  down 
2/3 
2/3 

1/3 

2/3 

half-way 

2/3 

final  formxila 

end 

half-way  down 

1/3 

line  9 

„    1 
half-way  down 
formula 
after  N  pole 
near  end 
top  of  diagram 
1/3  down 
Example 
1/3  down 


2/3       „ 
end 

near  end 
line  1 
1/3  down 


methylated 

»    §415 

»     §385 

»     + 

,,    §349 

„    §201 

,,  §201 
add  §  181 
»  §  280 
„  §  181 
„  §  391 
„  per  sec. 
alter  ¥\g.  136 
„  §955 
„  once 
„  pints 
„     §588 

, ,    presently  they  will  give 
„     CgHgaOe 
„     (d  +  iZ)2  to  right 
„     1665 
,,    Y 

„    sparkling 
„     850,000 
„     an  eighth  h.p. 
„    Fig.  360 
„    Fig.  362 
„    §884 


to  mjM. 
„  anhydrous 
,,§414 
„  §  386 

!,'  §  350 
„  §  202 
„  §  202 
„  §  265 
„  §  337 
„  §  265 
„  §  381 

„  Fig.  137 

„  §  959 

„  twice 

„  points 

„  §  630 

„  June  1935  they  gave 

„  CeHjjO, 

„  (d  -  W  to  right 

„  1666 

„  I 

„  sparking 

„  425,000 

„  a  car-8t€irter 

„  Fig.  361 

„  Fig.  363 

„  §  984 


§  395  „  §  396 

§  783  „  §  733 

§  885  „  §  886 

»    §876  ,,§887 

2/3       „  „    §886  ,,§885 

line  5  „     1874  „  1824 

2/3  down  „    §838  ,,§840 

1/3       „  „    §880  ,,§891 

1/4       „  „    §§877,653  ,,§§888,654 

end  „    §658  ,,§708 

„    §870  ,,§960 

1/3  down  „    §919  ,,§920 

line  6  „    §  557  „  §  559 

c.  Ill,  5  „     2-24,  0-228  „  4-48,  0-456 

§  195,  Ice  is  quickly  splintered  small  with  a  hat-pin. 
formula,  delete  the  third  fraction-line. 

Syren  is  more  correctly  Siren,  from  Gr.  aeireen,  a  word  of  origin  as 
doubtful  as  the  charmers  themselves,  syrinx^  a  reed  or  whistle 
(whence  syringe)  being  passed  over  by  the  etymologists.  What- 
ever its  parentage,  it  is  a  crying  child.  For  my  anglicism,  cf. 
Syren  and  Shipping  (9d.  weekly),  rhyme  (1550),  tyre  (tie-r),  and 
eke  Whyte. 
704     1  /3  down,  delete  in  this  case. 

797  end  of  §  978.  Later  in  1935  the  views  entertained  were  that  Sun 
and  Planets  '  consolidated  '  into  shape,  much  as  we  see  them, 
almost  simultaneously,  §  945 ;  and  that,  in  10,000  million  years 
or  less,  an  ageing  star  explodes  into  the  100,000  times  greater 
glory  of  a  '  New  Star,'  faded  in  four  months. 


CONTENTS 

MECHANICS 

CHAP.  PAOK 

I.      INTBODUCTOBY ;     UNITS  OF  TIME,   SPACE,  AND   MASS  1 

II.      MOTION   OF   A  PABTICLE  .....  7 

III.  NEWTONIAN       LAWS.  MOMENTUM       AND       FOBCE. 

GRAVITATION 17 

IV.  ENEBGY   AND   MECHANICAL   WOBK  ...  34 

V.      STATICAL   EQUILIBBIUM   OF   FOBCES  .  .  .41 

VI.      MOTION  IN  A  CUBVE,  CENTBIFUGAL  FORCE,  PENDU- 
LUM ;     BOTATION         ......  53 

VII.      FLUIDS,  PBESSUBE  AND   ITS  MEASUBEMENT,   FLUIDS 

IN   MOTION 67 

VIII.      FLOTATION,   AND    SPECIFIC   GBAVITY  ...         89 

IX.      ELASTICITY 99 

X.      THE  PBECISE  MEASUBEMENT  OF  LENGTH,  TIME,  AND 

MASS 110 

HEAT 

XI.      THEBMAL   EXPANSION   .  .  .  .  .  .122 

XII.      THEBMOMETBY,    TEMPEBATURE    AND    ITS    MEASUBE- 


MENT 


135 


XIII.  CALORIMETBY,  THE  MEASUBEMENT  OF  QUANTITY  OF 

HEAT 151 

XIV.  CALOBIMETBY   BY  LATENT  HEAT  METHODS        .             .  167 
XV.      COOLING.      CONVECTION  AND  CONDUCTION  OF  HEAT  163 

XVI.      THE   MECHANICAL   EQUIVALENT   OF  HEAT  .  .  177 

XVn.      CHANGE   OF   STATE.      MELTING,   OB  FUSION       .  .  182 

V 


vi  CONTENTS 

CHAP.  ^^^^ 

XVIII.      CHANGE   OF   STATE.      VAPORIZATION  .  .  .192 


XIX. 


HEAT   ENGINES   AND    COLD   ENGINES  .  .  .       209 


XX.      HYGROMETRY,  THE  MEASUREMENT  OF  ATMOSPHERIC 

MOISTURE 222 

XXI.      METEOROLOGY  AND   THE   WEATHER          .             .            .  231 

MOLECULAR  PHYSICS 

XXII.      THE   VISCOSITY   OF  FLUIDS    .....  250 

XXIII.  THE   LIQUID    SURFACE                .             .             .             .             .  258 

XXIV.  DIFFUSION.      OSMOSIS  .  .  .  .  .  .271 

MECHANICS   OF  PERIODIC  MOTION 

XXV.      PERIODIC   MOTION   OF   A  PARTICLE              .             .             .  288 

XXVI.      WAVE    MOTION,    INTERFERENCE,    REFLECTION,    AND 

REFRACTION,    OF   WAVES     .....  297 

SOUND 

XXVII.      THE   TRAVEL   OF   SOUND 318 

XXVIII.      MUSICAL  PITCH.      STRINGS 337 

XXIX.      ACOUSTIC  RESONANCE.      PIPES         ....  349 

XXX.      COMPLEX  VIBRATORS,  PLATES,  VOICE,   EAR,   ETC.       .  364 

LIGHT 

XXXI.      ILLUMINATION  AND  ITS  MEASUREMENT,  PHOTOMETRY  377 

XXXII.      THE  REFLECTION  AND  REFRACTION  OF  LIGHT              .  387 

xxxm.     LENSES  (thin) 403 

XXXIV.      CURVED   MIRRORS 420 

XXXV.      PRACTICAL  METHODS  OF  MEASUREMENT  OF  MIRRORS 

AND   THIN   LENSES     ......  428 

XXXVI.      THICK  LENSES 432 

xxxvn.    COLOUR 439 

XXXVIII.      ABERRATIONS   OF  MIRRORS   AND   LENSES             .             .  466 


CONTENTS 


CHAP.  PAOB 

XXXIX.      THE   EYE     ........       476 


XL.      OPTICAL  INSTRUMENTS 
XLI.      POLARIZED   LIGHT 


488 

535 


MAGNETISM 

XLII.      MAGNETS   AND   MAGNETIC   MATERIALS      .  .  .       643 

XLIII.      MAGNETIC   FIELDS.      THE   EARTH'S   MAGNETISM  .       553 

ELECTROSTATICS 

XLIV.      FRICTION AL   ELECTRICITY       .....       666 
XLV.      ELECTRIC  FIELD,  POTENTIAL,  CAPACITY,  AND  ENERGY      580 

MAGNETISM  AND  ELECTRICITY 


XLVI.      MAGNETIC   FIELDS   AND   ELECTRIC   CURRENTS,   ELEC 
TROMAGNETIC  INDUCTION 


XLVII.  THE   MEASUREMENT   OF   ELECTRIC   CURRENT 

XLVni.  RESISTANCE   AND   ITS   MEASUREMENT 

XLIX.  ELECTROMOTIVE   FORCE 

L.  ELECTRICAL  POWER  AND   ENERGY 

LI.  ALTERNATING   CURRENT 


594 
612 
623 
642 
655 
666 


ELECTRICITY 

LII.      THE  TRANSPORT  OF  ELECTRICITY  THROUGH  LIQUIDS  690 

LIII.      THE  TRANSPORT  OF  ELECTRICITY  THROUGH  GASES   .  719 

LIV.      X-RAYS 742 

LV.      RADIOACTIVITY 766 


LVI.      RADIATION 


RADIATION 
774 

SOLUTIONS  TO  EXAM  QUESTIONS        .  811 

INDEX 835 


PHYSICS 

MECHANICS 


CHAPTER   I 
UNITS 

AVE  FRATER! 

Four  and  twenty  years  ago  I  brought  out  a  physics  book 
for  the  use  of  students  of  science  and  medicine  alike ;  it  has 
helped  many  on  their  way,  but  now  I  count  it  out  of  date. 

For  while  it  was  a  joy  to  recognize,  in  the  showcase  labelled 
'  Apparatus  over  100  years  old,'  in  the  Faraday  Centenary  Exhibi- 
tion of  1931,  so  many  originals  of  once-famihar  woodcuts,  yet  when 
one  prefers  to  derive  one's  elucidations  of  physical  principles  from 
the  numberless  applications  of  them  that  surround  us  in  present- 
day  life,  the  rate  of  change  becomes  amazing. 

So,  with  some  few  years  more  experience  of  all  sorts  and  sizes 
and  sexes  of  students  of  varied  ages  and  aims,  I  take  up  my  pen 
again  (it  is  a  pencil,  and  4000  miles  of  sea  lie  ahead),  with  a  great 
deal  more  diffidence,  to  essay  another  book,  this  time  for  medical 
students  only.  For  as  regards  Natural  Philosophy,  the  study  of 
the  nature  of  things  inanimate,  I  have  learnt  a  difference  between 
them  and  their  cousins  in  science  :  the  medical  averages  half  the 
brains  and  double  the  humanity.  He  is  inclined  to  take  the  Socratic 
view  of  those  who  spend  their  lives  Trcpi  tov  (^uaeox;,  the  matters 
they  ask  him  to  pry  into  seem  so  valueless  in  the  market-place,  so 
divergent  from  the  ends  he  has  in  view.  Maybe  there  is  something 
in  the  way  our  wares  are  spread :  I  have  tried  to  recapture,  in  this 
strictly  modern  book,  something  of  the  attraction  that  Natural 
Philosophy  held  before  the  false -imagined  *  needs '  of  exams  had 
gnawed  it  nearly  to  the  bone :  the  result  is  a  volume  '  of  comfort- 
able proportions,'  but  I  never  could  feel  that  osseous  remains  were 
really  companionable. 

Indeed,  we  have,  expUcitly,  nothing  to  do  with  that  Principle  of 
Life  which  he  wishes  to  make  his  special  study,  which  dwells,  or 
dwelt,  in  all  the  structures  with  which  he  feels  pre-ordained  to 
deal ;  yet  day  by  day  the  worker  in  medical  research  comes  to  the 
physicist  with  puzzling  problems,  perpetually  cropping  up  in  the 
course  of  his  intricate  investigations. 

Nay,  not  only  is  one's  every  conscious  act  unconsciously  an  ex- 
perience in  practical  physics,  but  all  the  inner  processes  of  life  move 
this  way  or  that  under  the  most  intimate  physical  control. 


2  MECHANICS  [§  2 

§  2.  If  you  have  already  learnt  to  look  at  physics  from  the  usual 
unfortunate  school  standpoint — that  of  an  outlet  for  mathematics — 
you  will  doubtless  find  this  hard  to  believe  ;  but  take  a  hand,  let 
me  pass  on  to  you  the  frequent  kindly  grasp  of  them  that  taught 
the  men  who  taught  your  teachers  physics,  and  let  us  walk  together 
in  those  paths  of  Natural  Philosophy  that  will  not  lead  you  to 
blind  ends.  We  will  not  meander  among  mazes  of  mathematics, 
nor  will  we  feed  upon  a  farrago  of  formulae  ;  Physics  is  not  of 
these,  '  Science  is  organized  common- sense ' ;  we  will  look  at  com- 
mon things,  and  exercise  upon  them  the  good  sense  we  share 
between  us.  First,  actions  appealing  to  our  muscles,  then  to  our 
temperature  feeling,  next  to  ear  and  eye  ;  taste  and  smell  we  will 
let  alone,  but  then  we  shall  have  to  explore  a  region  in  which  the 
organism  has  as  yet  evolved  no  special  sense  whatever. 

Omissions  there  may  be,  and  obscurities  doubtless  a  many,  but 
may  I  be  spared  that  vastly  most  common  cause  of  all  such,  in  all 
books  :  that  the  writer  himself  has  really  no  clear  idea  of  what 
he  is  talking  about.  And  may  I  never  say  a  thing  is  '  obvious,'  for 
that  means  you  stumble  over  it  in  the  dark,  and  too  often  hides  an 
inabihty  to  offer  an  explanation — besides,  in  Natural  Philosophy, 
the  obvious  is  almost  always  wrong.  '  Evident  '  I  shall  permit 
myself,  for  that  means  '  to  be  seen  for  the  looking  '  ;  and  pray  you, 
keep  on  looking,  look  around  you  as  we  go,  and  learn  by  all  you 
see — ^you  shall  find  directions  enough  to  keep  you  from  straying 
from  your  course — get  out  of  the  old  school  way  of  thinking  that 
only  the  dry  and  the  stodgy  and  the  indoor  are  any  good  to  you  : 
we  don't  live  on  biscuit  aboard  this  ship. 


§  3.  And,  speaking  now  for  my  brother  Physics  Lecturers  and 
Examiners  as  for  myself,  least  of  all  things  do  we  desire  to  load  the 
storehouse  of  your  brain  with  perishables  for  consumption  within 
the  year.  We  would  have  you  clear  and  set  aside  a  well-lit  corner 
of  it,  for  a  workshop,  whereto  you  will  always  bring  your  problems, 
to  worry  to  bits  and  work  over  with  tools  of  the  mind,  kept  ready 
and  keen  by  daily  use.  In  our  line,  to  show  you  how  it  has  been 
done,  and  is  done,  and  how  you  can  do  it,  we  pick  for  you  specimens 
which,  in  our  close  contact  with  your  hospitals,  we  think  most 
likely  to  be  of  assistance  in  your  future  work  ;  they  are  simple  and 
above-board  matters  compared  with  which  a  test-tube  changing 
colour,  or  a  pulse  missing  a  beat,  are  puzzles  involved  indeed.  We 
try  to  show  you  how  to  make  the  best  of  a  problem,  to  arrive  at 
conclusions  of  minimal  doubt  and  ambiguity,  and  to  evaluate 
those  conclusions  for  yourself. 

Learn  this  habit  now,  develop  it  through  your  college  course, 
make  all  your  work  an  exercise  of  the  intellect — ^thereby  easing  the 
monotonous  strain  upon  memory,  lowest  of  the  mental  functions — 
maintain  it  throughout  your  medical  career,  walk  with  it  always, 
even  unto  the  End  of  the  Road. 


§  4]  UNITS  3 

§  4.  In  pastoral  times  of  old,  when  shepherds  watched  their 
flocks  by  night  on  open  hillsides,  when  all  the  boundary  of  their 
land  was  a  little  heap  of  stones  and  a  curse,  but  when  they  had 
charted  the  starry  skies  with  the  giant  figures  of  the  hunted 
and  their  hunters  of  forgotten  ages,  of  monsters  grown  mythical  and 
heroes  risen  to  godhead,  then  the  philosophical  ideas  of  Space  and 
Time  took  shape  in  men's  minds  ;  Space,  of  the  Universe,  which  the 
astronomers  have  never  ceased  to  enlarge,  until  at  the  present 
day  they  are  calling  upon  it  to  expand  most  furiously — whether 
inimitably  or  not  does  not  concern  us  here — and  Time,  marked 
nightly  by  the  passing  of  the  constellations  in  their  seasons,  and, 
very  much  more  obviously,  by  the  periodic  appearance  and  changing 
of  the  Moon. 

The  lunar  month  was  the  unit  of  time  of  the  Chaldaeans,  the 
'  year  '  of  Methuselah  :  allowing  thirteen  of  them  to  each  of  our 
years,  that  venerable  patriarch  was  about  seventy-five,  quite  old 
enough  for  a  wealthy  polygamist  to  have  around  him  a  progeny 
to  hold  his  name  in  honour  for  a  few  generations,  yet  an  age  in 
keeping  with  his  great  descendant's  dictum,  and  with  our  own 
experience  still.  The  division  of  time  by  the  march  of  the  Sun 
across  the  sky  is  plain  to  every  living  thing,  but  his  longer  journey 
the  whole  round  of  the  stars  was  quite  difficult  to  observe,  for 
always  he  obliterates  the  half  of  them ;  yet  when,  in  Egypt,  agri- 
culture developed,  and  all  prosperity  hung  upon  a  flood,  the  priest- 
hood published  a  solar  Year  of  360  days,  and  kept  up  their  sleeve 
the  discrepancy  of  5  days,  thereby  remaining  the  arbiters  of  the 
life-giving  rising  of  the  Nile.  In  temperate  regions,  with  their 
sharply-marked  seasons,  the  solar  year  is  readily  accepted  as  the 
great  unit  of  time,  witness  the  many  temples  sighted  upon  either 
the  Mayday  or  the  Midsummer  sunrise,  best  known  to  most  of 
us,  Stonehenge,  in  the  hollow  of  its  plain. 

The  smaller  Unit  of  Time  used  in  Physics  is  the  Second  {never 
the  minute),  the  mean  solar  second  which  is  the  86,400th  part  of 
one  day.  The  Astronomer  Royal  doesn't  observe  the  sun  at  all, 
but  a  star,  and  then  allows  for  the  sun  making  the  round  of  all 
the  stars  in  a  year,  365J  days.  For,  as  the  earth  goes  round  the 
sun  in  an  elliptic,  and  not  in  a  regular  circular,  orbit,  the  sun 
appears  to  us  to  wobble  considerably  in  his  time-keeping ;  he  gets 
to  bed  early  on  November  afternoons,  and  you  know  how  dark  the 
mornings  persistently  keep  when  you  return  to  work  after  the 
Christmas  holiday ;  the  sun  is  then  J  hour  late  in  rising ;  so  that 
the  sundial  is  quite  badly  out  in  winter,  though  never  more  than 
5  minutes  in  summer  (corrected  for  longitude,  of  course).  Those 
who  were  brought  up  within  sound  of  a  grandfather  clock  know 
what  a  second  is  ;  for  less  fortunate  people,  one  may  describe  it  as 
a  little  longer  than  a  quiet  pulse- beat. 

Practically,  we  observe  time  as  a  distance  on  a  scale,  across  a 
sundial,  round  a  clock  face,  or  on  a  travelling  chart.  Or  we 
invert  this,  especially  on  a  journey — '  It's  four  hours  farther  on.' 


4  MECHANICS  [§  6 

§  5.  Space  is  three-dimensional :  it  is  measured  by  length  right 
and  left,  up  and  down,  front  and  back.  The  early  unit  of 
length  was  human ;  I  find  in  the  prehistoric  shelter-towers  of 
Sardinia  the  doorway  is  the  height  of  a  smallish  man,  the  thick- 
ness of  the  boulder-built  walls  is  the  span  of  his  arms — ^there  was 
no  measuring-rod  for  the  dog  to  chew  the  end  off.  Our  yard  was 
the  arm  of  a  man  notably  large  in  his  day,  a  king,  and  we  have 
his  *  foot,'  his  '  hand,'  and  his  inch  thumb-width. 

In  the  French  Revolution  these  things  had  to  go,  and  the  over- 
grown Metre  was  laid  down  as  the  ten-millionth  of  the  distance 
from  the  North  Pole  to  the  Equator,  a  desperately  scientific  reaction 
from  the  horrors  of  history,  but  strictly  illogical,  because  one  end 
of  the  defined  distance  was  quite  out  of  reach.  The  meridian  of 
Paris  was  measured,  from  Dunkerque  to  Barcelona,  and  a  bar  of 
hard  unrusting  metal  was  inscribed  with  the  length  deduced  from 
the  geodetic  observations.  Later  and  more  extended  measure- 
ments showed  that  it  is  not  exactly  the  desired  ten-millionth,  as 
the  actual  shape  of  the  earth  was  not  determined  well  enough, 
but  by  that  time  nobody  bothered  about  that,  and  the  marked  bar, 
reposed  in  melting  ice,  remains  the  standard  Metre. 

Only,  as  it  is  known  nowadays  that  metals  are  prone  to  slow 
crystallization,  involving  change  in  size  and  shape,  safety  has  long 
been  sought  by  multiplying  and  distributing  copies  of  it,  and 
periodically  comparing  them.  Even  then,  if  any  discrepancy 
appears,  it  is  rather  a  toss-up  which  is  to  be  trusted,  but  by  now 
the  value  of  the  metre  has  been  measured  in  terms  of  the  wave- 
length of  the  red  light  emitted  by  electrified  incandescent  cadmium 
vapour,  a  unit  very  small  indeed,  but  of  a  constancy  compared 
with  which  man  and  all  his  works  are  ephemeral.  The  centimetre 
used  in  Physics  is  the  hundredth  of  the  metre  :  the  Frenchman 
always  talks  in  metres  and  millimetres. 

A  hollow  cube  made  with  one- tenth-metre  sides  was  to  be  the 
Unit  of  Volume,  the  Litre,  and  a  thousandth  of  it  the  cubic  centi- 
metre, or  c.c.  Again,  modern  refined  methods  have  disclosed 
experimental  error — the  standard  litre  now  in  use  is  not  exactly 
a  cubic  decimetre ;  so  that  the  '  Millilitre,'  ml.,  is  a  more  precise 
expression  than  c.c.  But  the  difference  is  away  among  the 
millionths,  too  small  for  use  in  this  book ;  only,  the  new  word  has 
just  to  be  introduced. 


§  6.  The  Litre  (as  then  determined)  of  air-free  distilled  water, 
at  a  temperature  where  it  caused  no  trouble  by  attempting  to 
expand  or  contract  appreciably,  4°  C,  was  made  the  Mass  of  a 
Kilogramme ;  and  the  unrusting  lump  of  hard  metal  then  pre- 
pared, and  ever  since  preserved  in  Paris,  is  the  metric  Unit  of 
Mass.  Again,  any  discrepancy  is  too  small  to  concern  us  ;  nor 
does  the  discovery  that  '  pure  '  water  is  a  mixture  call  for  any 
changes  among  units. 


8]  UNITS 


1  metre 
* 

=  1093633056  yd. 
=  3-281  ft.  =  39-37  in, 

1  cu.  m. 

=  1-308  cu.  yd. 

*1  gramme 
1  kgm. 

=  15-432  grains. 
=  2-2046  lb.  avdp. 

And  within  1%  : — 

*1  Utre 
4J  litres 
1  mg. 

(milligram) 
1000  kg. 

=  1 J  pint. 
=  1  gallon. 
=  gr.  1/64. 

=  1  ton,  less  1  stone. 

(kilograms) 

§  7.  The  centimetre-gramme-second  (c.g.s.)  system,  now  used  by 
everybody  in  physical  measurements,  was  devised  in  England  in 
1873.  It  is  an  unlovely  thing,  but  it  does  do  away  with  swarms 
of  inconsistent  and  easily  forgotten  numerical  factors,  and  self- 
contradictory  names,  such  as  1  fluid  ounce  =  1-732  cubic  inches ;  it 
is  far  ahead  scientifically  of  miscellaneous  national  lacks-of- systems, 
and  it  has  lent  itself  readily  to  the  construction  of  all  sorts  of 
electrical  and  other  '  derived  '  units.  Incidentally,  you  will  very 
likely  learn  your  Pharmacy  in  grams  and  millilitres. 

Table  of  Equivalents 

I  yard  =  91-44  cm. 

1  foot  =  30-48  cm. 

*1  inch  =  2-540  cm. 

*1  mile  =  1-610  km. 

1  sea  mile  =  1-853  km. 

1  sq.  ft.  =  929  sq.  cm. 

1  sq.  in.  =  6-451  sq.  cm. 

1  cu.  ft.  =  28-315  litres. 

1  cu.  in.  =  16-386  c.c. 

1  oz.  avdp.  =  28-35  gm. 
1  fl.  oz.  =  28-35  c.c. 
1  lb.  avdp.  =  453-6  gm. 

1  metre  contains  1,552,734-5  wave-lengths  in  vacuo  of  Cadmium  red  light, 
or  1,553,163-8  waves  measured  in  fresh  air  at  15°  C.  and  760  mm.  baro- 
metric pressure. 

1  micron,  /tx  =  1  /lOOO  mm.  =  10~*  cm.,  the  microscopist's  unit. 

1  Angstrom  unit  =  10~^  cm.,  the  atomist's  unit. 

1  bilHon  =  miUion  million,  10^^. 

Micro-  means  millionth,  mega-  means  million  times. 

1  year  contains  3-15  million  seconds. 

1  Radian  =  57-296°  =  206264-8  seconds  of  arc. 

Light  travels  in  vacuo  2-9986  x  10"  cm. /sec.  =  6  billion  miles  per  year. 


The  main  table  is  accurate  at  least  as  far  as  the  figures  go  ;  the 
few  at  the  end  serve  for  some  practical  purposes.  You  may  reason- 
ably be  expected  to  know  the  starred  numbers  ;  the  rest  are  put  in 
for  reference. 

There  is  Uttle  else  to  memorize  in  the  foregoing  chapter,  but  you 
see  how  physical  things  have  groum  in  the  minds,  and  into  the  Uvea, 
of  men. 

K  §  8.  But  do  notice  right  here  the  final  0  put  on  to  the  inch  and 
Hfcmile  figures.  This  is  a  declaration  that,  so  far  as  that  decimal 
^Bplace,  there  is  nothing  to  add  (and  that  the  next  figure  is  less  than  5). 
|rlt  is  of  more  practical  scientific  value  than  the  long  string  of  figures 


6  MECHANICS  [§  8 

for  the  metre-yard,  for  there  the  next  generation  of  investigators 
is  sure  to  alter  the  last  figure.  Whenever  you  record  a  numerical 
result,  do  please  put  in  just  as  many  figures  as  you  feel  pretty 
sure  about,  and  no  more.  It  is  part  of  your  training  for  all  scientific 
work,  to  examine  for  yourself  how  far  you  can  trust  your  method, 
to  work  up  to  mutually  consistent  limits  of  accuracy,  and  to  express 
the  results  accordingly. 

For  instance,  in  determining  specific  heats,  when  you  get  a  rise 
of  temperature  of  5°  and  your  thermometer  cannot  be  read  closer 
than  1/20°,  and  when  a  drop  of  water  weighs  about  1/10  gm.,  it 
is  no  use  weighing  on  a  fine  balance  to  ten  times  that  supposed 
accuracy,  or  more,  nor  do  you  get  additional  credit  for  handing  in 
a  result  like  0-032165,  because  you  haven't  possibly  worked  to  an 
accuracy  of  even  1  in  320,  and  can't  be  sure  of  the  1.  Of  course 
*  some  of  the  figures  may  be  right.' 

On  the  other  hand,  don't  leave  it  at  0-03,  because  that  means  that 
you  neither  know  nor  care  whether  the  next  figure  is  0,  or  4,  or  even 
9 ;  you  declare  to  the  examiner  a  possible  error  of  30% ,  and 
he  rewards  you  accordingly.  Learn  that  right  here  and  now ;  if 
not,  and  by-and-by  your  patients  deduct  that  much  discount  from 
your  bills,  thank  yourself. 

And  don't  quote  '  recurring  decimals,'  1/6  is  0-167  within  J%, 
or  0-1667  within  |  per  mille. 

The  0  put  in  front  of  the  decimal  point  is  just  a  type-setter's 
device  to  make  sure  that  that  important  little  item  doesn't  drop 
out  of  print. 


CHAPTER   II 


MOTION 


§11.  Three  kinds  of  motion  are  possible  to  a  body  : 

(a)  Deformation,  it  alters  in  size  and  shape ;  as  a  sponge 
squeezed  in  the  hand. 

(6)  Rotation,  it  turns  or  spins  about  a  centre. 

(c)  Translation,  it  moves  from  place  to  place  without  either 
(a)  or  (6)  ;  as  a  boy  sliding,  a  pen  writing,  a  ship's  compass. 

Any  or  all  can  go  on  continuously ;  or  stop  and  go  back  periodi- 
cally as  an  oscillation.  Chap.  XXV. 

The  most  general  motion  consists  of  all  three  at  once  ;  as,  for 
instance,  a  smoke-cloud  curling  out  of  a  chimney. 

In  a  rigid  body  (a)  is  impossible,  and  (6)  and  (c)  combine  to  the 
most  usual  motion  ;  as  a  cricket  ball,  or  the  bat  swung  to  meet  it. 

What  little  we  shall  have  to  say  about  Rotation  must  be  deferred 
for  the  present ;  we  are  now  going  to  take  only  (c)  the  Linear  Motion 
of  a  rigid  body.  Since  all  parts  perform  equal  and  parallel  paths, 
it  suffices  to  consider  only  one  particle,  negligibly  small  in  size,  but 
supposed  endowed  with  the  whole  mass  of  the  body. 

§  12.  Linear  motion  of  a  particle.  If  it  can  move  in  one  straight 
line  only,  then  calling  motion  one  way  +  and  the  other  way  — , 
the  result  of  its  motion,  or  its  '  resultant  displacement,'  is  the 
algebraic  sum  of  all  its  '  component  displacements.' 

But  if  successive  displacements  are  in  different  directions,  as  in 
Fig.  1  (i),  the  resultant  is  the  straight  line  AZ,  which  joins  the  last 
position  to  the  first,  and  completes  the  Polygon  of  Displacements  ABCZ. 

AZ  is  the  '  geometrical-,'  '  directed-,'  or  '  vector-  'sum  of  AB, 
BC,  etc.,  each  of  which  is  a  Vector,  i.e.  a  line  representing,  by  its 
length  and  direction,  some  quantity  which  possesses  definite 
magnitude  and  direction. 

For  only  two  motions,  the  polygon  becomes  the  Triangle  ABC  (ii). 
The  closing  side  is  the  resultant  of  the  other  two. 

By  redrawing  with  the  component  motions  in  different  succes- 
sion, you  can  assure  yourself  that  this  ultimately  makes  no  differ- 
ence, nor  does  it  if  they  are  broken  up  into  small  steps  and  applied 
alternately,  as  in  Fig.  1  (iii).  And  this  also  shows  that  the  Diagonal 
of  a  Parallelogram  is  the  same  as  the  Closing  Side  of  a  Triangle. 

§  13.  Velocity  is  the  distance  travelled  in  a  unit  of  time  in  a  given 
direction.  It  is  a  vector  quantity.  (Speed  is  distance  travelled  in 
a  unit  of  time  without  specifying  direction,  i.e.  it  is  only  a  '  scalar  ' 

7 


MECHANICS 


[§13 


quantity.)  Suppose  two  blows  given  to  a  particle  P,  one  of  which 
would  drive  it  to  Q  in  a  second,  and  the  other  alone  to  R,  Fig.  1 
(iv).  The  result  is  that  the  particle  is  driven  to  S,  where  PS  is  the 
diagonal  of  the  parallelogram  PQSR,  or  the  closing  side  of  the 
triangle  on  PQ  and  a  line  parallel  to  PR. 

Now  the  order  of  the  displacements  made  no  difference,  nor 
their  going  on  in  any  number  of  alternate  steps,  i.e.  virtually 
simultaneously.  Suppose,  therefore,  the  two  blows  simultaneous, 
the  velocities  combine  into  one  resultant,  found  by  the  vector 
parallelogram  or  triangle  exactly  as  before.  And  if  several  blows 
were  struck  on  the  particle  at  one  moment,  the  vector  sum  of  the 
velocities  they  produce  would  again  be  the  closing  side  of  the 
vector  polygon. 


Fig.  1. 


Examples  of  this  combination  of  velocities  abound.  The  fly 
crossing  a  moving  railway  carriage  in  2  sec.  southwards  is  mean- 
while carried  160  ft.  east,  and  actually  moves  a  little  faster  than 
the  train  in  a  direction  slightly  S.  of  E.  relatively  to  the  track. 
The  earth's  surface  has  carried  both  half  a  mile  nearer  the  rising 
moon,  and  earth  and  moon  have  travelled  some  30  miles  on  their 
journey  round  the  sun.  All  motion  is  relative  :  which  of  two 
things  moves,  and  the  way  it  moves,  is  a  matter  of  agreement 
with  the  neighbours.  When  you  were  very  young  the  fences  ran 
past  you  in  the  train,  now  you  regard  the  earth  as  fixed,  except 
when  thinking  astronomically. 

§  14.  To  specify  a  Velocity  you  must  quote  both  the  distance 
and  the  time  ;  the  velocity  =  the  distance  divided  by  the  time, 
V  =  S/T  ;  so  many  miles  per  hour  (miles  -^  hours) ;  so  many 
centimetres  per  second  (cm. /sec),  and  so  on. 


I 


§  16]                                              MOTION  9 

Then  this  simple  relation  ought  to  be  pretty  familiar  to  us  : 

Speed  at  which  you  travel  X  time  you  are  travelling^  .^jr^  _  ^ 
=  distance  to  go  j   V  i  —  a 

Or  put  it  in  another  way  : 

Tim^jourmy  will  take  =  <ii^t»^  Vo^  ^^etojo  ^  ^  S 
•^  speed  you  travel  at    '  V 

Or  yet  again  : 

o       7  ^7         distance  you  have  to  go  ^,       S 

Speed  you  must  make  =  j^ — ^, r^ — — ,  V  =  ^. 

time  allowable       *  T 

All  are  the  same  thing ;  sometimes  one  wants  one  form,  some- 
times another. 

There  is  only  one  speed  which  enjoys  a  name  of  its  own,  and 
that  is  the  Knot,  or  nautical  mile  per  hour.  The  Nautical  Mile, 
the  Sea  Mile,  6080  ft.,  in  which  all  distances  at  sea  are  quoted,  is 
the  one  really  sensible  unit  of  length,  being  the  surface  value  of  a 
minute  of  arc  at  the  earth's  centre.  The  old  log  line  had  a  miniature 
sea-anchor  at  its  end ;  when  this  was  thrown  overboard  it  stayed 
in  place,  and  dragged  the  line  aft  through  the  leadsman's  fingers 
as  the  ship  moved  on.  It  was  knotted  every  50  ft.,  i.e.  1/120  n.m., 
and  he  counted  the  number  of  knots  that  slipped  through  in 
4  minute  of  the  sand-glass,  and  that  was  the  speed  in  knots. 

If  vou  would  compare  speeds  at  sea  with  motor-car  speeds  on 
land,\ecollect  that  1  sea  mile  =  6080/5280  =  about  8/7  English 
land  miles  (never  used  at  sea),  so  that  to  convert  knots  into  m.p.h. 
add  l/7th,  or  m.p.h.  into  knots  subtract  l/8th. 

§  15.  *  Resolution  '  of  vectors.  Since  any  side  of  a  vector  triangle 
represents  the  resultant  of  the  other  two,  the  two  sides  of  any 
triangle  that  can  be  built  on  a  given  vector  as  base  are  possible 
motions  into  which  the  actual  motion  can  be  '  resolved.'  It  is 
often  useful  to  resolve  into  two  directions  at  right  angles ;  i.e.  a 
right-angled  triangle  is  built  on  the  vector  as  hypotenuse,  having 
its  sides  parallel  to  the  desired  directions,  e.g.  a  ball  thrown  up 
at  60°  at  40  ft.  per  sec.  has  at  start  a  horizontal  velocity  of  20  ft. /sec. 
and  a  vertical  of  34 J  (Fig.  1  (v)). 

Or  you  may  feel  safer  if  you  build  a  Parallelogram  '  of  velocities  ' 
on  the  vector  as  diagonal,  with  its  sides  in  the  desired  directions ; 
then  the  two  sides  starting  from  the  root  of  the  vector  are  the  two 
*  resolved  components  '  of  the  motion — as  they  are  drawn  in  (v) 
indeed. 

§  16.  Acceleration.  Velocity  rarely  remains  steady,  or  uni- 
form, for  any  length  of  time,  but  suffers  acceleration  to  higher 
speed,  or  deceleration  or  retardation  (negative  acceleration)  towards 


10 


MECHANICS 


[§16 


rest.  This  acceleration  is  measured  as  the  extra  velocity  acquired 
in  each  unit  of  time,  e.g.  a  body  falls  at  a  speed  which  exceeds 
by  32  ft. /sec.  the  speed  it  had  a  second  before,  its  speeds  at  the 
ends  of  successive  seconds  from  rest  being  32,  64,  96,  etc.  Thrown 
upwards  it  would  have  upward  acceleration  =  —  32. 

Change  of  velocity  per  unit  time  is  change  of  (distance /time)  per 
unit  time,  i.e.,  (distance /time)  -^  time, 

OR  distance  -:-  (time)^,  or  sji^,  so  that  one  must  quote  Acceleration 

as  (distance  per  sec.  gained  every  sec), 

or  (distance  per  sec.  per  sec), 
OR  (distance  per  sec. 2),  which  last  is  common  but  bad,  being 

brevity  at  the  expense  of  intelligibility. 

Acceleration  can  be  applied  in  directions  other  than  the  line  of 
motion,  and  then  alters  direction  as  well  as  velocity ;  or,  in  the 
particular  case  of  circular  motion,  where  it  is  at  right  angles  to  the 
motion,  it  alters  direction  only  ;  see  Figs.  15,  16. 


> 

H 

^<i 

o 
o 

B 

,<<^ 

rj 

s\\-K\ 

> 

^^<<X\ 

\\ib 

sV 

..<<f^^w^ 

x\\k^ 

.<<<5^!kVN^  ^§N 

wV 

t 

Fig.  2. 

§  17.  Distance,  time,  speed  and  acceleration.  As  we  have  already 
seen,  a  particle  moving  with  speed  v,  passing  over  v  units  of  length 
in  unit  time,  in  time  t  travels  a  distance  s  =  vt. 

[On  a  diagram  plot  times  t  horizontally  as  abscissae,  and  speeds 
V  vertically  as  ordinates ;  then  a  Distance  v^  is  represented  by  the 
rectangular  area  in  Fig.  2  (A).] 

If  v^  alters  steadily  to  v^  by  the  end  of  t,  the  average  speed  is 
\{v^  +  ^2)  [^^^  ^^®  distance  travelled  is  represented  by  the  whole 
area  of  A]. 

In  particular,  starting  from  rest  and  steadily  acquiring  a  final 
speed  V,  the  average  speed  is  |(0  -\-  v)  =  \v  [and  the  distance 
travelled  is  represented  by  the  triangular  area  B]. 

If  the  acceleration — that  is,  the  gain  of  speed  per  unit  time — 
is  a,  this  final  speed  v,  gained  in  t,  —  at 

.'.  the  average  speed  for  the  t  seconds  =  ^at 

distance  travelled  s  =  speed  ^at  x  time  t  [represented  by  area  B] 


.-.  S 


iat' 


§20]  MOTION  11 

the  distance  travelled  by  a  particle,  starting  from  rest,  with  steady 
acceleration  a,  and  travelling  for  time  t. 

For  instance,  a  stone  dropped  from  the  hand  is  accelerated 
downwards  by  gravity  32  ft.  per  sec.  every  second,  and  at  the  end 
of  the  third  second  would  be  travelling  at  a  speed  32  x  3  =  96 
ft. /sec,  and  would  have  fallen  a  distance  5  =  J  x  32  x  3^  =  144  ft. 

§  18.  It  should  be  said  here  at  once  that  acceleration  never  is 
really  '  steady.'  This  falling  stone,  as  it  gathers  speed,  suffers  an 
increasing  resistance  from  the  air,  which  discounts  its  acceleration 
more  and  more  every  second.  And  you  know  you  never  keep  your 
foot  steadily  on  the  accelerator,  nor  on  the  decelerator,  the  brake. 
That  means  that  the  slanting  line  in  the  diagram  varies  in  slope — 
is  more  or  less  humpbacked,  curved  differently  and  unaccountably 
in  every  single  case.  We  cannot  go  into  such  variations,  we  must 
generalize  and  simplify  by  considering  only  a  straight -line  growth, 
a  uniform  steady  acceleration.  It  is  the  best  we  can  afford,  we 
shan't  be  '  mathematically  exact,'  but  that  is  the  fault  of  the 
mathematics,  it  gets  too  complicated. 

Deceleration,  negative  acceleration,  means  a  downslope  in  Fig.  A, 
to  v.^  —  v^  —  at. 

§  19.  School  mechanics  books  here  blossom  forth  into  numerous 
Subsidiary  Formulae.  If  you  have  a  little  mathematical  ingenuity, 
you  can  make  them  all  up  for  yourself ;  if  you  have  not,  or  if  you 
are  of  those  who  throw  handfuls  of  shingle  at  a  mark,  you  can 
buy  a  book  and  learn  the  whole  bunch,  and  never  know  which  one 
to  use  when  the  time  comes — the  one  question  which  may  or  may 
not  crop  up  in  your  one  exam.  But  if  you  believe  in  aiming  a 
pebble,  master  the  little  '  average  '  argument  above,  and  use  it 
in  all  questions  of  starting  or  stopping.  If  you  do  happen  to 
recollect  s  =  \at^,  you  can  employ  it  as  a  half-way  stage,  or  just 
as  a  check  on  your  working. 

Here  is  another  trick,  connecting  speed  with  distance  travelled 
from  rest : — 

8  =  lat^  =  _-  aH^  =  }  v^    or  v^=  2as 
2  2a  2a 

And  you  can  verify  that  on  the  falling  stone  example  :  it  takes 
on  a  much  greater  significance  in  Chapter  IV. 

§  20.  You  have  now  struggled  through  a  chapter  of  abstract 
ideas,  and  you  may  or  may  not  have  found  them  simple  to  follow. 
This  book  is  no  child's  primer,  with  the  easiest  put  first.  Here 
follow  some  examples  which  should  help  you  to  master  what  you 
have  read,  and  prepare  you  to  face  anything  of  the  sort  you  will 
ever  meet.  When  you  have  tackled  those,  there  is  a  longer  and 
tougher  chapter  lying  ahead. 

But  beyond  that  are  other  chapters,  not  all  of  them  so  desperately 


12  MECHANICS  [§20 

distant  and  difficult  and  dry.  Ultimately  you  will  find  that  they 
all  fit  together  in  some  sort  of  order,  but  they  weren't  all  written 
in  that  order,  by  any  means,  and  if  you  feel  temporarily  disinclined 
to  plod  heroically  through  them  one  after  the  other,  there  is  not 
the  slightest  objection  to  your  skipping  and  dipping  where  some- 
thing catches  your  eye  that  you  seem  to  have  heard  of  before. 

When  an  open  wound  starts  to  heal,  granulating  points  appear, 
and  gradually  increase  in  size  until  they  coalesce,  and  then  the 
continuous  protective  tissue  so  formed  improves  steadily  in  thickness 
and  in  texture.  Apply  the  same  principle  to  this  book ;  look 
through  it,  and  establish  such  points  of  interest  as  you  can  find, 
pursue  them  a  little,  until  more  appear,  and  when  you  find  them 
linking  up  together  you  are  getting  a  grip  of  the  subject. 

Personally,  I  can  conceive  no  more  futile  and  disheartening 
occupation  than  slogging  away  at  a  subject  for  which  one  can  see 
no  use,  and  therefore  never  learns  to  use  :  it  becomes  a  tool  to 
be  thrown  aside  at  the  earliest  moment,  the  time  already  spent  on 
it  grudged  and  wasted,  and  all  its  possible  future  usefulness  gone 
by  the  board.  That  any  one  should  sit  down  and,  with  a  view 
to  pleasing  examiners,  deliberately  memorize  any  of  the  instances, 
illustrations,  and  applications  that  I  have  been  able  to  gather 
together,  is  to  me  a  distressing  idea  ;  what  is  hoped  is  by  interest 
to  lead  on  to  understanding — not,  of  course,  without  effort,  but 
let  it  be  an  effort  to  understand,  not  to  memorize — for  of  a  thing 
understood  memory  will  take  care  of  itself,  and  keep  it  ready  for 
use,  not  stored  under  protest.  Let  what  you  read  call  to  mind 
something  already  in  your  own  experience,  pursue  that,  and  get 
that  clear,  in  preference  to  anything  you  are  merely  told ;  so  give 
Natural  Philosophy  a  chance  to  get  well  rooted  in  your  mind. 
Then,  when,  after  its  brief  season  of  intensive  culture,  you  cut  it 
back  to  make  room  for  the  growth  of  other  things,  you  will  have  a 
trustworthy  support  for  them  all,  in  the  background,  where  supports 
ought  to  be,  and  better  than  any  number  of  adventitious  props. 

§  21.  There  are  no  pictures  in  this  book:  the  intention  is  that 
you  shall  see  and  handle  apparatus,  etc.,  in  the  Laboratory,  and 
wherever  you  can,  for  that  is  vastly  better  than  any  amount  of 
gazing  at  flat  pictures. 

I  have  drawn  my  Diagrams  as  nearly  to  scale  as  may  be,  having 
no  liking  for  making  things  as  broad  as  they  are  long.  Redraw 
them  as  you  will,  simplifying  and  emphasizing  to  suit  yourself. 

Charts,  such  as  96,  223,  etc.,  are  usable  with  confidence  as  closely 
as  their  scale  permits ;  and  Reference  Tables  I  have  gone  to  some 
pains  to  render  as  reliable  as  such  abbreviated  lists  can  be. 

There  are  no  *  Summaries  '  at  the  ends  of  Chapters.  The  book  is 
your  own,  and  the  paper  is  good,  mark  it  as  you  go,  and  therefrom 
draw  up  your  own  summaries,  aided  perhaps  by  the  hints  that 
precede  the  Questions.  Inexpert  though  you  may  be,  you  will  suit 
your  own  mind  better  than  any  one  else  could  :   it  would  be  sheer 


§  24]  MOTION  13 

presumption  on  my  part  to  attempt  it  for  you,  and  laziness  on 
yours  to  look  for  it. 

§  22.  It  is  assumed  that  you  use  an  English  Dictionary,  not  a 
costly  cross-word  spelling-book,  but  a  serviceable  one  that  will  give 
you  the  alternative  meanings  of  words  which  in  course  of  time 
grow  so  familiar  to  any  writer  that  he  forgets  to  define  them  :  and 
quite  a  bit  of  the  difficulty  of  a  fresh  subject  often  lies  merely  in 
its  words  and  phrasing. 

Two  harmless  words  of  Greek  origin,  common  in  natural  philo- 
sophy, but  never  used  in  ordinary  life,  are  missing  from  this  book. 
I  have  avoided  them  meticulously,  even  though  in  examination 
answers  their  frequent  repetition  appears  to  afford  as  much  hope 
of  salvation  *  as  that  blessed  word  Mesopotamia,'  or  to  lubricate 
writers'  thoughts  as  well  as  might  the  equally  blessed '  Isle '  of  Patmos. 

I  have  no  use  for  anything  that  differentiates  between  natural 
philosophy  and  daily  life. 

§  23.  And  pray  you,  never  be  content  to  say  that  such  and 
such  '  tends  to  '  do  so-and-so,  for  it  is  an  admission  of  ignorance, 
a  calling  upon  some  dusty  little  god  on  the  top  shelf  to  come  and 
get  you  out  of  it.  '  No  half  measures  ;  either  it  does  or  it  doesn't,' 
is  the  slogan  which  has  pushed  all  Science  forward  this  century  as 
never  before.  '  Tendencies  '  are  the  mob- sway ings  of  complex 
systems  the  inner  workings  of  which  baffle  us, 

Souvent  femme  varie 
suggests  observations  which  are  the  antithesis  of  the  direct  and 
fundamental  ones  of  Natural  Philosophy  :    if  a  piece  of  physical 
apparatus  starts  showing  a  *  tendency  ^ 

Fou  est  il  qui  s*y  fie 
and  to  bits  it  comes  in  the  workshop. 

*  There  must  be  some  kind  of  an  attractive  force  acting,  causing 
a  tendency,'  writes  the  student,  and  hopes  that  it  will  attract 
credit,  for  has  he  not  created  a  mystery,  and  does  not  that  entitle 
him  to  the  affectionate  regard  of  all  men?  Nay,  the  natural 
philosopher's  task  is  to  abolish  mysteries,  and  the  examiner  passes 
him  by  upon  the  other  side. 

§  24.  Now,  as  to  all  these  EXAMINATION  QUESTIONS.  When 
I  told  the  Principal  of  this  projected  book,  he  arose  in  haste,  dis- 
playing consternation,  commiseration,  warm  consideration,  and 
other  mixed  emotions,  and  finally,  summoning  his  Librarian, 
placed  their  every  resource  at  my  disposal.  Thereby  he  exemplified 
to  the  fullest  the  true  spirit  of  goodwill  with  which  the  University 
looks  out  upon  the  student  race ;  helping  into  its  fold,  in  due 
course  to  become  integral  with  itself,  all  who  prove  themselves  fit 
novices  for  the  training  it  exacts  and  controls. 

Your  thanks  and  mine  must  be  rendered  here,  not  only  to  him 
for  this  invaluable  assistance,  but  to  friends  of  all  sorts  scattered 


14  MECHANICS  [§  24 

everywhere,  who,  in  patient  answer  to  questions  often  of  the 
queerest,  have  sent  me  those  touches  of  personal  information  and 
everyday  experience  that  are  so  incomparably  better  than  any- 
thing one  ever  reads  in  print.  They  are  too  many  to  mention, 
and  they  would  hate  to  be  mentioned,  but  to  their  ready  kindness 
this  book  owes  not  a  little. 

And  so  it  does  to  him  of  your  own  clan,  equally  nameless,  whose 
painstaking  critical  reading  has  led  to  much  clearing  up  of  lurking 
obscurities. 

§  25.  Well,  I  have  helped  myself  to  every  London  question  set 
this  century,  sorted  them,  rolled  together  those  that  were  essentially 
identical,  and  put  the  lot  into  print  here.  Most  are  shortened  by 
leaving  out  the  preliminary  inquiries  for  the  meaning  of  two  or 
three  things  used  in  the  question  (which  earn  about  a  mark  apiece) ; 
and  a  good  deal  of  the  formal  verbiage  by  which  it  is  sought  to 
restrain  the  candidate  from  going  astray,  but  which  often  seems 
to  arouse  his  dire  suspicion,  has  been  cut  out.  The  various  numerical 
data,  too,  values  of  g,  H,  J,  densities,  specific  heats,  etc.,  wanted 
in  a  question,  and  almost  always  appended  to  it,  have  been  lopped 
off ;  for  it  will  do  you  more  good  here  to  make  out  what  you  want 
for  yourself,  and  look  it  up.  Redundant  data  appear  sometimes, 
and  timorous  creatures  spend  half -hours  trying  to  squeeze  them  in, 
fearing  a  trap.  Traps  are  simply  not  set,  though  it  never  does 
any  harm  to  keep  your  eyes  open ;  it  leads  to  answering  what  is 
asked  for,  not  necessarily  what  the  first  glance  led  you  to  think 
of.  Some  of  the  questions  have  been  edited  because  they  wanted 
it ;  a  few  of  similar  type,  all  '  possibles,'  have  been  filled  in  here 
and  there.  Those  that  have  appeared  more  than  once  are  marked 
at  the  end  ( x  no.  of  times). 

Dates  are  not  appended,  for  no  particular  change  of  character 
with  period  was  noticeable,  and  they  would  have  been  very 
misleading. 

The  Examination  Syllabus  is  not  reprinted  here  ;  that  is  for  the 
guidance  of  those  who  know  what  it  means,  and  these  questions 
interpret  it  for  you. 

§  26.  Solutions  to  numerical  questions  are  given  at  the  end  of 
the  book,  in  the  form  most  likely  to  be  helpful  to  you  ;  if  3^our 
result  doesn't  agree,  look  at  the  working.  Never  leave  them  in 
un-worked-out  form  in  your  exam,  for  it  is  your  job  to  work  them 
out,  correctly,  to  a  plain  answer ;  just  as  it  is  the  examiner's  job 
to  work  down  the  six-inch  stack  of  papers  in  front  of  him ;  and, 
whether  you  like  it  or  not,  he  is  not  going  to  interrupt  his  stride 
to  clear  up  any  unfinished  messes  of  your  making. 

You  can  work  them  out  by  plain  arithmetic,  which  is  safest,  and 
not  very  tedious  once  you  learn  how  to  cut  off  long  tails.  Or  you 
can  use  logs  (little  tables  of  which  are  obtainable  in  exam  rooms) ;  only 
mind  and  add  them  up  right,  for  that  is  where  I  find  most  mistakes. 


26] 


MOTION 


15 


Or  you  can  buy  now,  at  the  instrument  shops  or  the  big  stores,  a 
serviceable  English-made  Slide  Rule,  for  55,  (or  pup  size  'Ss.  9d.)  and 
learn  how  to  use  it  in  a  week  or  two  in  the  laboratory.  There- 
after, provided  you  take  care  of  the  decimal  points  for  it,  it  will 
do  most  of  your  calculations,  for  life  (except  adding  up  patients' 
accounts),  showing  you  plainly  all  the  time  exactly  to  what  degree 
of  accuracy  you  are  keeping,  e.g.  l/50th  in.  false  setting  anywhere 
on  a  5-in.  scale  is  an  error  of  1  in  250.  All  calculations  in  this  book 
have  been  done  by  slide  rule  :  calculations  assume  quite  a  different 
aspect  when  their  completion  is  only  a  matter  of  pushing  a  slip  of 
wood  to  and  fro. 

Since  these  words  were  written,  the  question  has  been  taken  up, 
and  you  are  now  definitely  permitted  to  use  your  own  slide  rule  in 
all  medical  exams  in  London. 


EXAM   QUESTIONS,   CHAPTER   II 


1.  Express  miles  per  hour  in  feet  per  second. 
1  mile  =  5280  ft.     1  hour  =  3600  sec. 

.-.  miles  hour  =  5280  ft./3600  sec.  =  22/15  in  ft. /sec. 

i.e.  15  m.p.h.  =  22  f.p.s.,  or  just  over  half  as  many  again. 
1  knot  =  1-7  f.p.s. 

2.  Express  miles  per  hour  in  centimetres  per  second. 

3.  Express  37  knots  in  miles  per  hoiu".     1  nautical  mile  =  6080  ft. 

4.  Express  knots  as  cm. /sec,  given  21,600  n.m.  =  40,000  km. 

5.  Find  in  how  many  seconds  a  train  jolts  over  a  number  of  30-ft.  rails 
[equal  to  its  speed  in  miles  per  hour.     Also  45-ft.  and  60-ft.  rails. 

6.  Explain  the  parallelogram  of  velocities.  How  would  you  find  the  speed 
;  of  the  train  by  the  track  of  falling  raindrops  on  the  carriage  window  ? 

7.  Explain  how  velocities  are  compounded. 
A  man  who  can  row  in  still  water  at  3  m.p.h.  wishes  to  cross  a  river  1/4 

[mile  wide,  flowing  at  1-5  m.p.h.  In  what  direction  must  he  row  to  reach  a 
point  on  the  other  bank  directly  opposite  his  starting  point,  and  how  long 
["Will  it  take  htm  to  cross  ? 

8.  If  points  are  moving  with  velocities  v  and  w  at  an  angle  A,  show  how 
[to  find  their  relative  velocity.  An  aviator  heads  for  east  at  70  m.p.h.,  but 
factually  travels  N.E.  at  50  m.p.h.;  find  speed  and  direction  of  wind. 

9.  A  man  rows  at  2  m.p.h.  relative  to  the  water,  at  right  angles  to  the 
direction  of  the  current  of  a  straight  river  flowing  at  2  m.p.h.     Another  man, 

^starting  from  the  same  point,  walks  along  the  bank  upstream  at  3  m.p.h. 
[How  far  apart  will  they  be  after  6  min.  ? 

10.  A  ship  is  heading  west  at  16  knots ;  she  is,  however,  in  a  current  flowing 
[towards  the  south-west  at  4  knots.     The  wind  is  blowing  from  the  north -west 

kt  12  knots.     Find  approximately  by  a  diagram  (a)  the  actual  course  of  the 
^«hip,  (6)  the  angle  between  her  keel  and  her  trail  of  smoke. 

1 1 .  Define  uniform  and  non-imiform  acceleration.  If  a  body  passes  three 
fequispaced  posts  with  velocities  of  20,  30  and  20  ft. /sec,  what  sort  of  accelera- 
ktion  has  it  ? 

12.  Define  velocity  and  acceleration.  A  tape  imwinds,  and  marks  are  made 
on  it  at  equal  times  :   how  would  you  find  its  varying  speed  ? 


16  MECHANICS 

13.  Express  an  acceleration  of  981  cm. /sec.  per  second  in  ft./sec.^. 

14.  How  far  should  a  body  fall  in  4  sec.  ? 

[By  the  argument  of  §  17,  «  =  ^t^  =  ^  X  981  X  16  =  7848  cm.] 

15.  How  long  will  it  take  an  electric  train  with  acceleration  2-5  ft. /sec. ^  to 
travel  100  ft.  from  rest  ? 

16.  A  has  an  initial  velocity  of  60  ft. /sec.  and  a  deceleration  4  ft. /sec. 2,  B 
has  initially  5  ft. /sec.  and  acceleration  2  ft. /sec. 2.  Which  travels  100  yds. 
first,  and  which  is  moving  fastest  after  1  min.  ? 

17.  Define  velocity  and  acceleration.  In  four  successive  seconds  a  body 
moves  10  cm.,  20  cm.,  30  cm.,  and  40  cm.,  respectively.  Calculate  its  accelera- 
tion and  its  velocity  at  the  end  of  each  second. 

18.  A  bullet  passes  in  succession  through  three  screens  1000  ft.  apart,  taking 
0-8  sec.  from  first  to  second  and  0*86  sec.  from  second  to  third.  Find  the  (nega- 
tive) acceleration.     (  X  2) 

19.  A  ball  is  flung  horizontally  at  15  m./sec.  from  the  top  of  a  20-m.  tower. 
How  long  before  it  reaches  level  ground,  and  where  ? 

[Horizontal  speed  does  not  affect  vertical  motion  at  all.  A  ball  rolled  off 
the  table  at  any  speed  falls  to  the  floor  just  as  soon  as  if  dropped  vertically 
from  the  edge.] 

20.  A  ball  is  thrown  up  at  40  ft. /sec.  and  60°  to  the  horizontal.  How  high 
does  it  go,  how  long  is  it  in  the  air,  and  how  far  away  does  it  strike  the  level 
ground  ? 

[Resolve  the  velocity  as  in  §  15  into  20  ft. /sec.  horizontal  and  34-5  vertical. 
These  are  now  quite  independent  of  each  other.  It  will  take  34-5  -^  32  sec.  for 
gravity  to  destroy  the  vertical  component,  the  ball  meanwhile  rising  at  mean 
speed  17-25  to  a  height  17-25  X  34-5  -f-  32  =  18-6  ft.  It  takes  as  long  again 
to  fall;  time  of  flight  =  2-16  sec,  during  which  it  travels  20  X  2-16  =  43  ft. 
horizontally.] 

21.  The  splash  of  a  stone  is  heard  2-7  sec.  after  dropping  it  down  the  well. 
If  sound  travels  up  at  1100  ft. /sec,  how  deep  is  the  well  ? 

22.  A  stone  falls  down  a  shaft  200  m.  deep.  How  long  after  letting  go 
will  the  sound  of  impact  be  heard  at  the  top  ? 

Velocity  of  sound  =  330  m./sec. 


CHAPTER   III 
MOMENTUM  AND  FORCE 


§  31.  Coherent  portions  of  matter  are  termed  Bodies.  A  body 
the  dimensions  of  which  we  wish  to  disregard  is  called  a  Particle. 
Any  aggregate  of  Matter  constitutes  a  mass,  and  Mass  is  the  measure 
of  the  quantity  of  matter. 

These  definitions  tell  you  nothing  fresh  about  ideas  that  you  have 
really  grown  up  with,  they  only  shut  ofE  the  multitudinous  other 
meanings  of  these  overworked  common  words. 

Definitions  are  all  very  well  at  school,  where  they  impose  definite - 
ness  on  the  wandering  youthful  mind,  but  to  be  useful  they  must  be 
remembered,  and  to  be  remembered  they  must  be  brief,  and  this 
very  brevity  is  apt  to  leave  them  as  half-truths.  The  physics 
examiners  you  are  going  to  meet  set  small  store  by  formal  definitions  ; 
but  they  do  expect  you  to  go  deeper  and  understand  the  thing 
properly.  Drop  the  schoolboy  outlook,  and  dig  on  your  own 
account,  and  you  will  soon  find  words  to  express  yourself  well  enough. 

§  32.  Momentum.  A  massive  body  is  naturally  looked  upon  as 
containing  a  greater  '  Quantity  of  Motion  '  than  a  light  one  at  the 
same  speed.  This  '  quantity,'  obtained  by  multiplying  the  mass  and 
speed  together,  mv,  is  called  the  momentum  of  the  body.  Like  v,  it 
is  vectorial.  A  2-oz.  bird  flying  S.  at  32  ft. /sec.  possesses  momentum 
equal,  but  at  right  angles  to,  that  of  a  4-lb.  cat  ambling  W.  at  1  ft. /sec. 

Provided  we  leave  out  of  account,  for  the  time  being,  such 
complicated  contrivances  as  a  discharging  rocket,  which  keeps  on 
losing  mass,  or  a  moving  bus  with  people  jumping  on  and  off,  or 
catastrophic  collisions,  the  mass  of  a  moving  body  does  not  change, 
whatever  happens  to  its  speed. 

§  33.  Force.  Newtonian  Laws  of  Motion  I  and  II.  Variation  of 
velocity  means  variation  of  m  times  v,  of  mv,  the  momentum  of  the 
moving  mass  ;  and  the  product  of  m  and  the  change  of  v  per  second, 
i.e.  mass  X  acceleration,  ma,  means  the  extra  momentum  acquired 
per  second,  or  the  rate  of  change  of  mx>m£,ntum. 

What  does  this  mean  ?  What  causes  the  change  ?  Sir  Isaac 
Newton  laid  down  three  Laws  of  Motion,  of  which  the  first  two  are  : 

I.  Every  body  continues  in  its  state  of  relative  rest  or  motion  in  a 
straight  line  except  when  compelled  by  Force  to  change  it. 

This  is  a  statement  of  the  inertness  or  Inertia  of  Matter. 

II.  Force  is  measured  by  the  quantity  of  motion  (Momentum)  it 
produces  or  destroys  per  second  in  its  own  line  of  action. 

17 


18  MECHANICS  [§  33 

Our  muscular  sense  informs  us  that  we  have  to  exert  force  to  set 
ourselves,  or  anything  else,  in  motion,  or  to  check  its  speed,  or  to 
persuade  it  to  come  round  in  a  curve.  We  argue  that  our  own 
experience  holds  good  generally,  and  that  whatever  affects  the 
motion  of  a  body  is  exerting  Force  on  it.  This  second  law  quanti- 
tatively connects  Force  and  Motion. 

You  have  seen,  and  probably  performed,  parlour  tricks  dependent 
on  the  Inertia  of  Matter  ;  such  as  knocking  a  penny  out  of  a  pile  of 
them  with  a  table-knife,  without  upsetting  the  rest.  At  any  rate, 
Newton  had,  and  this  First  Law  is  how  he  summed  up  his  experiences, 
that  dead  things  don't  move  unless  force  is  applied  to  them  ;  you 
hit  the  one  penny  hard  and  it  moves  out  quickly  ;  apart  from  a  little 
friction  no  force  reaches  the  others,  and  they  just  don't  move. 

'  The  ball  no  question  makes  of  ayes  and  noes, 
But  right  or  left,  as  strikes  the  player,  goes.' 

Don't  get  the  idea  that  '  Inertia  '  is  a  remarkable  new  property ; 
it  is  just  a  classic  word. 

Don't  imagine  that  the  rocket  and  the  bus  elude  these  Laws  : 
they  are  only  calculatory  nuisances  :   outlaws  there  are  none. 

Learn  these  two  Laws.  A  girl  student  of  mine  listened  to  a 
'  friend  '  and  was  persuaded  that  they  were  out  of  date,  because 
Einstein  in  his  Relativity  Theory  had  done  away  with  Force.  They 
came  up,  as  they  often  do,  and  down  she  went.  Of  course,  if  you  are 
fonder  of  a  vast  amount  of  mathematics  of  the  most  difficult,  than  of 
exercising  your  muscles,  follow  Einstein. 

§  34.  This  Law  II,  which  says  Force  =  change  of  mv  per  second, 
that  change  of  momentum  proves  that  force  is  acting,  and  that 
its  amount  per  second  measures  the  force — learn  it,  as  you  learnt  the 
Battle  of  Hastings,  it  is  the  primal  statement  of  mechanics  ;  do  not 
be  of  the  multitude  of  those  who  flounder  among  formulae  in  a 
foredoomed  effort  to  recover  it. 

You  may  have  come  across  a  '  definition  '  of  Force  as  '  that  which 
produces  or  tends  to  produce  motion  in  a  body.'  That  would  better 
define  a  wasp- sting  :  physically  it  is  a  futile  indefinition.  Read 
§  33  again  ;  and  then  say,  if  you  like,  '  Force  is  that  which  produces 
motion  of  a  body  in  the  absence  of  countervailing  force.' 

Consider  a  simple  instance  :  try  to  lift  your  lO-stone  friend  with 
one  hand.  He  doesn't  lift  :  why  not  ?  Because  Mother  Earth  is 
pulling  him  down  all  the  time  with  a  force  of  10  stone  weight,  and 
that  is  more  than  you  can  exert.  Then  why  doesn't  Earth's  pull 
move  him?  Because  Earth's  solid  surface  presses  him  up,  see 
§  44  ;  but  if  he  faints,  he  falls.  How  is  anyone  to  know  how  much 
you  were  '  tending  to  '  lift  him  ?  Nohow  :  you  go  red  in  the  face, 
of  course,  but  then  you  may  be  an  easy  blusher.  Not  until  he 
actually  lifts  off  the  ground  can  anyone  know  that  a  lifting-force 
was  acting — and  then  the  speed  with  which  he  rises,  the  height  to 
which  he  jumps,  gives  a  shrewd  measure  of  it. 


§  37]  MOMENTUM   AND   FORCE  19 

But  in  water,  Earth's  pull  down  on  him  is  almost  counterbalanced 
by  Water's  buoyant  lift,  §  132,  and  with  one  hand  you  can  hft  or 
lower  or  move  him  just  as  you  please,  and  the  force  you  exert  is 
apparent  to  all  by  the  speed  with  which  he  does  move.  If  he  wishes 
not  to  move,  he  must  kick  or  splash,  i.e.  make  the  water  move  instead, 
and  then  the  measure  of  your  force  is  the  vigour  of  the  splash  which 
countervails  it,  i.e.  always  a  motion  of  mass. 

But  this  talk  of  '  tending  to,'  in  natural  philosophy,  is  mere 
weakness  in  the  knees — or  in  the  head. 

Force  =  change  of  mv  per  second. 

=  m  X  change  of  v  per  second. 

—  m  X  v/t. 

=  m  X  a,  mass  x  acceleration. 

§  35.  Suppose,  therefore,  we  take  a  Triangle,  or  Parallelogram,  of 
Velocities,  such  as  Fig.  1,  (v)  :  divide  each  line  in  it  byj^ ;  that  only 
means  changing  the  scale  we  measure  them  by,  it  does  not  change 
the  shape.  But  now  it  has  become  a  Triangle,  or  Parallelogram, 
of  {vlt)s,  i.e.  of  Accelerations ;  so  that  one  combines  or  resolves 
accelerations  by  the  Vector  Triangle  or  Parallelogram,  as  with 
velocities. 

Or  else,  multiply  each  line  by  m ;  that  only  means  using  yet 
another  scale  to  measure  them  by,  the  shape  doesn't  change  ;  but 
now  it  is  scaled  in  masses  X  velocities,  or  Momenta. 

Or,  divide  by  t  and  multiply  by  m ;  this  means  using  yet  a  third 
scale,  the  shape  doesn't  change,  and  now  you  have  a  Triangle,  or 
Parallelogram,  of  masses  x  accelerations,  i.e.  of  Forces.  So  that 
Forces  combine  or  resolve  by  the  vector  law  :  we  shall  use  this  often. 

§  36.  Sir  Isaac  Newton  was  born  of  a  line  of  farmers  at  Wools- 
thorpe  in  Lincolnshire,  on  Christmas  Day  1642,  the  year  of  the  death 
of  Galileo.  He  became  the  foremost  of  natural  philosophers,  and 
will  be  refen-ed  to  again  and  again  in  this  book.  He  was  annually 
re-elected  President  of  the  Royal  Society  from  1703  until  his  decease 
in  1727.  His  remains  lie  in  Westminster  Abbey,  and  his  statue 
dominates  the  ante-chapel  of  Trinity  College,  Cambridge. 

§  37.  A  Natural  Law.  His  are  the  first  Natural  Laws  that  we 
have  come  across.  Now,  you  will  be  nearly  as  foolish  as  that  girl 
student  if  you  say  that  such  and  such  happens  '  because  somebody's 
Law  says  it  must.'  That  is  no  explanation  at  all,  it  is  a  mere  putting 
the  cart  before  the  horse,  although  it  appears  to  satisfy  a  great 
many  people  who  would  regard  the  pious  poet's  perfectly  accurate 
one,  that  '  'tis  their  nature  to,'  as  a  truism  beneath  contempt. 

A  Law  in  Natural  Philosophy  is  an  expression  of  the  originator's 
beUef ,  founded  on  the  gradually  accumulating  evidence  of  observa- 
tion and  experiment  (often  very  indirect),  that  under  prescribed 
circumstances  matter  invariably  behaves  in  the  manner  stated, 
provided  that  all  disturbing  influences  are  got  rid  of.  As  time  goes 
on,  reliance  comes  to  be  placed  on  the  law  according  to  the  way  it  is 


20  MECHANICS  [§  37 

supported,  or  not,  by  further  experiment.  Every  success  increases 
its  probability,  and  every  failure  diminishes  it,  but  always  it  remains 
a  probability  :  '  So  far  as  is  known  at  present  *  is  the  unwritten 
preface  to  every  accepted  Natural  Law.' 

§  38.  And,  while  we  are  at  it,  as  to  the  meaning  of  a  Theory. 
Don't  get  exercising  the  eyebrow  muscle,  the  '  levator  labice  superioris 
aloeque  nasi'  or  other  little  facial  muscles  that  you  will  be  picking  at 
in  a  year  or  two,  as  soon  as  you  see  the  word  Theory.  Nor  let  it  give 
you  *  a  sinking  feeling ' ;  it  is  not  a  thing  to  make  difficulty  for  you, 
but  to  ease  it. 

In  most  cases  the  function  of  a  Theory  is  to  give  us  a  mental 
working  model,  built  up  of  easily  realizable  notions  of  things  we 
can  see  and  feel,  which  in  its  action  shall  imitate  and  help  us  to 
forecast  the  stranger  and  more  recondite  processes  going  on  among 
entities  of  which  our  appreciation  is  mainly  intellectual. 

Originally  a  mere  suspicion  in  some  active  mind,  it  has  been  put 
forward  as  a  '  working  hypothesis,'  and  been  found  to  fit  in  with  the 
results  of  large  numbers  of  experiments,  until  it  has  become  the 
familiar  Theory  to  which  men's  thoughts  are  almost  unconsciously 
moulded  ;  and  for  a  while  it  controls  the  progress  of  its  branch  of 
science.  But  when  many  new  facts  come  to  Hght  which  it  cannot 
explain,  and  it  is  shown  that  a  quite  different  supposition  agrees  with 
them  and  also  with  the  facts  on  which  the  former  theory  was  based, 
then  let  it  pass  away — ^with  the  honours  of  war,  for  maybe  it  will 
become  a  useful  ally  again  as  fuller  knowledge  develops. 

We  need  not  entertain  the  conceit  that  in  it  we  have  hit  upon  the 
ultimate  truth,  as  the  newspapers  do  the  moment  they  hear  of  it ; 
we  are  but  exploring  what  promises  to  be  the  next  turning  on  the 
way  to  that  far-distant  goal. 

§  39.  The  Unit  of  Force  must  logically  be  that  force  which  produces 
unit  change  of  momentum  in  1  second, 

i.e.  which  gets  1  gramme  moving  at  a  speed  of  1  cm.  per  second 

after  pushing  on  it  for  a  second, 
or    which  increases  the  velocity  of  1  gm.  by  1  cm.  per  second  every 

second, 
or    which  gives  1  gm.  the  unit  acceleration  of  1  cm.  per  second  per 

second. 

Choose  which  statement  you  understand  best ;    it  defines 
the  Dyne. 

By  experiments  to  be  described  later  it  is  found  that  the  Earth 
exerts  on  a  gramme  mass  an  attraction  which  increases  its  speed 
about  981  cm. /sec.  in  each  second  of  its  motion. 

That  is,  the  weight  of  a  gramme  mass  is  about  981  times  the  unit 
of  force,  the  dyne. 

The  dyne  is  thus  a  trifle  more  than  a  milligram  weight ;  its  small- 


§41]  MOMENTUM  AND   FORCE  21 

ness  involves  big  numbers,  which  the  physicist  writes  in  powers  of  10  ; 
e.g.  981  millions  =  9-81  X  10^.  And  small  fractions  in  negative 
powers,  0-000033  =  33  X  lO-^. 

Force  W  dynes  =  mass  M  gm.  x  g,  the  general  symbol  for  the 
acceleration  of  gravity. 

Force  W  dynes  =  M  gm.  x  981  cm.  per  sec.  per  sec. 

The  unit  of  Force  in  English  Measure,  Foot-Pound-Second,  is 
called  the  Poundal,  and  it  accelerates  one  pound  mass  by  one  foot 
per  second,  every  second. 

Force  W  poundals  =  M  lb.  X  32  ft. /sec.  per  sec. 

§  40.  Thus,  you  see,  the  Gram  Weight,  which  naturally  suggests 
itself  to  most  people  as  a  Unit  of  Force,  is  really  some  queer  number  of 
times  the  unit  that  follows  logically  from  our  Laws  of  Motion. 

What  is  worse,  this  queer  number  is  variable. 

The  gramme  mass  is  constant  enough,  wherever  it  be  found  in  all 
Space,  but  the  force,  the  gramme  weight,  the  pull  of  the  Earth  on  the 
gramme  mass,  depends  on  how  near  it  is  to  the  centre  of  mass  of  the 
Earth.  Now,  the  Earth  is  not  a  smooth  sphere  at  rest,  but  a  nobbly 
spheroid  spinning  round  fast,  bulged  by  its  speed,  and  the  weight  of 
the  gram  at  any  particular  place  is  merely  the  difiference  between  the 
gravitational  pull  of  the  earth  at  that  particular  distance  from  its 
centre,  and  the  centrifugal  force  with  which  the  gram  would  fly  ofif 
the  spinning  earth  if  gravity  suddenly  let  go.  This  deduction  gets 
larger  the  nearer  to  the  equator,  where  the  radius  is  greatest  and 
the  motion  is  fastest,  and  the  981  reduces  to  978,  whereas  it 
increases  to  983  at  the  poles,  where  there  is  no  centrifugal  force  : 
further  values  are  given  in  §  47,  showing  this  Variation  with 
Latitude. 

Variation  as  much  as  this,  a  penny-farthing  in  the  pound,  wouldn't 
be  tolerated  in  commerce,  and  cannot  possibly  be  in  the  very 
foundations  of  science.  So  the  exact  quotation  of  a  Force  must  be 
given  in  Dynes,  and  remains  the  same  anywhere  in  Space. 

If,  for  everyday  convenience,  one  wishes  to  bring  it  into  grams 
weight,  then  : 

Force  in  grams  weight  =  force  in  dynes  -^  value  of  gravity  at  place 
of  experiment, 

and  this  g  is  approximately  981  at  sea  level  in  our  latitude.  Of 
course,  for  lots  of  common  purposes,  forces  quoted  straightaway  in 
grams  weight,  as  understood  in  the  temperate  zones,  are  quite  near 
enough. 

§  41.  A  digression  on  Friction.  We  never  see  the  first  phrase  of 
the  laws  of  motion  obeyed.  Moving  bodies,  unaided  by  applied 
power  or  downhill  slope,  always  slow  do^vn  and  stop.  Our  laws  bid 
us  look  for  forces  acting  always  to  reduce  momentum.     We  know 


22 


MECHANICS 


[§41 


that  reducing  the  roughness  of  surfaces  in  contact  prolongs  the 
motion,  e.g.  rolling  the  green,  or  ironing  the  cloth.  We  have 
reduced  Friction. 

Friction  is  a  Force  which  always  destroys  momentum.  It  breaks 
down  the  motion,  as  in  all  those  contrivances  misspelt  '  brakes.' 
It  must  be  subtracted  from  any  force  applied  to  increase  speed. 

Force  applied  —  friction  =  increase  of  momentum  per  second  in 
direction  of  force, 
or  Force  applied  =  friction  +  ditto. 

Friction  helps  every  retarding  force. 

§  42.  Friction  between  dry  surfaces.  In  Fig.  3  a  hanging  weight  w 
pulls  a  loaded  board  W  along  the  table  ;  the  object  is  to  study  the 
friction  between  board  and  table-top.  As  you  gradually  increase  w, 
giving  the  load  W  a  little  tap  along  each  time,  because  starting 


[YA 


m- 


FiG.  3. 


friction — '  stiction  ' — is  almost  always  a  trifle  greater,  you  presently 
find  that  W  will  slide  slowly  and  continuously  along. 

When?  this  happens,  w/W,  i.e.  the  fraction  of  the  force  W  pressing 
the  surfaces  together  which  has  to  be  applied  parallel  to  them  to 
cause  slipping,  is  called  the  Coefficient  of  Friction. 

This,  the  friction  between  dry  surfaces,  depends  very  much  on 
their  nature  and  smoothness,  e.g.  hard  wood  on  planed  deal  0-22, 
rubber  tyre-tread  on  dry  concrete  road  surface  J  to  f . 

If  you  overload  w  only  a  little,  the  speed  of  sliding'  increases 
altogether  disproportionately  :  you  infer  that  the  speed  of  sliding 
nmkes  very  little  difference  in  dry  friction. 

If  you  double,  treble,  etc.,  W,  you  will  find  that  w  has  to  be  doubled, 
trebled,  etc.,  to  cause  shpping,  and  the  fractional  coefficient  remains 
unchanged ;  i.e.  the  friction  between  dry  surfaces  is  proportional 
to  the  load  pressing  them  together. 

If  you  look  at  the  two  surfaces  critically,  edge-on  to  the  light,  you 
will  see  that  the  plane  has  left  them  by  no  means  perfectly  flat, 
but  that  contact  is  only  on  streaks  and  blurs,  which  must  be  quite 


§  43]  MOMENTUM   AND   FORCE  23 

different  from  place  to  place  along  the  table,  and  there  is  no  means  of 
ensuring  that  the  area  of  contact  is  the  same  everywhere.  This 
made  no  difference  :  you  infer  that  friction  between  dry  surfaces  is 
independent  of  the  area  of  contact. 

Liquids  and  lubricated  surfaces  follow  very  different  laws, 
for  which  see  §  335. 

Friction  is  often  a  bore,  but  seeing  that  without  it  most  things — 
instance  only  clothing  and  furniture  and  houses — would  immediately 
fall  to  pieces,  while  we  should  be  unable  to  keep  our  feet  anywhere — 
we  must  put  up  with  it,  and  merely  devise  means  of  minimizing 
it  when  necessary  ;  wheels  and  rollers,  etc. 

A  very  general  way  of  doing  away  with  friction  in  apparatus,  etc., 
is  to  set  up  Vibration,  when  the  sticking  surfaces  conceivably  hop 
entirely  free  of  each  other,  momentarily,  and  the  moving  part  enjoys 
brief  instants  of  perfect  freedom.  For  instance,  one  just  taps  a 
hydrometer  jar,  or  a  compass,  or  a  weather-glass  :  you  will  see  the 
jockey  '  float  '  along  the  monochord  wire  when  this  takes  up  a  note, 
just  as  books  float  off  the  seat  of  a  jarring  railway  carriage,  and 
fondly  tightened  nuts  vanish  from  a  motor-bike. 

§  43.  The  Newtonian  Law  III.  III.  Activity  and  reactivity  are 
equal  and  opposite.  Or  action  and  reaction,  or  put  and  take,  but 
these  words  have  so  many  meanings  that  here  I  prefer  an  unusual 
word,  which  demands  explanation. 

Suppose  you  start  to  run  ;  you  push  off,  and  away  you  go.  But 
suppose  the  ground  is  unexpectedly  slippery  ;  '  the  coefficient  of 
friction  is  small ; '  your  foot  slips  back,  and  you  don't  get  any 
forwarder.  So  in  skating,  if  you  don't  contrive  to  get  a  grip  on  the 
ice,  you  can't  start.  Anjrway,  why  should  you,  by  pushing  back- 
wards, start  moving  forwards  ?  Answer,  you  push  actively  back  on 
the  ground,  and  the  ground  reactively  pushes  forward  on  you,  and 
that  is  why  you  move  forward.  The  Law  says,  these  two  opposites 
are  equal. 

Standing  on  the  ice,  your  weight  presses  the  ice  down,  bends  it, 
and  so  calls  into  play  its  elastic  strength,  and  its  buoyancy,  and  it 
presses  equally  up  on  you.  If  not,  you  fall  through,  gravity  pulls 
you  in.  Walking  across  the  floor,  at  each  step  your  weight  presses 
down  the  floor,  which  gives  elastically,  developing  an  equal  upward 
force,  and  bears  you  up.  Unless  the  floor  does  give  a  little,  it 
develops  no  supporting  force  ;  the  mere  fact  that  your  tread  can 
be  heard  in  the  room  downstairs  means  that  the  floor  is  jumping 
up  and  down,  vibrating,  sending  sound-waves  through  the  air 
below. 

Suppose  you  put  the  brakes  on  a  car,  asking  the  wheels  to  press 
forward  on  the  ground  instead  of  kicking  it  away  behind,  and  the 
road  is  slippery,  with  a  coefficient  of  friction  only  1/20  instead  of  J  ; 
it  does  not  push  back  on  you  hard  enough,  and  you  go  skidding  on, 
checked  sometimes  more  on  one  wheel,  sometimes  on  the  other, 
yawing  about  helplessly. 


24  MECHANICS  [§  44 

§  44.  We  have  been  talking  about  Force.  Where  does  it  come 
from  ?  From  your  strong  arms,  of  course ;  what  do  you  know  of  force 
beyond  their  strength  ?  What  real  notion  have  you  of  the  weight  of  a 
ton  ?  Jump  into  a  skiff,  and  pull  hard  at  an  oar,  and  the  reactive  pull 
of  the  oar  promptly  slips  you  off  your  seat.  So  your  strong  arms 
haven't  been  so  much  use.  Adjust  the  stretcher,  and  now  pull  as 
hard  as  you  like,  while  your  feet  react  equally  hard  on  the  stretcher. 
Stand  on  slippery  ice  and  push  another  man  hard  :  you  can,  but  not 
for  long,  because  you  slip  rapidly  backward,  and  are  soon  out  of 
reach,  and  your  strong  arms  are  of  no  further  use.  The  reaction  to 
your  push  has  accelerated  your  mass,  giving  you  momentum  = 
JForce  X  time  of  pushing.  You  keep  sliding  back  until  friction  on 
ice  X  time  of  sliding  =  your  initial  momentum ;  if  you  strike  a 
rough  patch,  friction  increases,  and  time  is  shortened. 

Since  1  dyne  produces  unit  momentum  per  second,  F  dynes  acting 
for  t  seconds  on  a  mass  m  grams  will  give  it  a  momentum  Ft  =  mv, 
where  v  is  the  speed  which  it  acquires  provided  frictional  force  is 
kept  from  interfering  (and  a  ball,  e.g.  starts  from  your  hand  with  but 
little  friction).  But  suppose  friction  is  great ;  press  with  force  F  on 
a  wall,  and  the  wall  doesn't  move,  yet  you  are  giving  it  F  units  of 
momentum  (they  have  no  name)  every  second  ;  but  Friction  between 
the  wall  and  the  earth  can  far  exceed  your  feeble  F.  And  for  you 
to  push  the  wall,  your  feet  must  push  on  the  ground  the  other  way, 
so  as  much  momentum  as  you  give  the  wall  and  the  attached  earth 
one  way,  you  are  also  giving  the  earth  the  other  way,  the  algebraic 
sum  is  zero,  and  nothing  moves.  But  push  off  from  the  wall  and 
run,  you  and  the  earth  have  equal  and  oppositely  directed  momenta  ; 
as  you  run  the  earth  continuously  moves  back,  being  massive  it 
does  not  move  fast,  mY  +  M^?  =  0.  To  stop  running,  your  feet 
exert  backward  force  for  a  time,  and  the  earth  presses  on  your  feet, 
and  the  two  opposite  momenta  destroy  each  other. 

All  Force,  you  see,  is  between  Masses. 

You  have  long  since  found  out  how  useless  it  is  to  try  to  nail 
together  thin  boards  that  persistently  give  way  to  the  blow  ;  holding 
them  together  by  hand  while  you  strike  is  no  good  at  all.  But  just 
hold  a  brick,  or  an  iron  block,  behind  and  lightly  in  contact  with  the 
farther  loose  board,  and  you  can  hammer  the  nail  in  perfectly  well, 
the  inert  mass  jumping  only  a  little  at  each  stroke. 

Every  force,  whatever  its  exciting  cause,  must  be  anchored  on  a 
mass.  To  the  old  catch,  what  happens  when  immovable  mass  meets 
irresistible  force  ?  the  answer  is,  that  the  non-existence  of  the  one 
implies  the  impossibility  of  the  other. 

Here  is  a  special  application  of  the  Third  Law  for  you  :  every 
muscle,  as  it  contracts,  pulls  equally  at  both  ends.  So  when,  in 
Anatomy,  you  do  your  '  bones,'  and  find  a  muscle  attachment,  recall 
at  once  on  what  bone,  and  where,  the  other  end  pulls,  and  then 
reflect  what  happens  at  both  ends,  on  contraction.  If  that  doesn't 
halve  your  toil  it  will  double  its  efficiency  :  verb.  sap. 

Here  on  the  foredeck  of  this  lively  ship  I  am  getting  abundant 


§44]  MOMENTUM  AND  FORCE  25 

illustration  of  accelerative  forces,  but  a  motor-bus  provides  them 
perfectly  well.  '  Hold  tight,'  says  the  conductor  as  he  rings  the 
bell,  and  hold  tight  and  pull  yourself  forward  you  must,  or  the  bus 
will  leave  you  behind ;  the  forward  accelerative  force  comes  to 
you  through  your  arms,  soon  giving  you  very  considerable  momen- 
tum (Law  II) .  This  you  retain,  as  you  stand  or  sit  at  ease  at  uniform 
speed,  neither  pushed  nor  pulled ;  you  are  obeying  Law  I.  You 
turn  back  to  leave  the  bus  as  it  slows  to  a  stop,  and  you  have  to 
haul  yourself  along  back  to  the  door,  you  are  getting  rid  of  your 
forward  momentum  as  fast  as  you  can  ;  if  then  you  jump  before  the 
bus  stops,  you  come  down  heavily  forward  on  your  feet  on  the 
pavement,  and  the  earth  pushes  the  rest  of  your  momentum  out  of 
you  very  shortly,  else  you  must  run  on. 

Or  board  a  train,  and  hark  at  the  mighty  puffs  of  the  engine  as  she 
accelerates  you  into  speed,  her  pull  F  =  mv  given  to  the  train  per 
second  +  frictional  retarding  force.  This  frictional  retarding  force 
is  small  at  first,  being  merely  well-oiled-axle  friction,  but  as  speed 
increases  air  resistance  develops  enormously.  Presently  the  driver 
links-up,  and  quiets  the  puffing — the  boiler  has  not  steam  enough 
for  that  great  effort  for  long  ;  the  train  moves  at  uniform  speed  on 
the  level  track,  for  though  the  engine  is  pulling  all  the  time  and 
adding  momentum,  the  air  and  the  axle-boxes  are  dragging  just  as 
hard,  and  robbing  it  of  all  this  additional  momentum.  You  can 
apply  a  similar  analysis  to  the  family  car. 

Raindrops  of  the  largest  size  fall  no  faster  than  25  ft. /sec,  the 
earth  pulls  them,  but  the  air  resists,  and  their  momentum  remains 
unchanged.  This  means  that  the  two  forces  are  equal  and  opposite, 
and  the  body  moves  on,  obeying  Law  I ;  the  earth's  force  is  entirely 
spent  in  giving  momentum  to  fresh  quantities  of  air  which  the  drop 
disturbs  on  its  way  down.  A  snowflake  of  the  same  weight  falls 
slower  still ;  it  is  a  broad  thing  which  necessarily  accelerates  out  of  its 
way  a  greater  bulk  and  mass  of  air,  per  foot  of  fall,  than  did  the 
plump  smooth  drop,  so  that  the  momentum  mg  given  it  per  second 
by  the  earth's  pull  is  spread  over  a  large  invisible  mass,  M,  which 
consequently  has  only  a  small  acceleration,  and  speeds  must  remain 
slow.  A  falling  man  gains  speed  up  to  250  ft. /sec,  when  the  air 
resistance  =  his  weight,  and  he  then  moves  uniformly  by  Law  I, 
until  he  pulls  the  cord  and  spreads  his  parachute,  which  now 
disturbs  and  accelerates  from  rest  a  very  large  mass  of  air,  per  foot 
of  fall,  and  again  a  great  M  can  only  be  given  a  small  a  by  the  avail- 
able force  mg. 

In  vacuo,  with  no  air  to  be  moved,  snowflakes  and  feathers  would 
fall  as  fast  as  hailstones  or  guineas,  a  classic  experiment. 

In  a  tug-of-war,  when  everjiihing  is  balanced,  somebody  slips, 
and  the  whole  mass  begins  to  accelerate  towards  the  enemy.  He 
regains  his  footing  and  pulls  as  hard  as  before ;  the  acceleration 
ceases,  but  the  acquired  momentum  is  not  destroyed,  and  the 
ominous  drift  continues,  at  uniform  velocity.  He  and  his  side  must 
pull  harder,  and  decelerate  the  mass,  destroy  that  momentum  ;  keep 


26  MECHANICS  [§44 

on  pulling  harder,  and  now  the  extra  pull  goes  on  producing  visible 
momentum  his  way,  the  whole  mass  accelerates  towards  safety. 
Always,  strange  to  say,  the  pull  in  the  rope  is  equal  both  ways  ; 
only,  when  it  exceeds  one  team's  foot-grip,  this  difference  accelerates 
them  into  danger,  and  that  acceleration  can  be  checked  only  by 
increased  foothold. 

In  a  lift,  you  stand  on  the  floor  with  your  weight  W  dynes  = 
M  X  gf  dynes.  Now,  if  the  lift  starts  down,  i.e.  drops  away  from  you, 
with  an  acceleration  a,  the  force  with  which  your  feet  press  on  the 
floor  reduces  to  M  X  (g^  —  a)  dynes  [if  the  lift  were  let  fall  freely 
a  would  =  g,  and  you  would  cease  to  press  on  the  floor  at  all].  But 
this  '  sinking  feeling  '  ceases  when  the  lift  attains  its  steady  full  speed 
downwards,  i.e.  a  =  0,  and  your  feet  carry  Mg  again.  Then, 
towards  the  bottom,  the  lift  decelerating  towards  a  standstill  with 
acceleration  —  6,  your  feet  press  down  with  force  M{g  +  b)  dynes. 

a  X  duration    of  starting  =  full  speed  of  lift  =  6  x  duration  of  stopping 

Whence  Ma  x  h  =  Mv  =  M6  x  t^, 

Loss  of  wt.  x  time  of  start  =      momentum      =  gain  of  wt.  x  time  of  stop 

Nowhere  is  the  inter-relation  of  force  and  momentum  better 
illustrated  than  when  a  tug  on  a  rapid  river  {e.g.  the  Rhone  at 
Avignon)  brings  her  tow  alongside  to  tie  up  for  the  night.  Her 
engines  are  eased  until  the  momentum  that  her  beating  paddles  fling 
into  the  water  is  only  just  equal  to  that  which  the  moving  water  of 
the  stream  flings,  ceaselessly,  against  the  bows  of  the  vessels  at  rest 
relatively  to  the  bank.  Helms  over,  they  drift  in,  ropes  are  thrown 
from  each  craft  and  made  fast  to  the  bollards  on  shore,  but  remain 
slack.  Easing  still,  her  engine  throws  less  and  less  '  quantity  of 
motion  '  into  the  water,  but  the  balance  of  momentum  must  be 
kept  up  somehow,  the  mooring-ropes  slowly  tighten  as  the  engines 
sink  to  rest,  and  the  flotilla  sleeps  all  night  with  F,  the  silent  steady 
invisible  pull  of  the  ropes,  exactly  balancing  m,  the  mass  of  water 
surging  noisily  against  the  bluff  bows  every  second  X  v,  the  check 
to  its  downstream  speed  ;  force  =  momentum  destroyed  per  second 
in  the  stream. 

This  has  been  a  long  spell  of  Newtonian  Philosophy  of  Motion, 
but  your  examiners  are  oft-times  curious  about  it.  Treat  these 
dozen  instances  as  so  many  puzzles,  try  one  and  another  until  you 
find  you  are  getting  the  knack  of  them,  and  the  rest  will  follow  with  a 
rush.     Puzzles,  not  mental  shackles. 

§  45.  Impact.  Impulsive  forces.  In  the  colUsion  or  impact  of 
two  bodies  the  small  force  exerted  between  them  increases  to  a 
very  large  one  as  they  squeeze  each  other  out  of  shape.  In  plastic 
substances — lead,  clay,  etc. — or  fluids,  the  force  then  diminishes 
as  they  cease  to  squeeze  closer,  and  the  particles  become  accom- 
modated to  their  displaced  positions  ;  the  bullet  flattens  on  the 
target,  the  quoit  sticks  in  the  clay,  the  water  checks  your  rushing 
dive  and  bears  you  quietly. 


§  46]  MOMENTUM  AND   FORCE  27 

In  elastic  bodies,  it  decreases  as  they  move  apart  again,  for 
their  particles  have  no  choice  but  to  move  back  where  they  came 
from. 

The  whole  process  takes  only  a  small  fraction  of  a  second,  but 
at  every  instant  equal  and  opposite  forces  are  exerted  as  the  two 
bodies  change  their  motion,  one's  gain  of  forward  momentum 
is  equal  to  the  other's  loss,  and  the  whole  momentum  of  the  system 
(the  two  together)  remains  unchanged. 

This  is  called  the  Principle  of  the  Conservation  of  Momentum. 

The  impulse  is  measured  by  the  forward  momentum  imparted 
to  one  body  :  i.e.  it  can  be  expressed  as  the  average  force  which, 
acting  for  1  sec,  would  produce  the  same  change  as  the  varying 
and  enormous  force  which  acts  for  perhaps  0-0001  sec.  Part  of  this 
forward  momentum  may  be  used  in  destroying  existing  backward, 
as  in  a  cricket -bat  striking  and  reversing  the  direction  of  motion  of 
the  ball. 

Most  of  you  will  sooner  or  later  play  golf ;  let  us  consider  what 
happens  when  a  golf  ball  is  struck  by  a  driver.  The  ball  gets  short 
notice  to  move  on  quickly,  much  momentum,  50  gm.  X  V  of  flight, 
has  to  be  given  to  it  by  average  force  F  acting  for  time  t 

¥t  =  50V 

and  as  t  can  be  only  a  very  brief  fraction  of  a  second,  F  must  be 
large — you  wouldn't  care  to  hold  your  thumb  between  the  ball  and 
the  club — so  large,  in  fact,  that  spark- photographs  show  the  ball 
squashed  in  by  quite  1  cm.  Club  and  ball  at  that  moment  cease  to 
move  closer,  i.e.  they  are  temporarily  moving  at  the  same  speed. 
The  momentum  of  the  club  is  now  spread,  without  change  of  total 
quantity,  over  club  and  ball.  But  this  squashed  rubber  ball  is 
pushing  hard  on  the  club  in  its  elastic  effort  to  regain  its  shape,  and 
this  reactive  force  is  accelerating  it  away  from  the  club.  Supposing 
it  is  perfectly  elastic,  it  will  go  in  inverse  order  through  exactly  the 
same  succession  of  deformations,  before  it  leaves  the  club,  as  it 
suffered  while  the  club  moved  from  first  contact  to  maximum  ; 
i.e.  the  same  average  F  will  act  on  it  for  the  same  time  ;  it  therefore 
gains  as  much  again  momentum,  and  flies  off  at  double  the  speed  with 
which  the  club  follows.     (See  Ex.  15.) 

§  46.  A  muscle  exerting  what  we  consider  a  steady  force  is  receiv- 
ing from  10  to  40  nerve  stimuli  per  second,  and  can  be  heard  to 
vibrate.  When  tired,  the  stimuli  are  less  frequent,  and  the  force 
becomes  visibly  unsteady,  trembling,  an  obvious  sequence  of 
momentum-giving  impulses. 

On  the  Kinetic  Theory  of  Matter  all  substances  consist  of  mole- 
cules swarming  in  rapid  motion  ;  the  pressure  of  a  weight  on  the 
table  becomes  the  momentum  imparted  per  second  by  the  myriad 
impacts  of  one  molecular  swarm  on  another.  If  you  doubt  that 
great  solidity  can  arise  in  this  way,  spin  a  bicycle-wheel  fast  and  try 


28 


MECHANICS 


[§4e 


to  put  your  finger  through  it — and  that  is  only  a  few  dozen  impacts 
per  second. 

Thus  the  distinction  between  a  steady  force  and  momentum  of 
visible  motion  bridges  over,  and  our  way  of  measuring  forces  is 
justified.  J 

§  47.  Gravitation  is  the  mutual  attraction  of  all  masses. 
The  theoretical  method  of  measuring  forces  is  to  let  them  act  for 
1  sec.  on  a  mass  and  find  the  momentum  they  have  given  it.     Using 
1  gm.  the  velocity  it  acquires  in  the  second  (its  acceleration)  is  equal 
to  the  force  in  dynes. 

Practically,  one  weighs  the  force  against  the  gravitational  attrac- 
tion of  the  earth  on  a  known  mass.  Now,  this,  the  weight  of  the 
mass,  varies  a  little  from  place  to  place,  §  40.  Hence  the  weight 
of  a  gramme  cannot  be  made  a  primary  standard  of  force,  and  for 

accurate  scientific  purposes  we 
must    be    ready    ta   find    how 
many  djrnes  it  represents  locally. 
This  is  called  g,  the  force  of 
Gravity  at  a  place,  the  force  that 
people    associate    with  Newton 
and    the    apple.     Doubtless  he 
had  lain  awake  sultry  autumn 
nights,   listening  to   the  rustle 
and  plop  of  apple  after  apple  in 
his  orchard,  undone  by  the  dry- 
ness of  the  season  ;  wondering, 
like  all  good  apple-growers,  that 
these  things   should  be ;    won- 
dering, in  his  case,  how. 
Plainly,  being  the  dynes  that  are  acting  on 
the  gramme,  g  is  the  acceleration  of  a  falling 
gramme,  or  of  every  individual  gramme  in  a 
falling  body,  and  hence  these  methods  of  find- 
ing it. 

1.  Free  fall.  Fig.  4.  Things  fall  fast,  but 
measurements  may  be  made  with  a  tuning-fork  as  timekeeper.  A 
smoked-glass  strip  drops  from  the  dotted  position,  and  the  pointer 
on  the  fork  marks  on  it  one  complete  wave  for  each  vibration, 
occupying  the  very  short  time  P.  (How  P  is  found  see  Chapter 
XXVIII.)  Then  distance  s,  measured  from  the  starting-point, 
which  contains  n  waves,  has  been  fallen  in  time  nP,  and  s  =  \at^ 
becomes  s  =  Jgr(nP)2,  hence  g.  The  one  difficulty,  calling  for  steady 
handling,  in  this  experiment,  is  to  keep  a  clear  trace  at  the  starting 
place. 

2.  Atwood's  machine,  Fig.  5.  Atwood  {ca.  1790)  slowed  the 
speed  of  fall  of  a  weight  by  making  it  drag  along  inactive  masses. 
Equal  masses  MM  balance  on  a  light  frictionless  pulley.  On  one 
m  is  laid,  and  the  force  mg  dynes  pulling  it  down  has  now  to  move 


n 


M 


Fig.  5. 


Fig.  4. 


p 


48]  MOMENTUM   AND   FORCE  29 

the  whole  lot,  m  +  2M,  so  that  the  acceleration,  which  is  (force  -^ 
mass),  is  necessarily  reduced  in  the  same  ratio  as  the  mass  has  been 
increased,  i.e.  it  is  reduced  to  the  fraction  ml{m  -f  2M)  of  g. 

The  time  t  seconds  of  fall  from  rest  through  a  cm.  is  observed ; 
in  these  t  seconds  the  speed  steadily  rises  to  t  times  the  gain  in  an 
individual  second,  i.e.  to  gt  X  ml{m  -\-  2M),  and  therefore  averages 
half  this  value  over  the  whole  time  t. 

This  average  Igt  x  m/im  +  2M),  lasting  for  t  seconds,  carried  the 
system  through  the  permitted  distance  s  cm.  =  speed  x  time, 


it 


s  =  t  X  igt  X  ml{m  +  2M) 


which,  of  course,  is  the  s  =  ^at^  of  §  17. 

In  practice  I  use  an  ordinary  aluminium  ball-bearing  pulley, 
the  finest  plaited-silk  fishing-line,  and  one  M  a  trifle  heavier  and 
readjusted  before  use  until  it  just  feebly  crawls  down  when  given  a 
start,  indicating  that  inevitable  friction  has  been  neutralized.  It 
is  loaded  with  m,  the  pulley  hauled  up  the  wall  until  they  are  at  the 
desired  height,  and  the  other  M  let  go  from  the  table  as  the  clock 
ticks.     It  works  better  than  elaborate  machines. 

3.  Pendulum.  Method  1  is  hasty  and  2  is  grievously  affected 
by  friction.  The  ever-falling  pendulum  gives  by  far  the  most 
accurate  method.  From  §  84  its  time  of  swing  =  27T:\/length  -^  g. 
.'.  g  =  ^TiH  -i-  t^.  It  is  best  swung  in  vacuo,  but  in  the  laboratory 
you  will  use  a  simple  heavy  ball,  on  a  thin  plaited  line  which  will 
not  untwist  and  lengthen  as  you  watch. 

Newton  used  a  pendulum  with  a  bob  filled  with  wood,  wheat,  etc., 
to  satisfy  himself  that  the  Earth  attracts  all  substances  proportion- 
ally to  their  masses.  In  Gravitation,  nothing  matters  but  mass  and 
distance. 

Some  values  of  gr  are:  Equator  978-1,  Lat.  45°  980*6,  Greenwich 
981-17,  Edinburgh  981-54,  Pole  (calc.)  983-1  dynes  to  the  gram 
weight,  or  acceleration  in  cm./sec.^  The  altitude  of  Everest  reduces 
it  by  l/800th  part.  For  the  importance  of  these  variations,  see 
§§40,  113,  etc. 

These  being  in  cm.,  to  get  the  value  in  English  measure,  poun- 
dals  to  the  pound,  or  ft./sec.,^  divide  by  30-48  cm.  =  1  ft. 

§  48.  From  astronomical  considerations  Newton  was  led  further 
to  enunciate  the  Law  of  Universal  Gravitation  :  Any  two  masses 
attract  each  other  with  a  force  proportional  to  their  product,  and  inversely 
to  the  square  of  the  distance  between  them.  The  gradual  firm  establish- 
ment of  this  law  was  a  highly  technical  proceeding. 

By  building  up  a  5-ton  sphere  of  lead  with  its  centre  50  cm.  below 
a  1 -kilogram  ball  hanging  by  a  long  wire  from  a  balance,  it  was 


30  MECHANICS  [§  48 

found  that  its  downward  attraction  added  to  the  weight  of  the  ball 

by  i  dyne, 
mu      1  J                  5,000,000  gm.  X  1000  gm.  ,,  ,    ,    ,  , 

Thus  J  dyne       =  -^ 50  x  50  cm. ^  (whatever  may  be 

the  value  of  the  attraction  between  gram  and  gram  1  cm.  apart) 

.*.  this  latter  '  Newtonian  constant '  =  1  fifteen  millionth  of  a  dyne. 
Applying  this  to  the  attraction  of  1  gram  to  the  Earth 

_  Pia-ss  of  Earth  x  1  gram  1 

981  dynes  -  (^37000,000  =  radius)2   ^  15,000,000 

and  the  Mass  of  the  Earth  works  out  to  6000  trillion  tons,  6  x  lO^^ ; 
and  its  Mean  Density  to  5-5,  about  double  that  of  its  crust. 

By  its  pull  on  the  Earth,  the  Sun  calculates  out  to  be  330,000  times 
as  massive,  and  of  mean  density  1-4,  while  the  Moon  is  only  l/80th 
Earth,  and  its  mean  density  3-3. 

Planets  gravitationally  perturb  one  another  ;  it  was  the  perturba- 
tions of  Uranus  which  led  to  the  discovery  of  Neptune,  and,  re- 
examined recently,  to  the  photographing  of  distant  Pluto. 

A  few  double  stars  lent  themselves  to  gravitational  calculations ; 
and  thence,  by  devious  ways,  the  masses  of  almost  all  stars  are  now 
very  shrewdly  estimated. 

§  49.  If  unit  mass  were  taken  to  Mars,  or  to  the  Moon,  it  would 
be  attracted  to  those  centres  with  forces  very  different  in  value  from 
the  attraction  to  the  Earth,  on  account  of  the  lesser  mass  of  these 
spheres,  i.e.  its  weight  would  be  less. 

Your  own  weight  would  be  only  a  third  as  much  on  Mars,  or  one- 
sixth  on  the  Moon,  which  suggests  athletic  championship  meetings. 
Unfortunately,  it  works  both  ways  ;  you  would  get  no  '  second 
wind,'  for  Mars  has  been  able  to  hold  down  only  a  small  fraction  as 
much  atmosphere  as  the  Earth  has,  while  the  little  Moon  has  lost 
every  trace. 

§  50.  Gravity  surveys.  Mining  prospectors  now  quite  commonly 
employ  an  extremely  sensitive  '  torsion  balance,'  which  is  able  to 
detect  the  presence  of  any  large  mass  of  denser  material,  such  as 
metallic  ore,  hidden  in  the  earth,  to  right  or  left  of  the  apparatus, 
by  means  of  the  differential  gravitational  attraction  it  exerts  on  two 
masses  hanging  at  different  levels  from  the  beam  of  the  machine. 
It  is  set  up  at  selected  spots  all  over  the  site  to  be  surveyed,  and  each 
observation  takes  about  half  an  hour. 

Electrical  attraction  and  gravitation.  Wildly  swinging  pith  balls 
are  commonplace,  while  but  few  will  have  seen  the  gravitational 
attraction  of  masses  actually  demonstrated.  Why,  then,  does  one 
ignore  electrification  as  a  possible  astronomical  tie  ? 

Electrical  attraction  acts  on  the  surface  only,  gravitation  acts  on 
every  'particle,  however  deeply  buried.     The  surface  of  a  pith  ball  is 


§  51]  MOMENTUISI   AND  FORCE  31 

dozens  of  times  its  mass,  but  the  mass  of  a  core  boring  through  the 
earth  is  7  x  10®  the  area  of  its  end,  off  which  rubbed  sealing-wax 
might  lift  a  little  dust. 

§  51 .  Relativity.  As  hinted  in  §  33,  Emstein  has  '  done  away  with 
force,'  and,  according  to  his  purely  mathematical  theory,  the 
universe  hangs  together  '  for  'tis  its  nature  to,'  because  matter 
modifies  the  shape  of  '  space-time.'  The  full  analysis  leads  to  results 
which  differ  a  little  from  the  Newtonian  theory  of  mechanics  and 
gravitation,  but  the  difference  is  inappreciable  except  in  fast-moving 
masses.  Taking  the  fastest  planet,  Mercury,  travelling  at  fifty 
times  the  speed  of  a  naval  shell.  Relativity  has  succeeded  in  clearing 
up  a  discrepancy  of  1  part  in  many  millions  between  observation  and 
gravitational  computation.  The  mathematics  of  relativity  is  no 
joke,  and  astronomers,  most  accurate  of  men,  remain  content  to 
calculate  by  Newton,  and  make  a  minute  correction  for  Einstein. 

The  modest  medical  student,  dealing  with  speeds  seldom  exceeding 
those  of  a  tennis  ball,  can  dispense  with  the  correction. 


EXAM   QUESTIONS,   CHAPTER  III 

That  is  the  promised  long  tough  chapter  :  dealing  as  it  does  with  the  simplest 
things,  and  the  most  abstract  ideas,  there  is  probably  no  tougher  in  the  book. 

Give  it  the  once  over,  and  see  what  you  can  do  at  these  questions,  and  then 
pass  on,  while  it  soaks  in  a  bit.  But  don't  fail  to  come  back  before  long  for 
another  cut  at  it,  for  you  must  soon  get  muddled  if  you  are  shaky  over  the 
fundamentals — think  of  the  Norman  towers  that  have  collapsed  in  England 
or  of  the  baby  learning  to  walk — and  yet  they  are  the  hardest  to  make  sure  of. 

§  39  you  must  have,  §  40  tells  why.  §  42  you  do  in  the  lab.,  §  44  will  remind 
you  of  experiences  of  your  own  ;  recollect  them,  not  mine.  §  45  crops  up  in 
the  next  chapter.  §  47  you  do  in  the  lab.  §§  48-51  you  can  please  yourself 
about. 

The  questions  overlap  abiuidantly.  There  is  no  objection  to  using  formulae 
to  solve  them  provided  you  have  learnt  in  the  first  instance  how  they  were 
obtained  :  if  not,  you  are  sure  to  use  the  wrong  one,  and  should  stick  to  first 
principles  every  time.     No.  25  is  for  lab.  reference  only. 

1.  State  the  Newtonian  laws  of  motion  and  give  simple  instances  of  their 
application. 

A  gun  fired  a  shell  horizontally  and  recoiled;  the  shell  presently  burst. 
Trace  the  various  changes  in  momentum  of  gun,  earth,  and  shell  fragments, 
from  the  instant  of  firing  imtil  all  had  come  to  rest. 

2.  State  Newton's  third  law  of  motion.  A  man  sitting  in  a  loop  on  the 
end  of  a  rope  running  over  a  pulley,  pulls  himself  up  by  hauling  on  the  other 
end;  and  a  tug-of-war  team  pulls  its  opponents  over  the  line  without  pulling 
harder  on  the  rope  than  they  do.     Show  how  the  law  applies. 

3.  Explain  the  exact  significance  of  the  terms  acceleration  and  force. 

The  muzzle  velocity  of  a  projectile  of  mass  100  lb.  fired  from  a  gun  is  2200  ft. 
per  second.  If  this  is  imparted  in  0006  sec,  calculate  the  average  acceleration 
and  force  acting. 

4.  What  force  is  required  to  give  sua.  electric  train  of  150  tons  an  acceleration 
of  2-5ft./sec.2? 


32  MECHANICS 

[Force  =  gain  in  momentum  per  second  =  150  X  2-6  '  ton  '  units,  or  re- 
ducing to  English  gravitational  measure  by  dividing  hy  g  =  32'2, 

Force  =  150  X  2-5  -^  32-2  =  11-65  tons  weight. 

Or  in  dynes,  assimiing  1  ton  =  1,000,000  gm.. 

Force  =  150  X  10«  X  (2-5  X  30-5)  =  1-14  X  lO^"  dynes.] 

5.  What  force  is  required  to  stop  in  3  sec.  a  2 -ton  motor-car  travelling  at 
15  m.p.h.  ?    [=  loss  of  mv  per  second,] 

How  far  does  it  travel  with  brakes  on  ?     [Average  speed  X  3  sec] 

6.  If  the  coefficient  of  friction  is  ^,  and  half  the  weight  of  a  car  is  on  the 
hind  wheels,  calculate  the  maximimi  speed  attainable  in  100  m.  from  rest, 
on  the  level.  Also  the  minimum  time  for  stopping  (a)  on  the  level,  (6)  on  a 
downgrade  of  1  in  15. 

You  cannot  count  on  greater  friction  than  this  on  any  road. 

7.  What  units  of  force  are  in  common  use,  and  how  are  they  defined  ? 
Calculate  the  force  which  would  be  required  to  bring  to  rest  a  motor-car 

weighing  800  km.  and  travelling  at  a  speed  of  50  km.  per  hour,  in  a  distance 
of  10  m. 

8.  Find  the  force  on  a  wall  when  a  hose  delivers  100  gal.  of  water  per  minute 
perpendicularly  on  it  at  50  ft. /sec. 

[Force  =  mv  destroyed  per  second 

100  X  10  ^  _„  „     -^        .,  50,000  „„  „        . 

=         60         X  50  Ib.-ft.  units  =  60  ^  3^.^  =  26  lb.  wt. 

or    =  100  X  10  X  454  X  50  X  30-5  -^  60  =  1-15  x  10^  dynes. 

Splashing  back  would  increase  this,  as  the  wall  is  imparting  backward 
momentum.  By  the  third  law  it  is  also  the  force  with  which  the  fireman  must 
hold  up  the  hose.] 

9.  Find  the  force  exerted  on  a  water-wheel  struck  by  500  kg.  of  water  per 
second  travelling  at  4  m./sec,  wheel  moving  at  half  this  speed,  and  water 
dropping  off  dead. 

10.  A  stream  of  water  from  a  fire  hose  delivered  at  the  rate  of  5  lb.  per  second 
strikes  a  wall  perpendicularly  at  speed  60  ft.  per  second.  What  is  the  momen- 
tum of  the  water  arriving  per  second  ?  Assuming  it  not  to  rebound,  what 
force  acts  on  the  wall  ? 

11.  A  hose  which,  when  vertical,  throws  water  64  ft.  high,  is  turned  hori- 
zontally and  discharges  200  gal.  per  minute  against  an  earthen  bank,  where 
it  falls  '  dead ' ;  what  is  its  force  ? 

12.  Show  how  Newton's  Law  connecting  momentum  and  force  enables 
us  to  define  a  unit  of  force. 

A  cage  loaded  with  coal,  weighing  in  all  2  tons,  is  raised  vertically  by  a 
winding  engine.  It  reaches  a  height  of  360  ft.  in  12  sec.  Find  the  uniform 
pull  of  the  engine  during  this  acceleration,  and  its  pull  when  the  speed  becomes 
constant. 

13.  A  20-gm.  bullet  moving  at  700  m./sec.  embeds  itself  in  a  suspended 
100-kg.  log  ;   find  joint  speed. 

[Total  momentum  unchanged  /.  20  X  (700  X  100)  =  100,020  X  x] 

.*.  a;  =  14  cm. /sec. 

Find  also  the  force  exerted  to  give  the  bullet  its  speed  in  0-002  sec,  and 
the  average  force  it  exerts  in  penetrating  the  wood  17-5  cm.  deep. 
Starting  force  =  20  x  (700  X  100)/0-002  dynes. 

[Average  speed  during  penetration  =  (70,000  -f  0)/2  =  35,000  cm. /sec 
.*.  loses,  1,400,000  units  of  momentmn  in  17-5/35,000  sec. 
.'.  loses  at  rate  of  2-8  x  10®  per  second  =  dynes  force.] 


MOMENTUM   AND   FORCE  33 

14.  A  25-gm.  bullet  moving  at  300  m./sec.  stops  after  penetrating  3  cm. 
of  bone.     Calculate  the  average  force  it  exerted. 

15.  A  50-gm.  golf  ball  is  struck  by  a  200-gm.  club  and  flies  off  at  60  m./sec, 
which  in  simple  theory  is  twice  as  fast  as  the  club  follows.  Calculate  mininium 
striking  speed  of  club. 

16.  An  elastic  pellet  of  1  gm,  bounces  at  1000  cm. /sec.  between  plates 

2  cm.  apart.     Find  force  on  the  plates. 

Strikes  each  plate  1000/(2  x  2)  =  250  times  per  second. 

At  each  impact  V  changes  from  1000  up  to  1000  down  =  2000 

.*.  Momentum  given  up  per  second  =  250  X  1  X  2000  dynes  =  0-5kg.  approx. 

17.  Upon  what  does  the  amount  of  Friction  depend,  and  not  depend? 
How  do  you  define  and  measure  a  coefficient  ? 

18.  What  is  the  relation  between  force  and  momentum?  A  1-5-kg.  mass 
lies  on  the  table,  with  coefficient  of  friction  0-3 ;  attached  to  it  and  running 
over  a  pulley  on  the  edge  of  the  table  is  a  string  on  which  hangs  a  half  kilogram. 
How  far  will  the  system  move  in  5  sec.  when  let  go  ?  • 

19.  Define  the  term  coefficient  of  sliding  friction. 

If  the  resistance  to  motion  of  a  railway  carriage  on  level  rails  is  1  /200  its 
weight,  with  what  acceleration  would  the  carriage  nm  down  an  incline  of 
1  in  100  and  what  velocity  would  it  have  acquired  in  1  min.  ? 

20.  How  would  you  show  that  the  acceleration  of  a  falling  body  is  inde- 
pendent of  its  mass  and  its  material  ?     Describe  a  method  of  measuring  it. 

21.  Describe  the  most  accurate  method  you  know  for  the  determination 
of  the  acceleration  due  to  gravity.  How  does  this  quantity  vary  over  the 
earth's  surface  ? 

22.  What  do  you  understand  by  the  '  acceleration  of  Gravity  '  or  the  '  force 
of  Gravity  '  ?  In  what  units  do  you  express  it  and  how  do  you  measure  it  ? 
Why  does  it  vary,  and  what  is  the  scientific  importance  of  its  measurement  ? 

23.  A  smoked  plate  fell  in  front  of  a  tuning-fork  making  256  vibrations 
per  second  and  32  complete  waves  were  counted  in  7-7  cm.  from  the  start. 
Calculate  g. 

24.  A  plate  fell  in  guiding  grooves  past  a  fork  making  540  vibs./sec.  and 
90  waves  were  counted  in  5  in.  from  rest.  Calculate  g  and  observ-e  the  per- 
nicious effect  of  friction  in  the  grooves. 

25.  If,  as  usual,  starting-point  on  plate  is  blurred,  how  proceed  ? 

Mark  off  two  successive  sets  of  m  waves.  Fig.  5,  and  measure  their  lengths 
BC,  CD.     The  time  spent  on  each  is  mP,  and  if  t  is  spent  before  reaching  B — 

AB  =  \gt^      AC  =  lg{t  +  mP)^  AD  =  \g{t  +  2mP)2 

Subtracting,   BC  =  gtmV  +  Igm^V^  CD  =  gtrriP  +  %gm^V^ 

:.  CD  -  BC  =  ^m2P«  .-.  gr  =  (CD  -  BC)  -^  m'P« 

26.  The  masses  on  an  Atwood's  machine  each  weighed  228  gm.  When 
one  was  overloaded  with  3  gm.  it  fell  290  crn.  in  9-5  sec.     Calculate  g. 

8  =  ^at^,  290  =  ^a(9-5)2  /.  acceleration  6-42 
a  --=  mgr /total  mass  /.  6-42  =  3sr/459  .*.  g  =  982 

27.  Define  '  velocity  '  and  '  acceleration.'  To  the  ends  of  a  string  passing 
over  a  frictionless  pulley  are  attached  masses  of  88  and  90  gm. ;  find  the 
acceleration  with  which  they  move,  and  the  distance  travelled  in  the  first 

3  sec.  of  the  motion,  g  being  981  cm./sec.^, 

28.  Calculate  the  velocity  of  Atwood  masses  of  500  and  600  gm.  after 
moving  1  m.  from  rest. 


In  the  PRACTICAL  EXAM  the  coefficient  of  friction  is  asked  for,  either 
by  weight  and  pulley,  or  by  inclined  plane. 
C 


CHAPTER   IV 
ENERGY  AND  WORK 


§  61.  In  a  gun,  when  the  charge  is  fired,  the  powder-gases  press 
equally  hard  on  breech  and  base  of  shot — ^the  Third  Law  insists 

upon  that — and,  although 

^     M  their    actual    forces    may 

^    ^/y//////////////////y///f//y^^  vary  from  instant  to   in- 

Fig.  6.  t  the  shot  is  in  the  gun. 

That  is,  Yt  exerted  on  the 
gun  =  Yi  exerted  on  the  shot,  and  each  is  the  momentum  given 
to  the  mass, 

i.e.         Yt  =  M!o=^Yt  =  mV, 

.  where  M  is  the  mass  of  the  gun,  m  of  the  shot, 

V  is  the  recoil  velocity,  and  V  the  muzzle  velocity. 

Evidently  the  shot  will  travel  a  lot  faster  than  the  kicking  gun ; 
the  equation  easily  accounts  for  that  common  observation.  But, 
on  the  face  of  it,  it  does  not  account  for  the  further  common  observa- 
tion, that  the  shot,  when  it  hits,  can  do  a  lot  more  damage  than  the 
gun.  Not  that  a  gun  can't  kick  your  shoulder ;  still,  we  all  prefer 
to  be  at  that  end  of  it,  for  choice  ;  and  if  you  have  the  sense  to  hold 
it  tight  up,  your  shoulder's  mass  adds  in  with  the  gun's,  and  no 
harm  is  done. 

Looking  again,  however,  we  note  that  while  the  gun's  final  velocity 
is  V,  and  has  therefore  averaged  \v  from  rest,  the  shot's  is  V,  averaging 
JV,  and  that,  as  these  average  speeds  have  lasted  for  the  same  time 
t,  the  distances  moved  by  the  recoiling  gun  and  the  emerging  shot 
are  as 

the  same  force  F  having  pushed  through  these  two  distances. 

Is  it,  therefore,  the  circumstance  that  the  Force  has  pushed  the 
shot  a  far  greater  Distance  that  enables  it  to  do  so  much  more 
damage — damage  from  the  point  of  view  of  the  hunted,  useful 
work  perhaps  from  the  hunter's  ?  Or,  turning  to  a  more  prosaic 
avocation,  is  there  any  difference  between  pushing  the  garden-roller 
a  few  hundred  yards  and  just  leaning  on  it  hard  ? 

When  a  force  pushes  forward  it  is  said  to  do  Work,  and  the 
mechanical  Work  done  is  the  product  of  the  Force  and  the  distance  it 
pushes  forward  in  its  oum  line 

W  =  Fs. 
34 


§  63]  ENERGY   AND   WORK  35 

The  force  is,  of  course,  pushing  a  mass ;  force  can  act  only  on 
mass.  Sideways  motion  is  inoperative,  no  work  is  done  by  the 
weight  of  a  ball  rolling  on  the  billiard- table .  The  force  must  advance , 
and  this  is  true  in  more  ways  than  the  mere  mechanical.  Think 
as  hard  as  you  may,  you  do  nothing  unless  you  progress  along  the 
line  of  thought.  Sticking  too  long  at  the  hard  parts  does  not  pay. 
Thought  without  action,  speech,  or  writing,  may  perhaps  gain  you 
Nirvana,  but  neither  money  nor  credit  in  this  world. 

Now,  the  distances  gun  and  shot  move  in  time  t  are,  of  course, 
proportional  to  their  speeds  v  and  V,  so  the  Work  done  on  them  is 
proportional  to  their  speeds.  But  these  are  inversely  as  their 
masses,  consequently  the  shot,  with  a  hundredth  the  mass  of  the 
gun,  gets  100  times  as  much  Work  put  into  it,  and,  armed  with  this, 
can  do  100  times  the  damage. 

§  62.  It  is  a  matter  of  common  observation  that  a  moving  mass 
pushes  back  a  resisting  force  for  some  distance  before  it  can  be 
brought  to  a  standstill.  Hence  it  is  said  to  possess  energy  of  motion 
or  Kinetic  Energy. 

This  can  be  expressed  in  terms  of  its  mass  and  speed.  Let  all 
its  momentum  mv  be  due  to  a  force  F  having  acted  on  it  t  sec. 
Then  F  =  mv  -^  t.  Its  speed,  having  increased  steadily  from 
0  to  V,  has  averaged  \v  for  the  t  sec,  i.e.  it  has  been  pushed  forward 

a  distance  s  =  \vt.     .'.  the  work  done  on  it  F5  =  —  x  ^vt  =  Jmv^. 

t 

And  as  a  matter  of  experiment,  allowing  for  inevitable  friction, 

as  much  work  can  be  obtained  from  it,  as  it  is  stopped.     Hence 

stored  up  in  mass  m  moving  at  speed  v  is  Kinetic  Energy  equal  to 

half  the  product  of  the  mass  into  the  square  of  the  speed,  imvK 

Thus  in  the  gun,  the  shot  of  1 /100th  the  mass,  set  moving  there- 
fore at  100  times  the  speed  of  recoil  of  the  gun,  has  (1/100)  X  100  X 
100  =  100  times  the  kinetic  energy  of  the  gun. 

'Fs  shows  that  it  is  measured  in  dynes  x  centimetres  or  ergs. 

An  erg  of  work  is  done  by  1  dyne  pushing  forward  1  cm. 

It  is  a  small  unit,  roughly  the  work  done  by  a  diminutive  1-mg. 
fly  crawling  1  cm.  up  the  window-pane.  981  ergs  lift  1  gm.  1  cm., 
the  gram-centimetre  of  work. 

Ten  million  ergs  (10^)  is  the  Joule,  a  more  sizable  unit,  used  in 
electrical  measurements,  and  about  three-quarters  (0-737)  of  the 
foot-pound.  The  latter  is  the  work  done  in  lifting  1  lb.  1  ft.,  and 
is  the  gravitational  unit  used  by  all  engineers  here  and  in  the  U.S. 

The  absolute  English  unit  would  be  the  foot-poundal,  and  the 
no.  of  ft.-lb.  =  no.  of  foot-poundals  -^  gravity  (English)  at  the  place. 

§  63.  Impact.  We  must  return  to  this  subject  :  we  saw  in  §  45 
that  the  total  Momentum  remains  unchanged,  but  now  we  shall 
see  that  this  is  by  no  means  the  case  with  the  Energy  :  unless  the 
bodies  are  perfectly  elastic — and,  in  practice,  none  are — there  is 
always  loss  of  energy,   in  crushing,   vibration,   noise,   heat,   etc. 


36  MECHANICS  [§  63 

Take  the  simple  case  of  a  bullet  fired  into  a  100  times  heavier  log 
at  rest.  Conservation  of  Momentum  tells  us  that  100  times  the 
mass  moves  at  1/100  the  speed,  Imv^  is  replaced  by  J  X  (100m)  X 
(v/100)2  =  \mv^  X  100/10,000  =  0-01(Jmi;2),  and  the  remaining 
9Q%  of  the  energy  has  gone  in  mutual  destruction. 

The  loss  of  energy  upon  impact  is  not  calculable  a  priori  ;  one 
may  rain  blows  from  a  light  hammer  upon  a  big  nail,  and  do  nothing 
but  uselessly  batter  its  head ;  two  or  three  blows  from  a  heavier 
hammer  will  overcome  its  resistance  and  drive  it  home,  while  a 
sufficiently  great  force  would  push  it  in  silently  without  any  shock. 

Another  common  case  of  Impact  is  a  car  collision  ;  the  total 
momentum  may  remain  unchanged,  but  the  useless  expenditure 
of  energy  is  only  too  frequently  and  unprofitably  brought  home 
to  the  medical  practitioner. 

Shock  is  always  wasteful :   see  e.g.  §  295. 

§  64.  Every  sudden  collision,  indeed,  reassures  us  of  the  reality 
of  Kinetic  Energy,  yet  we  never  buy  energy  in  that  visible  form. 
But  we  will  pay  to  be  carried  up  a  hill,  to  have  heavy  clock- weights 
wound  up,  for  steam,  for  electric  energy,  for  water  under  hydraulic 
pressure,  for  food,  coal,  or  cartridges.  These,  whether  '  things  ' 
or  not,  we  value  for  the  energy  of  motion  of  ourselves,  of  machinery, 
shot,  etc.,  which  we  can  get  from  them.  For  in  lifted  weights, 
in  steam,  in  combustibles,  etc.,  is  hidden  '  what-may-become- 
energy,'  or,  as  we  call  it,  Potential  Energy. 

It  is  useful  to  regard  this  as  simply  another  form  of  real  energy, 
convertible  into  or  from  Kinetic  Energy.    Take  instances  : — 

The  energy  Imv^  of  motion  of  a  ball  throvm  vertically  up  gradually 
diminishes  to  zero  at  the  top  of  its  path.  Here  the  ball  is  at  rest, 
storing  as  gravitational  potential  energy  all  the  work  (less  air 
friction)  done  in  lifting  it,  ^mv"^.  Lifted  slowly  to  the  same  height  s 
against  the  earth's  pull,  its  weight  mg,  the  work  would  be  mg.s. 
Equating  these,  \mv'^  =  mg.s.,  an  equation  fraught  with  much 
more  meaning  than  the  balder  form  v^  =  2as  established  in  §  19. 

We  say  that  its  total  energy  remains  unchanged  all  the  while ; 
(kinetic  +  potential)  =  constant. 

The  energy  is  all  kinetic  again  by  the  time  the  ball  strikes  the 
ground,  and  is  then  quickly  converted  into  potential  energy  of 
elasticity  as  the  ball  is  squeezed  out  of  shape,  to  be  just  as  quickly 
reconverted  into  kinetic.  The  diminished  rebound  shows  that 
the  ball  has  lost  part  of  this  energy,  but  this  we  can  account  for 
in  air  friction,  and  in  the  heating  of  the  imperfectly  elastic  rubber, 
as  evidenced  in  motor- tyres. 

A  watch  balance-wheel  bends  or  unbends  the  hair-spring,  and  is 
thereby  stopped  at  each  end  of  its  swing.  In  this  instance  of  a 
Conservative  (energy-preserving)  System  there  is  a  '  flow  '  of  energy 
from  one  part  to  another.  If  the  spring  were  unhitched  from  the 
wheel  when  most  '  wound  up,'  it  would  contain  all  the  energy  as 
potential,  and  the  wheel  would  remain  at  rest.     Half  a  swing  later 


§  66]  ENERGY   AND   WORK  87 

the  spring  would  have  remained  slack,  and  the  wheel  have  gone 
on  spinning  with  all  the  energy  kinetic. 

The  Clock  Pendulum,  rising  to  a  standstill,  and  falling  to  maximum 
speed,  is  the  simplest  instance  of  all. 

If  you  coast  down  a  hill  you  may  get  some  distance  up  the  other 
side  without  pedalling ;  but  if  you  cycle  out  against  a  head  wind 
and  the  wind  drops,  where  is  the  potential  energy  you  fondly  hoped 
you  were  accumulating  to  help  you  home  ?  Look  for  it  where  you 
invested  it ;  that  was  in  the  wind,  now  50  miles  away.  The  trustee 
has  bolted  with  the  funds,  in  the  shape  of  an  increased  violence  of 
air  motion  where  you  rushed  through  it,  by  this  time  mere  frictional 
heat.  The  energy  has  gone  to  a  distant  part  of  the  system,  it  is 
not  destroyed,  but  you  cannot  get  it. 

Investments  in  potential  energy  must  be  made  discreetly.  Even 
coal — bottled  sunshine — would  be  useless  without  air  to  burn  it. 

These  are  a  few  instances  leading  up  to  the  enunciation  of  two 
principles  which  we  believe  to  govern  all  processes,  both  physical 
and  vital,  the  Principles  of  the  Conservation  and  of  the  Dissipation 
of  Energy. 

§  65.  The  Principle  of  the  Conservation  of  Energy  states  that 
energy  is  indestructible.  It  may  be  transformed  in  all  ways  into 
any  sort  of  recognizable  kinetic  or  potential  energy — mechanical, 
luminous,  electrical,  chemical,  thermal — may  be  scattered  broad- 
cast or  hidden  in  ways  yet  unknown,  but  cannot  be  altered  in 
total  amount.  Fresh  suppUes  may  be  unexpectedly  discovered 
{e.g.  radioactive  substances),  but  they  are  not  fresh  creations. 
[See,  however,  §  951.] 

The  Principle  of  the  Dissipation  of  Energy  states  that  energy, 
although  indestructible,  becomes,  in  every  cycle  of  changes,  less 
available  for  use.  No  actual  transformation  of  energy  can  be 
exactly  reversed  so  as  to  restore  the  precise  conditions  at  the  start. 
Always  there  is  more  or  less  irrecoverable  loss — friction,  noise, 
electrical  disturbance,  all  ultimately  ending  in  heat  of  no  useful 
intensity.  The  engineer  is  imsparing  of  efforts  to  reduce  this  tax, 
both  in  heat  engines,  when  it  is  inevitably  heavy,  §  294,  and  in 
transmission  mechanisms,  now  become  very  efficient. 

Is  therefore  the  whole  Universe  coming  to  a  tepid  standstill? 
So  far,  it  seems  as  if  it  must  be,  but  we  hate  to  think  it  is.  Hence 
many  conjectures,  and  the  interest  physicists  are  taking  in  the 
intensely  energetic  '  cosmic  rays '  (§  947),  which  reach  us  in  scanty 
number  from  unknown  sources  in  outer  space. 

§  66.  Power.  The  rate  of  doing  work,  i.e.  the  amount  of  energy 
transformed  in  a  unit  of  time,  is  called  the  Power. 

An  engine  working  at  the  rate  of  1  horse-power  (Watt's  liberal 
estimate  of  a  Cornish  mine-horse,  in  warranting  his  pumping 
engmes)  suppUes  33,000  ft.-lb.  per  minute,  which  is  746  X  10'  ergs, 
or  746  joules,  per  second. 


38 


MECHANICS 


[§66 


A  power  of  1  joule  per  second  is  called  1  Watt.  .*.  1  h.p.  =  746  watts. 
The  kilowatt  of  electrical  engineers  is  thus  about  IJ  h.p. 
Work  =  force  X  distance. 
.*.  Power,  which  =  work  -^  time. 

=  force  X  distance/time. 
=  force  X  speed. 

[Stepping  back  again,  Work  =  power  X  time 

Heavy  pulls  at  slow  speeds  therefore  represent  no  more  Power 
than  light  pulls  at  high  speeds ;  the  resounding  puffs  of  a  loco- 
motive leaving  a  station,  or  the  violent  starting  of  an  electric  train, 
mean  great  Force  but  no  unusual  consumption  of  steam  or  '  watts.' 

The  Power  transmitted  by  a  rope,  etc.,  is  the  product  of  its  speed 
and  its  pull. 

Animals  excel  in  '  overload  capacity ' ;  a  man-power  is  about 
Jth  h.p.,  but  he  can  exert  J  h.p.  running  upstairs,  or  even  1  h.p. 
for  a  few  seconds. 


§  67.  Measurement  of  power  and  energy.  The  mechanical 
measurement  of  power  involves  that  of  a  force  and  the  speed  at 
which  it  moves.  The  total  work  done  is  found  by  multiplying  their 
product  by  the  time  of  motion. 

e.g.  the  hauling  up  of  a  3300-lb.  cage  at  1000  ft./min.  requires 

3300  X  1000  ft.-lb./min.  =  100  h.p. 
The  pumping  of  the  same  weight 
of  water  the  same  height  per 
minute  likewise  calls  for  100  h.p.  (in 
this  way  the  earliest  engines,  which 
were  pumps,  were  measured). 

The  Httle  '  Joule  mill '  com- 
monly employed  for  laboratory 
evaluations  of  the  mechanical 
equivalent  of  heat  (§  254)  will 
serve  as  a  miniature  example  of  the 
friction  dynamometers  used  for 
measuring  the  '  brake  h.p.'  of 
engines — ^in  this  case,  of  you.  In 
Fig.  7,  C  is  a  friction-clutch  consist- 
ing of  a  pair  of  diminutive  brass 
'  fiower-pots  ' ;  the  outer  (shown  in 
section,  and  contained  within  an  empty  outer  shell,  from  which  it  is 
thermally  insulated  by  black  hard  rubber)  is  rotated  by  cord  and 
handwheel,  and  its  revolutions  N  are  counted  by  a  100-tooth  wheel  ; 
the  inner  carries  the  large  wooden  wheel,  on  which  winds  a  thin 
line  running  over  a  guide  pulley  and  carrying  a  (200-gm.)  weight  M. 
If  the  clutch  did  not  sMp,  the  weight  would  be  lifted  I  cm.,  the  cir- 
cumference of  the  wooden  disc,  every  revolution,  and  a  total  work 


Fig.  7. 


§  67]  ENERGY  AND   WORK  39 

=  force  X  distance  =  Mgr  dynes  x  NZ  cm.  =  MglN  ergs  would  be 
done.  But  the  clutch  is  oiled,  and  slips ;  the  mill  looks  exactly  as 
if  it  were  winding  up,  and  exactly  the  same  work  is  done,  but  now 
all  of  it  is  wasted  in  frictional  heating  of  the  clutch.  This  is  coun- 
teracted by  supplying  cold  water  ;  in  engine  practice  a  hose  can  be 
kept  running  on  a  very  large  clutch  of  any  convenient  variety. 

Then  your  '  power  '  is  MgW  ergs  -f-  time  in  seconds  of  N  revs.,  or 
MglN  -^  10 'T  joules  per  second  =  watts,  and  dividing  this  by 
746  gives  horse-power  ;  or  by  1000,  kilowatts. 

Or,  direct  into  h.p.,  suppose  400  revs./min.,  I  =  2J  ft.,  M  J  lb. ; 
then  urn  =  250  ft.-lb./min.  =  250/33,000  =  1/132  h.p. 

For  larger  machinery,  and  for  collecting  the  heat  produced,  see 
§253. 

EXAM   QUESTIONS,  CHAPTER   IV 

This  chapter  deals  with  energy  and  agility  and  power,  and  is  of  the  utmost 
importance  physically.  Don't  learn  Fig.  7,  but  use  it  to  study  the  machine 
you  use  in  the  lab.  for  mechanical  equivalent  of  heat. 

1.  What  do  you  understand  by  (1)  energy,  (2)  momentum,  (3)  power? 
Give  suitable  illustrations. 

2.  Define  momentum  and  kinetic  energy  and  show  that  the  latter  involves 
the  square  of  the  velocity. 

Trace  the  changes  of  momentum  and  of  energy  in  the  system  consisting 
of  a  man  and  a  garden  roller  which  he  pushes  from  rest  up  to  a  uniform  speed 
over  the  earth. 

3.  A  curling  stone  weighing  20  lb.  is  projected  horizontally  along  a  sheet  of  ice, 
the  friction  being  constant.    Draw  a  graph  of  its  velocity  plotted  against  time. 

If  it  takes  10  sec.  to  pass  a  mark  100  yd.  from  the  thrower,  and  continues 
to  move  for  a  further  22-5  yd.,  find  from  the  graph  (i)  the  velocity  of  pro- 
jection, (ii)  the  velocity  after  10  sec,  (iii)  the  loss  of  kinetic  energy  after  10  sec. 

4.  Define  work  and  kinetic  energy.  Show  that  when  a  force  acts  on  a 
freely  moving  mass  the  work  done  by  the  force  is  equal  to  the  gain  in  kinetic 
energy  of  the  mass. 

5.  A  man  climbs  a  hiU.  Where  does  the  energy  come  from  and  go  to? 
Why  does  he  get  hot  ? 

6.  Trace  as  far  back  as  you  can  through  its  various  transformations  the 
energy  obtained  from  a  water-wheel. 

7.  Explain  where  the  energy  goes  to  when  you  expend  it  in  (o)  winding 
a  watch,  (6)  lifting  a  box  from  the  floor  on  to  a  shelf,  (c)  riding  a  bicycle  uphill, 
(d)  rowing  a  boat  on  a  still  pond,  (e)  rowing  upstream. 

8.  Distinguish  between  Force,  Power,  and  Energy.  Where  does  the 
energy  go  to  in  (i)  exhausting  the  air  from  a  vessel,  (ii)  '  tacking  '  a  boat  up- 
stream, wind  being  downstream,  (iii)  the  action  of  the  heart  ? 

9.  Define  and  distinguish  kinetic  energy  and  momentum.  A  mass  has 
momentum  500,  and  kinetic  energy  10,000  ergs ;  find  m  and  v. 

10.  A  mass  of  50  gm.  is  dropped  from  a  height  which  causes  the  kinetic 
energy  on  contact  to  be  6,250,000  ergs ;  what  was  the  momentum  ? 

11.  Explain  the  meaning  of  potential  energy  and  kinetic  energy,  and  find 
an  expression  for  the  kinetic  energy  of  a  body  of  mass  m  moving  with  velocity  v. 

A  gun  of  mass  10,000  kg.  fires  a  50-kg.  shot  with  a  muzzle  velocity  of  500  m. 
per  second.  Calculate  the  velocity  with  which  the  gun  begins  to  recoil,  and 
compare  the  initial  kinetic  energies  of  the  gim  and  the  shot. 

12.  What  is  meant  by  (a)  conservation  of  momentum,  (6)  conservation 
of  energy  ? 

A  mass  of  50  gm.,  with  a  velocity  of  110  cm. /sec.,  collides  with  a  mass  of 


40  MECHANICS 

20  gm.  moving  in  the  same  direction  at  65  cm. /sec.  If  they  stick  together, 
find  their  velocity  and  kinetic  energy  after  impact.  Account  for  the  difference 
in  total  kinetic  energy  before  and  after  impact. 

13.  What  is  meant  by  gr  =  980  ?  If  16  kg.  be  lifted,  find  the  work  done 
and  the  speed  of  fall. 

14.  What  are  Potential  Energy  and  Kinetic  Energy  ? 

A  ball  of  mass  100  gm.  is  dropped  from  a  height  of  5  m.  on  to  a  pavement, 
from  which  it  bounces  up  with  only  half  the  velocity.  To  what  height  will  it 
rise  and  what  loss  of  energy  has  it  suffered  ? 

15.  A  mass  of  100  gm.  is  released  at  the  top  of  a  sloping  track,  20  cm. 
vertically  above  the  lowest  point.  It  moves  down  the  track,  and  mounts  a 
plane  inclined  at  30°  to  the  horizontal. 

What  is  the  velocity  at  the  bottom,  how  far  up  the  plane  will  it  travel, 
and  how  much  work  is  done  in  motmting  the  plane  ? 

16.  A  40-kg.  mass  slides  40  m.  down  a  smooth  toboggan  slide  inclined  at 
30°,  it  loses  J  its  energy  in  turning  into  the  rough  horizontal,  and  then  comes 
to  rest  in  25  m. ;   calculate  the  average  coefficient  of  friction. 

17.  What  do  you  mean  by  force,  work,  potential  and  kinetic  energy  ? 
What  work  is  done  taking  1  kg.  up  a  1-m.  slope  at  30°,  if  (a)  it  is  smooth, 

(6)  it  has  a  coefficient  of  friction  0-5  ? 

18.  What  are  dynes  and  ergs,  why  are  all  scientific  measurements  based 
on  them,  and  how  are  their  values  foimd  ? 

19.  Define  gram,  dyne,  erg,  and  joule.  What  is  the  kinetic  energy  of  a 
5-kg.  shot  travelling  at  400  m./sec,  and  what  force  will  stop  it  in  1-2  m.  ? 

20.  A  140-lb.  man  climbs  a  40-ft.  vertical  ladder  in  1  min. ;  what  work 
does  he  do  and  what  power  does  he  exert  ?  Ditto,  if  he  ascends  instead  by 
a  slope  of  1  in  8  in  2  min.  ? 

21.  Two  men  meet  on  parallel,  but  oppositely -moving,  escalators,  and 
stop  to  talk  ;   what  mechanical  work  do  they  do  ? 

22.  Define  erg,  joule,  watt,  horse-power.  In  a  rope  encircling  the  8-ft. 
diam.  flywheel  of  an  engine  there  is  maintained  a  steady  pull  of  36  lb.  wt ; 
what  is  the  h.p.  at  300  r.p.m.  ? 

23.  At  what  horse-power  does  a  1-ton  car  work  when  climbing  in  10  at 
15  m.p.h.,  frictional  resistance  being  5  lb.  per  ton  ? 

[15  m.p.h.  =  22  ft./sec. 
.-.  climbs  1/10  X  22  =  2-2  ft./sec.  vertically. 

.'.  work  done  against  gravity  =  weight  X  lift  =  2240  X  2-2  ft. -lb. /sec. 
Add,  5  lb.  overcome  in  each  foot  of  travel  =  5  x  22  ft. -lb. /sec. 
Total  =  4928  +  110  =  5038  ft.-lb./sec.  =  9-16  h.p.  at  wheels.] 

24.  Define  energy  and  jsower,  stating  units.  If  a  15-cwt.  motor  takes  a 
hill  of  1  in  20  at  30  m.p.h.,  and  the  road  resistance  averages  1  lb,  per  cwt., 
what  is  its  horse-power  ? 

25.  If  a  car  engine  exerts  20  h.p.,  how  long  will  the  24-cwt.  car  take  to 
climb  a  hill  of  1  in  15,  500  ft.  high,  road  resistance  being  1  lb.  per  cwt.  ? 

26.  Define  unit  work  and  unit  power.  A  25-cwt.  car  climbs  1  in  22  at 
30  m.p.h.,  the  engine  exerting  14  h.p. ;  how  much  is  the  frictional  resistance  ? 

27.  Show  that,  with  equal  braking  force,  it  takes  the  same  time  to  check 
a  car  from  40  to  30  m.p.h.  as  to  stop  it  from  10  m.p.h.,  but  that  it  travels  7  tiwes 
as  far f  and  7  times  the  heat  is  generated  in  the  brakes. 

28.  For  fifty  years,  '  Big  Ben  '  was  wound  up  by  hand.  The  '  going- 
barrel  '  took  a  man  20  min.  twice  a  week.  Reckoning  a  man  as  1/8  h.p., 
and  allowing  a  third  of  the  time  for  breathing-spells,  calculate  the  horse- 
power of  the  great  clock. 

29.  The  '  hour-striking  '  took  a  man  5  hr.  twice  a  week.  The  4-cwt. 
hammer  falls  9  in.  vertical  height;    calculate  the  efficiency  of  the  machine. 

[The  quarter-striking  train  takes  about  the  same  power.  The  quarter 
bells  are  of  about  4,  1-75,  1-25,  and  1  ton;   the  hour  bell  13-5  tons.] 


CHAFrER   V 


STATICAL  EQUILIBRIUM  OF  FORCES 


§  71.  According  to  the  Newtonian  first  law,  a  body  unacted 
on  by  force  remains  at  rest,  or  else  moves  uniformly  in  a  straight 
line.  Any  application  of  force  upsets  this  condition.  Now,  we 
know  perfectly  well  that  every  body  on  earth  is  being  affected  by  at 
least  one  force,  the  Gravitational  pull  of  the  Earth,  and  every  moving 
thing  is  also  being  retarded  by  a  force  due  to  Friction.  Clearly, 
to  remain  at  rest,  a  body  must  be  constantly  acted  on  also  by  some 
other  force  which  just  neutralizes  the  pull  of  the  earth ;  and  to 
travel  at  uniform  speed  a  body,  e.g.  a  train,  must  in  addition 
be  constantly  acted  on  by  some  force  just  neutralizing  friction. 

Hence  when  an  actual  body  behaves  as  if  free  from  forces  alto- 
gether, it  is  said  to  be  '  in  equilibrium  '  under  the  action  of  all  the 
forces  actually  exerted  on  it ;  or  all  the  forces  concerned  form 
'  a  system  in  equilibrium.'     Their  study  constitutes  Statics. 

It  has  been  insisted  all  along,  however,  that  force  is  momentum 
supplied  per  second,  and  consequently  the  forces  acting  when  a 
body  is  visibly  changing  its  motion  in  speed  or  direction — a  falling 
stone,  a  stopping  train,  a  piece  of  revolving  wheel — form  just  as 
much  a  system  in  equilibrium  as  when  the  body  is  at  rest  or  moving 
steadily.  Only,  one  of  the  vectors  concerned,  one  of  the  arrows  in 
the  diagram,  happens  to  be  not  a  '  feelable  '  force,  but  its  equivalent, 
a  visible  change  of  momentum,  once  called  the  vis  inertias  of  the 
body,  its  mass  multiplied  by  (—  its  acceleration).  The  diagram  of 
vectors  is  perfectly  unchanged. 

Coming  to  the  simplest  possible  case,  the  Third  Law  assures  us 
that  every  single  force  forms  part  of  a  system  in  equilibrium,  for 
equal  and  opposite  to  it  is  a  reactive  force.  Your  weight  presses  on 
the  ground  and  the  ground  presses  on  your  feet,  the  air  drags  on  the 
train  and  the  train  drags  the  air  forward,  you  press  forward  on  the 
ball  and  the  ball  presses  equally  back  on  your  hand,  telling  you  that 
it  is  absorbing  momentum  for  flight. 

But  this  individual  treatment  of  forces  leads  nowhere ;  they 
must  be  grouped.  In  considering  the  equilibrium  of  a  body  it  is 
convenient  to  separate  all  the  forces  into  two  groups,  viz.  those 
exerted  by  the  body,  and  the  reactive  forced  on  the  body ; 
either  group  must  form  a  system  in  equilibrium  with  itself. 

V^ery  particular  care  is  necessary  to  avoid  mixing  up  members  of 
these  two  groups.  Your  weight,  for  instance,  is  the  pull  of  the  earth 
acting  on  you,  but  the  downward  pressure  of  your  feet  on  the  floor  is 
not  on  you,  what  comes  into  reckoning  now  is  the  reactive  upward 
pressure  of  the  floor.     When  you  jump,  it  is  this  that  lifts  you 

41 


42 


MECHANICS 


[§71 


(though  of  course  you  call  it  mto  being  by  first  of  all  compressing  the 
elastic  floor  harder  than  usual,  and  you  provide  all  the  energy) ; 
failing  the  reaction,  as  in  water,  you  cannot  jump.  This  increased 
reaction  shows  very  plainly  in  jumping  off  a  spring-board  or  a 
weighing-machine.  It  provides  the  force  ma  acting  on  the  body, 
which  is  directly  opposed  by  the  W5  ^n6r/^oB  mass  X  (—acceleration) 
already  referred  to.  It  is  only  during  acceleration  that  the  mass  of  a 
body  comes  into  account. 

§  72.  The  Equilibrium  of  a  particle  may  be  maintained  either  by 
forces  all  in  one  line  or  by  forces  in  different  directions. 

With  forces  in  one  line,  their  algebraic  sum  =  0,  any  one  is  equal 
and  opposite  to  the  algebraic  sum  of  the  others. 

With  forces  in  various  directions,  their  vector  or  geometrical 

sum  =  0,  any  one  is  equal  and  opposite  to  the  resultant  of  the 

others,  to  the  diagonal  of  the  parallelogram  drawn  on  them  as  sides. 

Any  number  of  forces  acting  at  a  point  can  be  reduced,  two  by 

two,  to  three  only ;  then  these  three  forces  acting  on  the  point  are  in 

equilibrium  when  any  one  of  them  is 
equal  and  opposite  to  the  diagonal  of 
the  parallelogram  drawn  on  the  other 
two. 

Thus  in  the  apparatus  of  Fig.  8, 
the  knot  settles  to  rest  when  the 
three  weights  A,  B,  C  are  exerting 
forces  on  it  along  and  proportional 
to  a,  —  6,  and  c,  or  similarly  for 
either  of  the  other  parallelograms. 

§  73.  Equilibrium  of  a  body.    A 

rigid  body  simply  provides  a  sort 
of  framework  to  which  the  forces 
can  be    attached   before   reaching 
Fig.  8.  their   common  point.     As  with    a 

particle,  if  they  are  all  in  one  Une 
their  algebraic  sum  must  vanish.  If  not  in  a  line,  then  when  prolonged 
to  meet  one  another  all  three  must  meet  in  one  point  and  there  obey  the 
foregoing  law.  But  there  is  now  a  third  case,  the  common  point  may 
be  '  at  infinity,'  the  forces  being  parallel  to  one  another.  Evidently 
the  algebraic  sum  must  be  zero,  but  this  is  not  now  a  sufficient 
condition  of  equilibrium,  and  the  Principle  of  Moments  must  be 
introduced  : 

In  Fig.  9  let  OA  =  a  and  OB  =  6  be  forces  with  resultant  OC  =  c. 
The  triangles  OAC  and  OBC  are  equal  in  area ; 

the  area  of  a  triangle  =  J  base  x  perpendicular  height ; 
hence  a  X  CD  =  6  X  CE  =  twice  the  area  of  either  triangle. 

The  product  of  a  force  and  its  perpendicular  distance  from  a  point 
is  called  the  Turning  Moment  of  the  force  about  the  point. 


§74] 


STATICAL   EQUILIBRIUM 


43 


y^ 

V          \ 

^ 

c 

E 

9^ 

^ 

Fig.  9. 


and  acts  at 
them  such 
equal    and 


Hence,  if  two  forces  are  actmg  at  a  point,  their  turning  moments 
about  a  point  in  the  line  of  their  resultant  are  equal  and  in  opposition. 
{Any  point  in  the  resultant,  for 
CD'  :  C'E'  =  CD  :  CE.) 

This  resultant  reversed  keeps 
the  point  O,  and  the  whole 
system,  in  equilibrium,  i.e.  a 
rigid  body  DCE  acted  on  by 
OA,  -  OC  and  OB  would  be 
kept  in  equilibrium,  as  in  Fig.  8. 

As  the   angles   between  the 
forces    diminish,    until   finally 
they    become    parallel,    DCE 
straightens  out,  and  the  equili- 
brating force   becomes  —  (the   sum  of  the  others), 
a    point    in    the     perpendicular    distance     between 
that  their  turning    moments    about    the    point    are 
opposite. 

Thus  fke  condition  for  the  equilibrium  of  a  body  under  the  action 
of  parallel  forces  is  that  their  algebraic  sum  is  zero,  and  that  the  algebraic 
sum  of  their  turning  moments  about  any  point  is  zero. 

We  don't  specify  three  forces  only,  for  it  is  easy  to  show  that  this 
appUes  to  any  number.     If  there  are  only  three,  simply  equate  the 
moments  of  the  outer  forces  about  a  point  in  the 
middle  one. 

Notice  particularly  that  it  is  the  distance  drawn 
from  the  point  perpendicularly  to  the  force  which, 
multiplied  by  the  force,  gives  its  turning  moment 
about  the  point.  For  instance,  in  the  fanciful 
crank  of  Fig.  10,  taken  from  a  knife-grinder's 
machine,  the  turning  effect  of  the  pull  in  the 
connecting-rod  is,  at  the  instant,  pull  X  CL,  and 
never  exceeds  pull  X  CK. 

§  74.  Centre  of  Mass.  Let  a  and  6  be  the  weights  of  masses  of  3  lb. 
and  1  lb.  attached  to  the  ends  of  a  2-ft  light  bar,  Fig.  II.  Their 
resultant  will  act  vertically  through  a  point  J  ft.  from  the  3  lb., 
since  the  moments  about  this  point  are  3  X  J  and  1  x  H  opposite 
ways.  At  this  point  the  bar  must  be  supported,  the  whole  weight  of 
4  lb.  appears  to  act  there  whatever  the  tilt  of  the  bar,  for  3  lb.  x  CE 
still  =  -  1  lb.  X  CD. 

This  point  is  the  Centre  of  Gravity  (e.g.)  or  Centre  of  Mass  of  the 
rigid  body.  At  rest,  or  moving  in  a  straight  line,  the  whole  mass  acts 
as  if  it  were  concentrated  at  this  centre  ;  supported  there  the  body 
rests  indifferently  in  any  position  ;  struck  there  it  moves  straight  off 
without  turning. 

Sometimes  symmetry  points  out  the  e.g.  It  is  at  the  geometrical 
centre  of  uniform  bars,  rectangular  blocks,  rings,  etc.,  and,  as  in  the 
last  case,  is  often  not  situate  in  the  solid  material  at  all. 


44 


MECHANICS 


[§74 


To  calculate  the  position  of  the  mass-centre  of  any  number  of 
masses  in  line,  take  the  sum  of  the  moments  of  their  weights  about 
any  point  in  the  line,  and  equate  this  to  the  moment  of  the  total 

weight  acting  as  at  the  c.  of  m. 
Thus  masses  arranged  on  a 
bar,  1  at  0,  2  at  1  ft.,  3  at 
2  ft.,  and  4  at  3  ft.  have  a 
total  moment  1x0  +  2x1 
+  3x2  +  4x3  =  20  lb.  X 
ft.,  about  the  point  where  the 
1  lb.  is  attached ;  and  this  = 
(total  10  lb.)  X  2  ft.,  .*.  c.  of  m. 
is  2  ft.  along  bar  from  the  1  lb. 
Draw  the  diagram  for  yourself. 
In  practice  the  body  {e.g. 
semicircle  of  Fig.  11,  lower)  can 
be  hung  by  a  thread,  which 
supplies  a  vertical  force  passing 
of  course  through  the  point  of 
support  and  the  e.g.  The  sum 
total  of  the  moments  of  all  the 
particles  in  the  left-hand  half 
about  any  point  in  the  plumb- 
line  =  ditto  of  right-hand  half. 
Then  hanging  from  another 
point,  the  new  plumb-line  cuts  the  first  in  the  e.g. 

§  75.  The  very  way  it  has  been  derived  shows  that  the  Principle 
of  Moments  is  not  confined  to  parallel  forces,  and  it  is  often  con- 
venient and  sufficient  to  use  it  with  forces  at  angles  rather  than  to 
draw  their  parallelogram  diagrams.  Levers  are  treated  in  both  ways 
below. 

Levers.  The  typical  lever  of  theory  is  a  rigid  bar  on  which  act 
three  forces  usually  called  the  '  weight,'  w,  the  '  reaction  of  the 
fulcrum,'  /,  and  the  '  pull,'  p.  More  or  less  disguised  levers  build  up 
the  greater  part  of  machinery,  and  our  own  limbs. 

Crowbar,  Fig.  12,  A.    Drawing  XYZ  perpendicular  to  the  forces 

w  X  YX  =  and  opposes  p  X  XZ 
(and  /  =  and  opposes  w  -{-  p) 
XY  being  short,  w  lifted  may  be  large. 

In  practice  the  forces  are  rarely  parallel,  then  : 
Either,  Fig.  B,  draw  XY,  XZ  perpendicular  to  the  two  forces 

w  X  XY  =  and  opposes  p  x  XZ, 

or,  Fig.  C,  producing  the  forces,  the  fulcrum  reaction  must  meet 
them  both  in  one  point,  hence  its  magnitude  and  direction  by 
the  parallelogram  law.     This  gives  the  fuller  information  that  / 


I 


§75] 


STATICAL   EQUILIBRIUM 


46 


is  not  simply  vertical,  but  can  be  resolved  into  vertical  and  horizontal 
components,  the  latter  of  which  makes  the  fulcrum-block  slip,  unless 
the  ground  is  rough,  or  you  press  your  toe  against  it. 

Bent  Lever,  Hammer  drawing  nails,  Fig.  D.     Draw  XY,  XZ 
perpendicular  to  resistance  of  nail  and  pull  of  hand, 

w  X  XY  =  and  opposes  p  x  XZ. 

Or  the  dotted  parallelogram  gives  the  same  result,  and  the  further 
information  that  the  reaction /is  its  (oblique)  diagonal. 


In  a  second  way  of  using  both  the  straight  and  bent  levers,  the 
fulcrum  is  at  the  end,  and  the  '  weight  '  in  the  middle,  producing 
what  are  sometimes  called  '  levers  of  the  second  order.' 

In  Figs.  E  and  Y,w  X  XY  =  and  opposes  p  X  XZ, 

and  w  =  and  opposes  f  +  p» 

These  two  uses  increase  force;  the  third  way  of  using  levers 
diminishes  force  and  increases  motion,  p  and  w  change  places,  see  Figs. 
G,  H  and  M.     Sometimes  H  is  called  a  '  lever  of  the  third  order.' 


46  MECHANICS  [§  75 

Of  these  types  are  the  levers  which  convert  the  small  movements  of 
strong  muscles  into  the  rapid  movements  of  our  extremities,  see  the 
remaining  figures,  K,  L  and  N. 

M.  The  old  Hanse  merchant's  instrument  of  barter  was  a  thick- 
ended  club  of  hard  wood.  From  a  notch  at  its  little  end  he  slung  the 
bundle  of  skins  brought  in  by  the  hunter,  then,  holding  all  aloft, 
he  slipped  the  suspending  thong  along  until  balance  was  reached  at 
one  of  the  numerous  rings  scored  closer  and  closer  on  the  yard. 
Their  weight  was  then  {fg  -^  the  shorter  arm)  times  the  weight  of 
the  wood,  acting  at  its  centre  of  mass  g. 

There  is  a  favourite  practical  exam  question,  where  you  are  asked  to 
find,  with  the  aid  of  a  50-gm.  weight,  the  mass  of  a  wooden  rod  rather 
like  this.  By  balancing  it  alone  you  find  g,  its  centre  of  mass,  where 
the  whole  pull  of  the  earth  on  it  is  centred,  then,  hanging  the  50-gm. 
weight  on  one  end  (in  place  of  the  pelts),  find  the  new  balance 
position  / ;  then  wt.  of  lever  is  to  50  gm.  inversely  as  their  two 
'  arms  '  from/. 

G.  In  market  centres,  the  Hanseatic  League  set  up  their  Steel- 
yard, the  whole  establishment  taking  its  name  from  the  more  trust- 
worthy weigh-beam  of  Roman  pattern,  still  in  use  in  most  weighing- 
machines  for  heavy  objects,  from  human  infants  to  railway  loco- 
motives, at  the  present  day.  Fig  G  suffices  to  represent  it ;  the  heavy 
load  hangs  on  the  hard  steel  prism-edge  p,  f  is  the  similar  fulcrum 
supporting  all,  at  w  hangs  a  hook  on  which  weights  can  be  placed. 
Suppose  wf  —  14  times /p  (shown  too  wide  in  sketch),  then  1  lb.  at 
w  balances  14  lb.  at  p,  or  J  lb.  a  half-stone.  Instead  of  using  smaller 
weights  than  this,  a  sliding  weight  is  moved  along  the  scale,  of  equal 
parts.  Suppose  this  weighs  10  oz.,  and  normally  remains  in 
position  at  0,  the  yard  being  suitably  counterpoised  to  carry  it  there. 
Now  move  it  out  to  X,  where  OX  =  fp ;  its  moment  about  / 
increases  by  10  oz.  x  fp  ;  i.e.  it  balances  an  additional  10-oz.  load. 
Moved  along  to  XX,  twice  as  far,  its  moment  increases  again  by 
10  fp  oz.,  and  it  now  balances  20-oz.  load  ;  and  so  on  until  when  at 
XII,  12  times  fp,  it  balances  120  oz.  =  7^-lb.  load.  Thus,  cutting 
10  notches  in  each/p  distance  along  the  bar,  we  have  now  a  scale  of 
equal  parts  reading  ounces,  and  with  range  enough  to  bridge  the  gap 
between  successive  |-lb.  increases  at  w. 

In  miniature,  this  reappears  as  the  Rider  Apparatus  on  fine 
balances  in  the  laboratory,  where  it  obviates  the  use  of  weights 
smaller  than  10  mg. 

L.  A  10-stone  man  stands  '  on  tiptoe  '  on  one  foot.  From  '  toe  '  to  ankle- 
joint  is  6  in.,  thence  to  attachment  of  tendon  of  Achilles  2  in.  Find  pressure 
at  joint  and  pull  in  tendon. 

Here  ty  presses  up  at  toe,  /  down  at  joint,  p  up  at  heel.  Taking  moments 
about  the  fixed  point  (on  floor) 

6"  X  /  =  and  opposes  8"  X  p.  :.  p  =  i  X  f 
Also  we  have  p  -\-  w  =  and  opposes  f..'.tu  =  ixf 
Hence  /  =  40  stone,  p  =  30  stone. 
This  is  a  tricky  problem,  because,  standing  flat,  /  is  your  weight  plus  any 


§  77]  STATICAL   EQUILIBRIUM  47 

little  tensions  in  the  front  and  back  muscles  of  the  leg,  and  one  is  apt  to  forget 
that  contraction  of  these  greatly  increases  /.  If/  were  vertical,  and  muscles 
relaxed,  heel  and  toe  would  carry  f  and  J  of  w,  but  before  you  can  safely  rise 
on  tiptoe  the  weight  has  been  transferred  forward,  and  the  calf  muscle  is 
already  pulling  hard.     Stand  up  and  try  it. 

§  76.  <  Virtual  Work.'  There  is  in  connection  with  the  other 
levers  of  our  anatomy  a  great  diiSiculty  in  saying  just  where  and 
in  what  direction  the  muscle  pulls  them.  In  more  complex  mechan- 
isms, too,  the  construction  of  force  diagrams  becomes  tedious.  The 
difficulty  can  be  escaped  by  using  the  so-called  Principle  of  Virtual 
Work.  Problems  relating  to  machines,  gearing,  etc.,  are  all  most 
easily  solved  by  it,  and  its  application  would  have  saved  the  labours- 
in-vain  of  the  many  unhappy  inventors  of  '  Perpetual  Motions,' 
power- for-nothing  contrivances,  whose  invariable  failure  was  part 
of  the  foundation  of  the  Principle  of  the  Conservation  of  Energy. 

Let  the  mechanism  make  a  small  movement,  so  that  one  of  the 
forces  pulls  and  does  work  on  it.  Then  the  machine  gives  out  an 
equal  amount  of  work  at  the  other  end  by  pushing  back  the  force 
there  through  a  distance  obtained  by  any  convenient  means  of 
measurement. 

Then  last  force  X  distance  moved  against  it 
=  first  fcyrce  X  distance  it  pulled 

or  the  forces  are  in  the  inverse  ratio  of  the  distances  they  move 
in  their  own  lines  of  action.  This  is  called  the  Velocity  Ratio  of  the 
mechanism. 

The  ratio  of  most  importance  in  a  primitive  mechanism  for  heaving 
or  pulling  on  great  weights  is  that  of  the  Force  it  enables  you  to 
overcome  to  the  Force  you  have  to  exert,  and  this  is  called  the 
Mechanical  Advantage. 

But  for  friction,  it  would  equal  the  Velocity  Ratio.  Friction 
brings  it  down  to  a  fraction  of  this,  called  the  Efficiency  of  the 
machine,  and  this  is  its  Output  of  Work  ^  Work  put  in. 

§  77.  Of  these  three  methods — ^parallelogram,  moments,  virtual 
work — sometimes  one,  sometimes  another,  happens  to  fit  the 
problem  easiest : 

.  P.  A  light  ladder  stands  on  a  rough  pavement  and  leans  against  a  smooth 
wall.  A  man  climbs  the  ladder.  Prove  that  its  tendency  to  slip  increases 
as  he  ascends. 

The  smooth  wall  can  exert  only  a  reaction  r  perpendicular  to  itself  (having 
no  component  capable  of  resisting  slip).  The  man's  weight  w  presses  on  the 
ladder  vertically;  the  reaction  at  the  foot  must  pass  through  the  common 
point.  It  therefore  slants  more  and  more  as  the  man  ascends,  and  may 
presently  require  a  larger  horizontal  component  (actually  =  r')  than  friction 
on  the  ground  can  supply.  (If  weight  of  ladder  is  taken  into  account  the 
vertical  force  acts  through  e.g.  of  man  and  ladder,  and  moves  slower  than 
he  does.)  .  •  u    u  ir 

Q  is  the  plan  of  a  three-legged  table  on  the  edge  of  which  is  a  weight  half 
that  of  table.  The  e.g.  is  plainly  at  G,  \  radius  from  centre.  To  find  the 
pressure  on  each  leg  : 

Either  by  levers— draw  CG  to  meet  AB  in  E,  then  EG/CE  of  weight  preesee 


48 


MECHANICS 


[§77 


on  C  and  CG/CE  at  E.     The  latter  again  divides  between  A  and  B  in  the 
inverse  ratio  of  the  distance  of  E  from  them. 

Or  by  virtual  work — Hft  each  leg  in  turn,  and  find  out  what  fraction  of  this 
distance  the  centre  of  mass  lifts.  (Simply  scale  perpendicularly  to  line 
joining  fixed  feet.)     Pressure  on  leg  ^  this  fraction  of  whole. 

With  a  four-legged  table  the  problem  cannot  be  solved  :  you  know 
how  chau-s  and  tables  and  pairs  of  steps  rock  until  you  wedge  up  the 
fourth  foot.  It  is  the  yielding  elasticity  of  table,  or  carpet,  or  car 
springs,  that  settles  how  much  weight  is  borne  by  each  leg  or  wheel. 


Stiff  steady  stands  have  to  be  tripods,  in  spite  of  the  drawback  of 
being  easily  knocked  over.  For  in  Q,  if  overloading  on  the  edge, 
or  tilting,  brings  G  to  the  line  AB,  the  least  touch  and  over  goes  the 
tripod,  unless  some  widespreading  accessory  support,  e.g.  a  ring 
ABC,  comes  into  contact  and  saves  disaster."  Notice  your 
Microscope  has  a  tripod  foot. 

A  solves  also  this  problem  :  If  two  men  carry  a  ladder  which  one 
would  carry  alone  at  X,  how  are  they  loaded  ? 

But  if  three  men  in  line  carry  a  ladder,  and  you  would  know  the 
load  on  each,  you  have  a  simple  problem  which  is  completely 
unsolvable. 

R.     A  man  pulls  an  oar  with  a  force  of  40  lb.  wt.,  rowlock  is  2  ft.  away  and 
naiddle  of  blade  5  ft.  beyond  it.     Find  forces  acting  on  oar. 
Let  the  oar  rotate  a  very  little  about  the  rowlock, 
the  '  Virtual '  Work  p  x  ZZ'  =  /  x  XX'. 
By  similar  triangles  ZZ'  =  0-4XX'. 
/.  /  =  0-4p  =  16  lb.  wt.  and  w;  =  /  -f  jo  =  56  lb.  wt. 


§79] 


STATICAL   EQUILIBRIUM 


49 


How  are  the  forces  used  in  the  boat  ? 

Forces  acting  on  boat  are  shown  in  double  line  :  56  lb.  forward  action  of 
oar  on  rowlock  and  40  lb.  backward  push  of  rower  on  stretcher  and  seat, 
leave  16  lb.  propulsive  force  (really  the  forward  reaction  of  water  on  oar). 
The  16  lb.  is  not  immediately  applied  to  the  water  resisting  the  boat,  but  is 
temporarily  partly  used  in  increasing  momenttun  of  boat  and  rower.  During 
the  return  stroke  this  momentum  is  being  dissipated.  The  rower,  who  took 
more  than  his  share  in  swinging  bow-ward,  i.e.  faster  than  the  boat,  now  gives 
it  up  by  moving  sternward. 

S.  What  are  the  forces  acting  on  an  aeroplane  ? 

The  plane  overhauls  per  second  air  at  rest  AB,  and  drives  it  away  down 
in  some  direction  such  as  BC,  giving  it  downward  momentum  by  means  of 
the  force  BD,  practically  perpendicular  to  its  smooth  surface,  with  which  it 
sits  on  it.  Equal  and  opposite  to  BD  is  the  reaction  BE  which  the  air  exerts 
on  the  plane ;  forward  is  the  thrust  p,  the  reaction  of  the  air  on  the  propeller ; 
and  downward  is  w,  the  weight  of  the  plane ;  together  building  the  equilibrium 
parallelogram  shown. 

T.  How  can  a  boat  sail  more  or  less  up-wind  ? 

Wind  coming  up  along  AB  has  to  spill  out  of  the  sail  in  some  such  di '  j«ion 
as  BC,  the  sail  having  given  it  momentum  more  or  less  aft  by  its  rt^\j5lo» 
against  it,  BD,  which  is  fairly  at  right  angles  to  its  surface.  The  equal  t'^ 
opposite  wind-pressure  on  the  sail  is  BE,  which  resolves  into  the  forward 
component  BF  propelling  the  boat  along  her  keel,  and  the  beam  component 
BM  causing  leeway.  If,  as  in  a  tub,  sideways  motion  through  the  water  is 
as  easy  as  forward,  the  whole  drifts  down-wind;  but  if  length,  leeboards, 
keel,  etc.,  make  sideways  motion  more  difficult,  the  beam  force  produces 
only  a  disproportionately  small  velocity  (dotted  vectors)  and  the  boat  makes 
a  course  only  a  little  to  leeward  of  her  nose. 

§  78.  The  so-called  pulleys  of  the 
human  anatomy  are  mere  eyelets 
through  which  runs  a  sinew.  Ordi- 
nary multiple  pulley-blocks  contain 
sets  of  two  or  three  independent 
wheels.  A  rope  is  rove  through  and 
through  each  block  alternately,  in  a 
way  famiUar  to  everybody.  It  is  only 
necessary  to  count  up  how  many  {n) 
portions  of  the  cord  are  pulling  on  the 
movable  block,  each  pulls  with  the 
same  p  (barring  friction),  then  w  =  np. 

§  79.  The  inclined  plane,  Fig.  14,  U. 
On  the  slope  of  a  hill  a  body  is  held  in 
equihbrium  by  its  Weight,  w,  the 
Reaction,  r,  of  the  plane  perpendicular 
to  its  surface,  and  a  Pull,  p,  up  the 
slope  (which  may  often  be  merely 
friction).  Their  parallelogram  of  forces 
has  its  triangular  halves  similar  (being 
entirely  at  right  angles)  to  the  large 
triangle  ABC. 


Fio.  14. 


60  MECHANICS  [§  79 

Hence : 

pull  along  slope  _  vertical  rise 

weight         ~  actual  length  of  slope 

Consideration  of  work  done  against  gravity,  which  is  the  same  whether 
dragged  up  the  slope  or  lifted  vertically,  also  gives  this  result. 

W.  Find  the  force  required  to  drag  a  ^-ton  wagon  up  an  incline  of  1  in 
20,  and  the  horse-power  at  7^  miles  per  hour. 

Force   =  1/20  X  1120  =  56  lb. 

Power  =  speed  X  force  =  11  x  56  =  616  ft.-lb./sec.  =  1-12  h.p. 

If  the  inclined  plane  is  driven  under  the  '  weight '  by  a  force 
parallel  to  its  base,  the  force  triangle  becomes  that  of  Fig.  14  V,  and 

horizontal  force height 

vertical  force    ~~  horizontal  length  of  plane 

This  is  the  action  of  a  wedge,  type  of  all  Nails,  Knives,  Axes,  and 
CM* "els  (lower  diagram). 

^]  rapping  the  slope  round  a  cylinder  gives  the  Screw.  The  pull 
iu  the  end  of  a  long  handle  moves  tangentially  round  in  a  circle  a 
distance  27rr  while  the  screw  advances  its  *  pitch  '  distance  ;  hence 
w  =  {p  X  27rr)  -^  pitch. 

y.  A  tablet-making  press  has  a  ^-in.  pitch  screw  and  two  handles  5  in. 
long,  14  lb.  is  applied  at  right  angles  to  each.  What  is  the  pressure  on  the 
tablet  ? 

«;=14x2x27rX5-^|  =  1760  lb.  wt. 

Z.  What  is  the  thrust  of  a  steamship's  propeller  which  absorbs  5000  h.p., 
has  an  effective  pitch  of  20  ft.,  and  makes  80  r.p.m.  ? 

w  X  20  X  SO  =  ft. -lb.  per  min.  =  5000  X  33,000 
w  =  103,000  lb.  wt. 


EXAM  QUESTIONS,  CHAPTER  V 

If  you  are  of  a  mechanical  bent,  these  little  problems  will  very  likely 
interest  you ;  but  if  you  are  not,  cut  them  out  and  devote  the  time  to  some- 
thing else. 

§§  71-75  and  79  should  certainly  be  studied. 

1.  Describe  one  simple  type  of  lever  giving  a  mechanical  advantage  (a) 
greater  than  1,  (6)  less  than  1.  Discuss  the  support  of  a  weight  on  the  out- 
stretched hand. 

2.  A  25-lb.  window  sash  3  ft.  wide  is  supported  by  two  sash-cords,  to  each 
of  which  is  attached  a  weight  of  10  lb.  If  one  of  the  cords  breaks,  where 
must  the  hand  be  placed  to  raise  the  sash  with  the  least  effort  ? 

3.  Explain  the  principle  by  which  the  resultant  of  two  forces  which  have 
not  the  same  direction  is  determined. 

The  bob  of  a  simple  pendulima  is  deflected  so  that  the  string  makes  an  angle 
of  60°  with  the  vertical.  Determine  the  direction  and  magnitude  of  the 
acceleration  of  the  bob  at  the  moment  when  it  is  released. 


STATICAL   EQUILIBRIUM  51 

4.  What  are  the  conditions  for  the  equilibrium  of  three  forces  acting  in 
a  plane  ? 

A  uniform  rod  AB  of  mass  10  lb.  is  hinged  at  A  and  hangs  down.  A  hori- 
zontal force  F  applied  at  B  deflects  the  rod  through  45°.  Find  the  value  of 
F,  and  the  magnitude  and  direction  of  the  reaction  at  the  hinge. 

5.  An  aeroplane  is  propelled  horizontally;  draw  a  diagram  of  equilibrivun 
between  propelling  force,  reaction  of  air  against  the  tilted  wing  planes,  and 
weight;  and  show  diagrammatically  their  relative  values.  Show  also  that 
if  the  first  and  second  forces  increase  the  aeroplane  will  rise. 

6.  An  aeroplane  wing  is  slanted  upwards  at  10° ;  show  in  a  diagram  the 
forces  maintaining  equilibriiun.  If  the  engine  be  shut  off  at  1600  ft.  and  150 
m.p.h.,  describe  the  downward  path. 

7.  In  what  circumstances  is  a  moving  body  in  equilibriiun  ?  How  would 
you  arrange  ejcperiments  to  illustrate  the  laws  of  equilibrium  in  the  case  of 
a  rigid  rod  subjected  to  a  system  of  parallel  forces  ?     What  are  these  laws  ? 

8.  A  bridge  rests  on  stone  piers  at  the  ends;  explain  how  the  forces  on 
them  vary  as  a  heavy  load  is  drawn  over  the  bridge. 

9.  How  would  you  weigh  a  30-lb.  bicycle,  without  taking  it  to  pieces,  with 
a  spring  balance  weighing  up  to  20  lb.  only  ? 

10.  Define  centre  of  gravity.  A  6-lb.  bar  3  ft.  long  is  supported  1  ft.  < "  ^ 
one  end ;  what  weight  hung  on  that  end  would  balance  it  ?  If  forces  k  U 
applied  at  the  other  end  what  is  their  '  mechanical  advantage  '  ?  ^ 

11.  A  12-lb.  block  of  stone  lies  6  in.  from  the  end  of  a  40-lb.  plank  14  ft. 
long ;  where  would  you  put  the  fulcrum  to  balance  ?  If  the  coefficient  of 
friction  be  J,  show  graphically  how  much  tilt  is  possible  before  the  block 
slips  off. 

12.  A  20-ft.  uniform  plank,  wt.  5  lb. /ft.,  rests  on  a  log,  and  projects  8  ft. 
beyond  it ;   how  far  can  an  80-lb.  boy  walk  along  it  before  it  tips  up  ? 

13.  Define  centre  of  gravity;  how  would  you  find  that  of  a  flat  plate 
experimentally  ? 

14.  A  rod  is  made  up  of  2  in.  of  iron,  3  of  brass  and  4  of  aluminium  joined 
end  on  end;   their  masses  per  inch  are  as  7-6  :  8*4  :  2-6;   find  the  e.g. 

15.  Discuss  the  laws  of  sliding  Friction. 

How  may  the  coefficient  of  friction  be  determined  ? 

If  a  non-uniform  rod  be  rested  horizontally  on  the  two  forefingers,  and  the 
fingers  slowly  brought  together,  the  rod  will  remain  balanced,  and  the  fingers 
will  meet  mider  its  centre  of  gravity.     Why  is  this  ? 

16.  Explain /orce,  moment  of  force,  couple. 

A  imiform  3-m.  plank,  weighing  15  kg.,  is  supported  horizontally,  with  one 
end  in  a  hole  in  a  wall,  by  a  rope  attached  to  it  at  2  m.  from  the  wall.  Deter- 
mine the  tension  in  the  rope  when  the  latter  is  (o)  vertical,  (6)  inclined  at  an 
angle  of  45°  to  the  wall  and  attached  to  a  point  vertically  above  the  hole. 

17.  A  uniform  horizontal  '  forearm,'  of  length  40  cm.  and  weight  1000  gm., 
pivoted  at  one  '  elbow  '  end,  is  held  in  the  horizontal  position  by  the  pull  of 
a  muscle  at  an  inclination  of  45°,  applied  at  a  point  5  cm.  from  the  elbow. 
A  kilogram  rests  on  the  forearm  30  cm.  from  the  elbow ;  find  the  pull  of  the 
muscle  and  the  reactive  force  on  the  elbow -joint. 

18.  A  15-ft.  ladder  resting  on  rough  ground,  friction  0-5,  leans  at  60°  against 
a  smooth  wall;  how  high  can  a  12-stone  man  climb  it  safely  ?  Neglect  the 
weight  of  the  ladder. 

19.  A  uniform  ladder  on  rough  groimd,  coefficient  of  friction  0-5,  leans 
against  a  smooth  wall ;  at  what  angle  will  it  slip  ?  If  you  immediately  put 
yoiu"  foot  on  the  bottom  rung,  slipping  will  stop ;  how  far  up  could  you  climb 
before  it  slipped  again  ? 

20.  A  ladder  leaning  between  wall  and  ground  at  45°  is  on  the  point  of 
slipping;  calculate  the  coefficient  of  friction,  which  is  the  same  for  both 
surfaces. 


'A' 


52  MECHANICS 

21.  Explain  what  is  meant  by  the  composition  and  resolution  of  forces. 
Find  an  expression  of  the  mechanical  advantage  of  an  inclined  plane  (o) 

when  the  effort  is  applied  parallel  to  the  plane,  and  (6)  when  the  effort  is  applied 
horizontally. 

22.  What  is  meant  by  saying  that  a  body  is  in  equilibrium  ? 

A  weight  rests  on  a  smooth  plane,  inclined  at  an  angle  of  30°  to  the  horizontal. 
It  is  kept  in  equilibrium  by  means  of  a  horizontal  force  of  15  kg.  and  a  force 
of  8  kg.  acting  downwards  parallel  to  the  plane.  Find  the  weight  and  the 
reaction  on  the  plane. 

23.  State  the  conditions  of  equilibrium  of  forces  in  a  plane. 

A  boy  drags  a  hand-cart,  of  the  same  weight  as  himself,  up  a  hillside. 
Neglecting  the  rolling  friction  of  the  wheels,  show  in  a  diagram  the  forces 
keeping  (1)  the  cart,  (2)  the  boy,  in  equilibrium. 

24.  A  roller  is  kept  from  running  down  to  the  bottom  of  the  sloping  garden 
path  by  a  boy,  of  half  its  weight,  standing  on  it.  Show  in  a  diagram  where 
he  must  stand.     How  must  he  move  to  make  it  roll  up  the  path  ? 

25.  Show  in  a  diagram  the  three  forces  which  keep  an  ordinary  microscope 
stage-clip  in  place,  and  prove  that  it  holds  fast  in  the  stage  unless  the  latter 

too  thick,  when  it  cannot  hold  at  all.     Find  the  limiting  thickness  if  the 

5'ng  is  2  in.  long  and  the  coefficient  of  friction  between  peg  and  hole  is  0-2. 

1^6.  Define  the  terms  Efficiency,  Mechanical  Advantage,  and  Velocity 
latio. 

Find  the  maximum  load,  in  grammes -weight,  which  can  be  raised  by  a 
machine  of  54%  efficiency  and  velocity  ratio  60,  when  the  applied  force 
is  a  million  dynes. 

Explain  how  this  may  be  done  without  violation  of  the  principle  of  con- 
servation of  energy. 

27.  With  a  machine  of  mechanical  advantage  5  and  velocity  ratio  8,  how 
much  work  must  be  done  to  lift  15  kg.  10  m.  ?     (  X  3) 

28.  The  screw  of  a  jack  has  a  ^-in.  pitch  on  a  cylinder  1-25  in.  diam.,  a 
20-in.  arm,  and  an  efficiency  40%  ;  what  load  will  a  pull  of  56  lb.  lift  ?  (  x  2) 

29.  Calculate  the  force  in  a  parallel  vice  or  press  producible  by  a  ^-in. 
pitch  screw,  with  the  hands  18  in.  apart  on  a  tommy  bar  and  each  exerting 
20  lb.     Will  the  actual  force  be  as  great  ?     (  X  2) 


In  the  PRACTICAL  EXAM  you  may  be  asked  to  find  the  mass  of  an 
irregular  lever,  given  a  50-gm.  wt.;  to  verify  the  parallelogram  law;  to  find 
the  mass  of  a  roller  by  aid  of  an  inclined  plane,  weight,  and  cord,  etc. 


CHAPTER   VI 


MOTION  IN  A  CURVE 


§81.  It  was  an  ancient  doctrine  that  'circular  motion  was 
perfect,'  but  now  we  hold,  with  the  first  Newtonian  law,  that  a 
body  departs  from  a  straight  line  only  because  it  is  given 
momentum,  i.e.  force  acts  on  it,  in  some  other  direction.  Con- 
tinuous supply  of  sideways  momentum  results  in  continuous 
change  of  direction,  Movement  in  a  Curve.  The  greater  the  rate 
of  supply,  the  sharper  the  curve,  but  the  greater  the  original 
momentum  the  less  is  the  disturbing  effect  and  the  flatter  the  curve. 

To  find  a  relation  between  initial  momentum,  transverse  supply 
of  momentum,  and  curvature  of  path,  Fig.  15. 

The  curvature  of  any  curve,  though 
it  gradually  varies,  is  always  that  of 
the  '  circle  of  curvature  '  which  just 
fits  it  very  near  the  spot  under  con- 
sideration. For  an  instant  the  par- 
ticle is  moving  in  a  circle,  though  it 
may  soon  change. 

It  therefore  suffices  to  study  a  cir- 
cular path  only,  which  has  the  con- 
stant curvature  1/r,  the  reciprocal  of 
its  radius.  The  particle  is  always 
moving  at  right  angles  to  the  radius 
joining  it  to  a  fixed  point,  the  centre. 

Take  a  '  particle,'  mass  m  (a  train, 
for    instance),    moving    at    constant 

speed  V  round  a  circular  curve,  radius  r.  Let  it  travel  AB  in  1  sec, 
AB  =  V.  Draw  the  tangent  BD  to  represent  wu  at  B,  resolve  this 
into  component  momenta  BE  perpendicular  to  AC,  and  BF  down- 
wards, parallel  to  AC. 

At  A  the  particle  had  no  downward  movement,  1  sec.  later  at  B 
it  has  downward  momentum  BF,  /.  a  force  has  been  acting  on  it 
=  BF/BD  of  mv.  Then  BFD  and  BGC  are  similar,  BF/BD  =  BG/BC. 

If  AB  is  a  very  small  arc,  BG  becomes  =  arc  AB  =  t;  and 
BF  points  very  nearly  to  C.  It  is  rather  an  outrage  to  make 
•a  statement  such  as  this  to  the  strictly-brought-up  young 
mathematician,  but  look  at  the  lower  part  of  the  diagram,  where 
the  time  has  been  reduced  to  1/7  sec.  ;  and  by  taking  it,  if  you 
like,  a  miUionth,  you  see  the  result  is  not  a  mere  approximation. 

BF 


Fio.  15. 


force  towards  centre 


BD 


mv 


BG  AB  m»» 

^rnv=^^mv=~- 


53 


64 


MECHANICS 


[§81 


That  is,  if  m  at  speed  v  be  acted  on  by  a  force  {i.e.  supplied  with 
momentum  every  second)  mv^/r  at  right  angles  to  its  motion,  and 
always  directed  to  a  fixed  point,  it  will  move  round  it  in  a  circle 
of  radius  r  with  unchanging  speed. 

And  to  compel  a  body  to  move  in  a  circle,  this  force  must  be  con- 
tinuously apphed,  say  by  a  string,  or  by  the  walls  of  a  cup  containing 
rotating  liquid,  by  the  grinding  together  of  rails  and  wheel-flanges, 
or  by  gravitational  or  any  other  pull. 

From  our  youth  up  we  know  '  Centrifugal  Force,'  and  we  all  say 
that  a  body  moving  round  '  exerts  centrifugal  force.'  By  all  means, 
but  recollect  that  a  body  will  not  move  in  a  circle  unless  it  is  forced. 

The  centrifugal  force  is  the  reaction 
of  the  inert  mass  to  this  active  force 
which  constrains  it  to  move  in  the 
curve. 

mv^/r  shows  that  increase  of  r,  as 
by  letting  the  string  slip  through  your 
fingers,  reduces  the  necessary  con- 
straining force.  Letting  go  altogether, 
the  body  moves  off  in  a  straight  line 
(r  infinite)  and  pays  no  more  heed 
whatever  to  the  original  centre.  The 
yam  of  the  farmer  who  crooked  his 
gun  and  shot  round  and  round  the 
stack  is  better  found  than  founded. 

§  82.  If  the  force  is  not  at  right 

angles  to  the  body's  motion  it  can 

be  resolved  into  two,   one  at  right 

angles  and  the  other  in  the  line  of 

motion.     The  former  curves  the  path, 

the  latter  alters  the  body's  speed  in 

Fig.  16.  it.     To  swing  a  weight  faster,  the 

hand  moves  in  a  small  leading  circle 

as  at  A,  Fig.  16  ;  slower,  in  a  lagging  circle  B.     Notice  how  bicycle 

spokes,  which  transmit  driving  or  braking  effort,   are   tangent  to 

just  such  a  small  circle. 

Fig.  16  C,  of  the  earth's  elHptical  orbital  motion,  illustrates  this. 
Notice  how  autumn  (below)  is  the  (accelerated)  '  fall '  in  more 
senses  than  one. 


§  83.  Particular   case    of    circular    motion.    Conical   pendulum. 

The  heavy  bob  of  a  '  conical  pendulum  '  goes  round  in  a  horizontal 
circle  with  speed  v  while  the  string  sweeps  out  a  cone.  Fig.  17, 
elevation  and  plan. 

The  pull  of  the  earth  on  B,  its  weight,  mg,  acts  vertically  down- 
wards, and  centrifugal  force  mv^/BC  horizontally  outwards,  and  the 
string  sets  itself  in  line  with  their  resultant. 


84] 


MOTION   IN   A  CURVE 


55 


Plainly,  by  similar  triangles, 

mg  AC 


or 


wuVBC 
BC2 

27r.BC 

V 


BC 

m.AC 
m.g 


=  2^V 


m.AC 
m.gr  ' 


Now,  27r .  BC  =  length  of  circular  path,  which  divided  by  v  gives 
the  time  of  1  revolution. 


=^-4 


m.CA 
m.g' 


(This  shows  that  if  T  is  diminished  by 
driving  round  faster,  CA  must  diminish,  i.e. 
the  bob  rises  and  opens  out  as  in  that  familiar 
example,  the  steam-engine  governor.) 

Now,  if  the  angle  A  is  very  small,  CA  is 
very  nearly  equal  to  CB  =  Z,  the  length  of 
the  pendulum. 

/.  for  a  small  cirde         T  =  2^-^^  —. 
''  ymg 

§  84.  Simple  pendulum.  Now,  notice,  this 
holds  nearly  true  for  any  sized  circle  provided 
it  is  still  so  small  that  the  vertical  rise  of 
the  bob  is  hardly  perceptible,  i.e.  CA  is  not 
appreciably  less  than  AB.  We  should  have 
to  watch  a  long  time  to  detect  any  difference 
in  the  times  kept  by  a  metre  pendulum 
swinging  in  a  5-cm.  circle  and  in  a  1-cm. 
circle. 

It  should  make  no  difference  therefore  if  the 
bob  changes  from  one  circle  to  another,  i.e. 
changes  its  distance  from  C,  during  the  swing. 
This  means  swinging  in  a  little  ellipse.     Nor  Fio.  17. 

should  it  matter  if  the  smallest  circle  touched 

(the  breadth  of  the  elhpse)  vanishes  altogether,  and  the  bob  travels  to 
and  fro  along  a  short  line. 

A  small,  heavy  bob  swinging  to  and  fro  on  a  fine  thread  (of 
insignificant  mass)  constitutes  a  Simple  Pendulum,  and  the  pre- 
ceding argument  gives  its  time  of  complete  small  swing  there  and 
back  in  seconds, 


Vmass  X  length  _  2      /length 
weight  Vgravi 


gravity 


56  MECHANICS  [§  84 

both  being  in  foot,  or  both  in  cm.,  units.  This  is  its  period  of 
vibration  or  oscillation. 

If  the  swing  widens,  and  the  bob  lifts  appreciably,  this  gets 
further  and  further  from  the  truth  :   the  '  circular  error.' 

Pendulum  swinging  in  whole  arc  of      5°  10°       20°       40° 

Loses  seconds  per  hour  0*43        1*75       7        27 

So,  in  your  '  practical,'  take  care  that  your  pendulum  does  not 
swing  in  too  wide  an  arc  ;  that  shown  in  Fig.  17  is  half  as  wide 
again  as  is  healthy  with  an  examiner  in  the  offing. 

§  85.  It  is  not  customary  to  ask  you  to  reproduce  either  of  these 
mathematical  investigations  in  your  exam,  but  the  pendulum 
relation. and  its  limitations  should  be  known,  as  graphs  are  often 
called  for  in  the  '  practical.' 

Squaring  it  up, 

^2  =  47r2  -      or      g  =  47r2 1  =  39-5 1. 

You  therefore  plot  a  graph  with  co-ordinates  I  and  t^,  and  starting 
from  the  origin  0,0,  which  in  this  instance  is  definitely  on  the  graph, 
rule  a  straight  line  as  fairly  as  you  can  among  all  the  experimental 
points  ;  any  point  on  this  then  serves  to  give  a  mean  value  of  g. 

The  length  of  the  seconds  pendulum  at  sea  level  at  Greenwich 
(where  g  =  32-19  ft./sec.^)  in  vacuo,  is  39-1393  in.,  and  for  30  years  this 
was  legally  regarded  as  the  means  of  recovering  the  standard  yard  if 
this  were  lost ;  but  the  difficulties  of  the  measurement  are  great, 
and  the  Act  was  repealed  in  1855  in  favour  of  multiplied  copies. 

The  isochronism  of  the  pendulum  was  first  described  by  Galileo 
in  1581.  From  the  high  roof  of  the  cathedral  at  Pisa  hang  row 
upon  row  of  lamps,  just  overhead,  which  the  sacristan,  reaching 
up  to  handle  and  light,  naturally  disturbs  and  leaves  swinging. 
Galileo  observed  that  these  twinkling  lights  all  swung  in  time  with 
one  another,  those  most  lately  disturbed  neither  gaining  nor  losing 
on  others  whose  solemn  stately  swinging  had  almost  died  away. 
The  swings  at  most  were  small,  the  pendulums  extremely  long,  and 
the  isochronism  was  perfect.  Verily  the  lamps  hang  there  to  this 
day,  and  you  may  repeat  for  yourself  the  observations  from  which 
this  first-year  medical  student  of  seventeen,  innocent  of  mathematics, 
soon  devised  a  means  of  timing  the  pulse  :  it  was  not  adopted  for 
controlling  clocks  until  fifty  years  later. 

§  86.  Tension  in  a  revolving  hoop.  '  Centrifugal  action  '  causes, 
in  the  rim  of  a  revolving  wheel  or  hoop,  or  in  the  driving-belt 
encircling  it,  a  considerable  tension.  Notice  how  a  boy's  hoop, 
broken  at  the  weld,  *  opens  out '  as  it  runs  faster  downhill,  or 
how  the  belt  driving  a  circular  saw,  taut  enough  when  at  rest, 
bulges  and  hardly  seems  to  touch  the  small  pulley  at  full  speed. 

Considering  a  very  small  piece  (say  1  cm.,  of  mass  m)  of  the 


§  87]  MOTION   m  A   CURVE  67 

circle,  as  in  Fig.  18,  the  force  that  holds  it  to  its  circular  path  and 

prevents  it  flying  on  straight,  is  the  pull  exerted  on  both  ends 

of  it  by  adjoining  portions  of  the  rim  or  rope.     The  two  must  have 

the  resultant  mv^/r  {=  PR)  towards 

the  centre  C.     They  are  tangents  at 

tlie  ends  of  a   1-cm.   arc,   and  are 

t  lierefore  inclined  to  each  other  at  a 

small  angle  =  arc  of  1  cm.  4-  radius 

)■  =  l/r  =  the   angle   PQR   in   the 

]  )arallelogram  of  forces  =  the  small 

•  arc  '  PR  4-  the  '  radius  '  PQ. 

Hence  the  tension  PQ  is  r  times 
mv^/r,  or  the  Tension  in  a  rim  or  a 

rope  travelling  at  speed  v  is  mv^,  or  [its  mass  per  cm.  X  square  of  its 
speed  in  cm. /sec]  dynes. 

Beyond  a  peripheral  speed  of  2  miles  per  minute  a  cast -iron 
rim  is  likely  to  fly  to  pieces  ;  ropes  and  belts  are  never  run  beyond 
1  mile  per  minute. 

This  investigation  will  be  useful  to  us  in  Sound,  §  394. 

§  87.  Apphcations  of  the  forces  attainable  by  centrifugal  action 
abound.  Simplest  of  all,  you  will  learn  to  swing  the  clinical 
thermometer  to  persuade  the  mercury  back  into  the  bulb  ;  having 
of  course  been  careful,  in  your  rush  to  your  patient's  bedside, 
not  to  take  comers  too  sharply,  lest  the  car  skid,  or  even  roll  over, 
outwards  ;  and  to  keep  your  eyes  on  the  road,  and  not  on  an  aero- 
plane banking,  or  looping,  in  the  blue ;  not  even  on  an  autogjnro, 
its  windmill  wings  held  stiff  by  centrifugal  force  only. 

You  may  have  occasion  to  take  turbid  fluid  from  the  patient  and 
spin  it  in  your  Centrifuge,  which  is  the  conical  pendulum  of  Fig.  17 
with  the  bob  replaced  by  a  glass  test-tube,  and  spun  so  fast  that  the 
height  CA  almost  disappears,  the  tubes  looking  like  a  horizontal 
haze.  You  will  be  assisting  the  sedimentation  of  the  deposit  by  a 
force  250  times  gravity,  and  there  are  power-driven  centrifuges 
capable  of  200  times  as  much. 

As  *  centrifugals  '  or  '  hydro -extractors  '  these  machines  are 
widely  used  in  chemical  works,  sugar-mills,  and  laundries  ;  the  wet 
masses  of  crystals  or  clothes  are  put  in  perforated  circular  metal 
baskets  and  spun  free  from  suds  or  mother- liquor.  And  in  the 
milk  separator  of  the  dairy  a  steel  cup  spins  at  7000  revs./min.,  the 
milk  '  takes  the  wall '  and  the  lighter  cream  rises  in  the  middle. 

Interesting  is  the  action  of  a  winding  river  :  constrained  by  the 
hollow  bank  on  the  outside  of  its  curve  to  change  the  direction  of 
its  momentum,  the  river  presses  hard  on  the  bank,  and  persistently 
washes  it  away.  .  The  water  is  piled  higher  here  by  this  centrifugal 
action,  and  now  contains  heavy  grit ;  the  result  is  the  establishment 
of  a  cross-current  below,  from  high  outer  side  to  lower  water  on  the 
inner  side  of  the  curve,  and  the  sand  is  carried  across  and  partly 
dumped  as  a  flat  spit,  which  grows  and  fills  the  hollow  of  the  river 


68  MECHANICS  [§  87 

bend .  Towns  stand  on  the  high  outer  banks,  where  the  stream  has 
scoured  deep  berths  for  shipping  :  only  comparatively  recently 
have  modem  drainage  systems  made  the  flats  across  the  stream 
healthily  habitable. 

There  are  two  tubes  in  your  Centrifuge,  at  opposite  ends  of  a 
diameter,  and  you  are  careful  to  fill  them  equally  with  fluid ;  for 
if  you  have  a  gram  more  in  one  side,  that  means  a  force  of  250 
grams -weight  tugging  opposite  ways  at  the  axle  50  times  a  second, 
and  the  friction,  vibration,  and  noise,  will  be  astonishing.  A  car- 
engine  runs  just  as  fast,  and,  in  common  with  all  fast  machinery, 
must  be  scrupulously  balanced ;  anjrwhere,  in  all  its  complication, 
a  single  ounce  unbalanced — '  statically,'  lever  fashion — becomes 
*  dynamically  '  a  furious  whirling  force  of  14  lb.  weight,  causing  an 
insufferable  vibration  throughout  the  car,  and  wear  and  risk  of 
breakage  in  the  engine. 

Rotation 

§  88.  Let  us  pursue  Rotation  a  little  further.  We  want  only  a 
very  small  bit  of  it  ahead  in  this  book,  and  what  interest  may  be 
found  in  the  chapter  is  general  and  not  examinational ;  but  the 
way  of  a  ship  in  the  sea  and  a  shell  in  the  air,  of  all  the  many  missiles 
of  our  games  of  peace — -physical  experiments  every  one—do  they 
not  appeal  to  us  in  our  island  home  ? 

In  the  Rotation  of  a  body  it  is  evident  that  the  different  portions 
of  the  mass  contribute  very  differently  to  the  total  momentum 
(and  total  energy)  of  the  motion,  for  those  near  the  fixed  axis  of 
rotation  move  much  less  than  do  the  outer  parts.  The  totals 
have  to  be  got  by  summing  together  the  mv  (or  the  ^mv^)  of  all  the 
individual  particles,  by  a  process  called  Integration,  effected  by 
the  devices  of  a  '  calculus  '  of  its  own,  quite  beyond  this  book. 

The  speed  always  quoted  in  Rotation  is  the  angular  speed,  q, 
with  which  any  radius  sticking  out  at  right  angles  to  the  axis  changes 
its  direction  of  pointing  ;  this,  of  course,  is  the  same  throughout  the 
body. 

Then  v  =  qr,  the  linear  velocity  of  a  particle  =  angular  velocity 
of  body  X  distance  r  of  particle  from  axis  of  rotation. 

Putting  V  =  1  and  r  =  1,  the  unit  q  is  that  which  causes  the  end 
of  a  1-cm.  radius  to  move  1  cm.  per  second,  =  1  radian  per  second. 

The  whole  distance  round  once  being  27r  radians,  q  =  27r  X  revolu- 
tions per  second  {e.g.  q  of  the  minute-hand  =  27r  X  1/3600  radians 
per  second). 

The  momentum  mv  of  the  particle  is  therefore  mrq. 

But  in  a  rotating  wheel,  half  is  moving  up  and  the  other  half 
down,  i.e.  the  total  momentum  in  any  direction  is  zero.  That 
merely  says  that  the  wheel  is  not  running  away  ;  but  what  we  are 
interested  in  is,  what  motion  can  the  engine  flywheel  store  and 
transmit  along  its  axial  shaft  to  the  car,  i.e.  we  want  an  expression 
for  the  total  turning  effect  of  all  the  individual  bits  of  momentum 


§89] 


ROTATION 


59 


about  the  axis.  This  is  got  by  multiplying  each  by  its  *  lever  arm  *  r 
from  the  axis,  and  then  summing  up  the  whole  lot,  so  that : 

The  turning  effect  of  the  momentum  stored  in  the  flywheel  about 
the  axis 

=  the  integral  sum  of  mrq  x  r,  i.e.  of  mr^  .  q. 

By  the  time  the  integration  is  done,  all  the  w's  have  lumped 
together  to  form  M,  the  whole  mass  of  the  wheel,  and  all  the  r^'s 
have  elected  a  representative  R^  (where  R  is  called  the  Radius  of 
Clyration)  so  that 

this  integral  sum  =  MR^  x  q  =  lq. 

:MR2,  the  Moment  of  Inertia  of  the  fl3rw^heel  about  its  axis,  being 
usually  written  I. 

R  comes  by  calculation  only,  you  cannot  measure  it  with  a 
foot-rule  ;  don't  mix  it  up  with  the  r's  below. 

By  analogy  with  §  62  we  can  write  at  once 

Energy  of  Rotation  stored  in  the  wheel  =  ^Iq^. 

§  89.  Moments  of  Inertia.  Some  values  of  the  integral  I  are, 
for  bodies  of  mass  M  rotating  about  a  fixed  axis  through  their  centre 
of  mass,  and  having  r  as  extreme  radius  from  it : 


Mr 


fMr 
Fig.  19. 


fMrVi^Md' 

Thin  hoop*  or  hollow  cylinder  about  usual  axis  perp.  to 

cu-cle  Mf2 

Disc*  or  solid  cylinder        „  „  „  Mr^  X  2/4 

Sphere  Mr^  x  2/5 

Thin  rod  about  centre,  rectangle  or  rectangular  block 

about  central  axis  parallel  to  edge  Mr*  X  2/6 

[**  half  as  much  when  rotating  about  a  diameter.] 
,  Lengths  along  the  axis  do  not  come  into  calculation. 


60 


MECHANICS 


[§89 


Rotation  about  an  axis  not  through  the  centre  of  mass  means 
that  the  body  as  a  whole  moves  forward,  turning  as  it  goes.  For 
instance,  a  flung  stick  has  moved  about  your  shoulder  as  centre  ; 
quitting  your  hand  it  analyses  the  motion  into  forward  flight  as  a 
whole  and  rotation  about  its  own  middle.  Apply  the  argument  over 
again,  and  you  see  that  there  will  be  a  new  and  larger  Moment 
of  Inertia  appUcable,  viz.  MR^  +  Mh^,  or  M(R2  +  h^),  where  h 
is  the  distance  from  the  central,  to  the  new,  axis  of  rotation. 

§  90.  An  instance  of  this  is  the  Compound  Pendulum,  which  is 
anytliing  that  is  swinging  and  is  more  complicated  than  a  heavy 
bob  on  a  thread ;  a  child  in  a  swing,  a  pivoted  bar ;  your  arm,  or 
leg,  or  bat ;  a  balance  beam,  a  rolling  ship,  etc.  Let  us  deduce  its 
time  of  swing  from  what  we  have  done  already  for  the  simple 
pendulum  : 

mg  \  mgl 

ml^  is,  plainly,  its  moment  of  inertia  about  its  point  of  support. 


T  for  the  simple  pendulum  =  2:? 


Fig.  20. 


mg  X  Z  is  the  maximum  turning- moment  of  the  controlling  force 
that  gravity  can  exert  on  it ;  when  it  is  stretched  out  horizontally, 
at  right  angles  to  g,  =  weight  mg  X  lever  arm  I. 

So  that  the  formula  becomes 


!-V- 


moment  of  inertia 


max,  possible  turning-moment  of  controlling  force 

And  in  these  compound  pendula  of  Fig.  20  where  •  is  the  centre 
of  mass  (easily  found)  and  x  is  the  centre  of  swinging,  distant  h 
from  it 


=  2^V 


M(R2  -f  h^) 
Mg.h 


§  91.  Spinning  tops.  Spinning  tops,  as  usually  made,  are  top- 
heavy,  which  complicates  matters,  and  the  top  of  Fig.  21,  which 
can  be  made  by  any  mechanic,  is  better.  Its  stem  can  be  screwed 
up  or  down,  and  locked  anywhere  by  the  lock-nut,  so  that  the  top 
can  be  made  to  just  exactly  balance  on  its  toe,  '  in  neutral  equili- 
brium,' neither  falling  down  nor  standing  up.     Spin  it,  by  finger 


§  ^>1] 


ROTATION 


61 


and  thumb,  on  the  slightly  cupped  top  of  a  little  steel  upright; 
i;  goes  on  spinning  in  whatever  direction  you  leave  it,  without 
(  hange.  For  there  is  no  force  available  to  change  it,  it  is  supported 
at  the  point  where  all  the  earth's  pull  on  it  is  centred,  its  centre 
of  mass,  and  the  earth  has  no  further  power  over  it.  Pointed  to  a 
star  and  kept  spinning,  it  should  continue  to  point  to  the  star,  its 
axis  fixed  in  space,  while  the  earth  rolls  round  beneath  it.  No  top 
or  gyroscope  has  actually  been  balanced  perfectly  enough  to  guide  an 
astronomical  telescope,  but  they  show  willing. 

Spin  up  the  top  with  your  right  hand,  so  that  AB  represents 
the  considerable  momentum  of  a  '  particle  '  of  brass  in  the  rim  facing 
\()u.  Now  tap  the  rim  downwards  at  A,  or  pull  its  stem  forward 
towards  you.  Your  force  gives  the  particle  the  small  amount  of 
momentum  AC  downwards. 

your  force  x  its  time  of  action  =  momentum  AC. 

These  two  momenta  combine  of  course,  by  the  parallelogram  law, 
into  the  resultant  momentum  AD,  A  moves  to  D  instead  of  to  C, 


(^^O 


^c 


Fig.  21. 


Fig.  22. 


which  means  that  the  top  leans  over  to  the  left,  in  the  plane  of  the 
paper,  at  right  angles  to  the  direction  out  of  the  plane  of  the  paper 
in  which  you  attempted  to  topple  it. 

To  right  the  top,  don't  tap  up  the  lowest  point  of  the  rim,  but 
tap  it  up  at  A. 

Now  touch  your  finger  against  the  axle,  and  it  runs  round  and 
round,  snuggling  tight  against  you  ;  you  can  lead  it  anywhere, 
but  you  cannot  push  it  over,  as  you  are  really  trying  to  do  all  the 
time. 


Screw  the  axle  down,  so  as  to  make  it  topheavy,  like  an  ordinary 
top ;  spin,  and  tap  over  as  before.  Now,  whichever  side  the  top 
leans  to,  its  own  overhanging  weight  provides  the  downward 
momentum  AC  ;  from  leaning  out  to  the  front  it  will  soon  lean  over 
to  the  left,  from  leaning  to  the  left  it  will  lean  back  behind  the 
paper,  from  leaning  to  the  back  it  will  lean  to  the  right,  tliat  will 
fetch  it  out  to  the  front  again,  and  so  on ;  this  parallelogram  of 
momenta  is  continuously  in  action  at  the  lowest  part  of  the  rim. 


62  MECHANICS  [§  91 

That  is,  the  top  '  precesses '  steadily  round,  its  axis  sweeping  out  a 
cone  of  constant  inclination,  and  in  the  direction  of  spin. 

As  the  top  slows  down,  the  momentum  AB  gets  shorter,  AD 
therefore  slants  more,  i.e.  the  conical  precession  of  the  axis  presently 
widens  out  visibly,  then  rapidly,  and  the  top  at  last  lurches  over. 

The  faster  the  spin,  the  larger  is  AB,  and  therefore  the  less  in- 
clined is  AD  ;  the  top  is  stiff,  and  precession  is  slow  and  proud. 

The  more  top-heavy  it  is  made,  the  larger  is  AC,  and  the  pre- 
cession is  faster  and  more  violent. 

If  instead,  the  axle  is  screwed  up,  so  that  the  top  persistently 
stands  up  when  at  rest,  it  will  precess  against  the  direction  of  spin. 

The  gravitational  attraction  of  the  Sun  on  the  centrifugal 
equatorial  bulge  of  the  Earth  is  constantly  trjring  to  pull  the  latter's 
axis  straight  up,  which  would  do  away  with  the  Seasons.  Instead, 
the  axis  precesses,  and  its  end,  the  Pole,  travels  round  among  the 
stars,  in  a  cone  of  23 J°  angular  radius  on  the  sky,  once  in  26,000 
years.  '  The  Ancients  '  fitted  constellations  to  all  the  sky  they 
could  see,  but  stars  too  near  the  S.  pole  never  rose  into  their  ken ; 
the  centre  of  the  round  patch  they  left  blank  was  the  S.  pole  of 
their  day,  and  both  dates  and  locates  them. 

*  Sleep  like  a  top,'  but  ours  didn't ;  its  point  was  too  dainty,  it  is 
the  stubby  little  peg-top  of  the  pavements  that  sleeps  so  straight 
and  sound.  Fig.  22  shows  the  toe  of  the  leaning  top ;  its  point 
of  support  is  not  in  the  axis,  but  lies  in  the  small  circle  OP,  which 
grinds  round  on  the  rough  floor,  the  frictional  resistance  of  which 
points  back  through  the  paper  at  P,  offering  to  trip  the  top  up,  and 
make  it  fall  forward,  out  of  the  paper.  That  is,  it  provides  mo- 
mentum AC,  just  exactly  as  you  did.  The  resultant  is  AD,  A  travels 
to  D  instead  of  to  B,  the  top  rises.  As  it  rises,  the  grinding  circle 
OP  gets  smaller,  and  ultimately  vanishes  when  AB  is  horizontal ; 
so  therefore  does  the  frictional  disturbing  force  AC,  and  the  top 
sleeps  upright  on  its  point. 

If  AB,  Fig.  23,  is  the  momentum  of  the  top  of  your  bicycle  front 
wheel,  and  you  begin  to  fall  to  the  right,  providing  momentum  AC, 
the  spinning  wheel  turns  along  AD,  runs  under  you,  and  saves  you. 
This  '  gyrostatic  '  action  is  usually  small,  but  increases  enormously 
at  high  speeds ;  running  away  downhill  it  takes  complete  charge^ 
jumps  ruts,  and  kicks  away  stones,  with  stiff  stability  ;  hands 
and  don't  interfere. 


§  92.  The  Gyroscope,  or  g3Tostat,  is  just  a  heavy  flywheel  spinning 
in  a,  suitable  supporting  frame,  so  that  it  can  be  handled  in  a  greater 
variety  of  ways  than  a  Common  Top.  Your  bicycle  front  wheel 
and  steering-head,  for  instance  ;  or  take  out  the  back  wheel,  screw 
in  the  step  for  a  handle,  spin  the  wheel,  and  you  can  carry  it  about 
by  the  finger,  crooked  on  the  end  of  the  step.  Its  overhang  makes 
it  precess;    if  you  turn  round  at  the  speed  it  suggests,  the  axle 


1 


§93] 


ROTATION 


remains  horizontal ;  if  you  grip  it  and  resist  precession,  it  slowly 
sinks  until  it  hangs  vertical ;  if  you  attempt  to  accelerate  precession, 
it  rises  and  stands  sleeping  on  your  hand,     try  this. 

As  a  torpedo  is  fired,  a  trigger  releases  a  wound-up  spring  which 
spins  up  a  1-lb.  brass  gyroscope  wheel,  standing  upright.  This  takes 
charge  by  the  time  the  horizontal  torpedo  has  found  its  prescribed 
depth ;  if  it  now  pitches  up  or  down,  the  wheel  swings  forcibly  at 
right  angles,  to  port  or  starboard,  actuating  a  slide-valve  which 
admits  pressure-oil  to  the  control- cylinder  of  the  depth-rudders, 
and  these  forthwith  correct  the  tilt. 

Similar  gyrostats  are  fitted  in  some  great  liners ;  when  the  ship 
rolls,  the  wheel  swings,  now  fore  and  aft,  and  starts  a  motor  which 
exaggerates  the  roll  to  a  50 -ton  gyroscope  wheel ;  this  precesses 
with  extreme  reluctance,  keeping  the  ship  to  a  decimal  of  the  wallow 
she  intended. 


Fia.  23. 


Fig.  24. 


Gyroscopes  are  used  for  many  purposes  in  aeroplanes ;  but, 
j  of  all  others,  the  Ship's  Gyro-compass  merits  mention.    Under  a 

compass-card  lies  a  1-lb.  gyroscope,  axis  horizontal,  electrically 
!  spun  at  7000  revs,  per  minute ;  from  the  ends  of  the  axle  hangs  a 
I  J-lb.  weight.  Suppose  the  axis  roughly  E.  and  W. ;  at  A,  Fig.  24, 
I  the  pull  of  the  earth  on  the  |  lb.  is  evenly  distributed  on  both  ends 
I  of  the  axle  :  the  Earth  rolls  on,  but  the  balanced  flywheel  tries  to 
I  maintain  an  unchanged  direction  in  Space.     That  brings  the  gravity 

pull  increasingly  on  to  the  eastern  end  ;  the  wheel  of  course  refuses 

to  tilt  down  in  the  E.W.  plane,  but  precesses  horizontally  out  of  it, 
'  towards  N.S.,  and  after  a  couple  of  hours'  exploring,  finds  peace 

from  the  terrestrial  roll  only  in  the  N.S.  direction ;  and  therefore 
I  reposes  in  it,  as  at  W.  If  the  ship  steams  N.,  round  the  curve  of 
!  the  earth,  the  compass  yaws  off  a  trifle  E.  and  W.,  but  setting  a 
I  knob  to  the  known  speed  displaces  the  '  lubber's  line  'the  right 
j  amount  to  correct  for  this  ;  there  are  none  of  the  uncertainties  that 
I  afflict  the  magnetic  compass.     The  machine  is  sturdy  and  can  work 

a  relay,  and  '  Metal  Mike  '  steers  the  ship. 

f    §  93.  The  diagram  for  your  front  wheel  serves  equally  well  for 
Bowls.     Here  AC  is  the  trifling  weight  of   the  bias  overhanging 


64  MECHANICS  [§  93 

one  side  of  the  tread  ;  so  long  as  AB,  the  forward  momentum,  is 
considerable,  the  bowl  runs  appreciably  straight,  but  when  the  speed 
dies  down  AB  shortens,  and  the  angle  BD  widens  out,  and  the  wood 
curls  in  to  the  jack.  Many  a  bowler  is  convinced  that  he  can  in- 
fluence the  run  of  the  bowl,  in  some  telepathic  way  or  other,  after  it 
has  left  his  hand.  If  you  find  this  notion  in  the  peppery  head  of 
some  elderly  and  well-to-do  patient,  it  may  be  discreet  to  humour 
it,  but  don't  entertain  it  for  a  moment  yourself,  for  it  is  nonsense. 

Just  in  the  same  way,  a  High  Diver  who  has  given  himself  rotary 
momentum  as  he  kicks  off  from  the  board,  has  no  further  power  to 
alter  its  amount  before  he  hits  the  water.  What  he  can  do,  however, 
is  to  alter  his  moment  of  inertia,  reducing  it  by  bunching  up,  or 
increasing  it  by  sprawling  out  his  limbs  at  length  ;  then  : 

Angular  momentum  =  moment  of  inertia  I  x  speed  of  rotation  q, 

so  by  doing  this  he  increases  or  diminishes  the  rate  q  at  which  he 
somersaults  over,  and  strikes  the  water  how  he  chooses. 

Nor  has  a  Cat  any  power  to  turn  over  bodily  when  you  drop  it 
carefully  upside-down  on  to  a  heap  of  straw.  But  while  its  tota 
angular  momentum  remains  zero,  what  it  can  do  is  to  bunch  ui 
one  pair  of  limbs  and  stretch  out  the  others,  and  then  twist  iti 
amazingly  flexible  back,  rotating  its  ends  opposite  ways,  but  througl 
quite  different  angles,  on  account  of  the  great  difference  of  thei 
moments  of  inertia ;  then  out  reaches  one  long  leg  and  touchei 
ground,  and  the  rest  is  easy.  At  least,  that  is  a  suggestion ;  th« 
physiologist  will  tell  you  three  things  that  happen  inside  the  cal 
as  it  decides  to  fall  right  way  up,  but  the  real  point  is,  what  happeni 
outside  ;   and  my  trials  with  kittens  leave  me  a  little  dubious. 

§  94.  A  solid  rolling  ball  has  energy  of  forward  motion  |Mt;* 
and  ^q^  because  it  is  turning.  I  =  fMr^  and  v  =  angular  speed  X 
radius  =  qr,  so  that  its  energy  of  rotation  is  |  X  fMr^  x  v^/r^ 
iMv^.  These  are  the  normal  supplies  of  energy  in  a  billiard  ball 
but  it  is  easy,  by  cueing  the  ball  out  of  centre,  to  give  it  extrt 
spin  so  that  its  rotary  energy  is  much  more  than  2/7  of  the  whole 
That  enables  it  to  go  on  and  do  a  good  deal  (most  simply,  '  follow 
through,'  when  cued  on  top)  after  its  visible  forward  energy  hi 
been  reduced  in  collision  :  the  best  idea  of  what  it  is  likely  to  do  i^ 
obtained  by  estimating  which  way  it  is  '  scraping  its  feet '  on  the  clotl 

It  is  remarkable  that  in  a  perfectly  uniform  sphere,  while  th| 
axis  of  spin  remains  fixed  in  space,  it  is  not  fixed  in  the  ball.  Thul 
if  the  earth  were  such  a  sphere,  the  pole  star  would  keep  true, 
but  the  polar  regions  would  wander  about  everywhere.  On  th< 
other  hand,  what  about  a  billiard  ball  the  shape  of  the  earth  ? 

§  95.  Quoits,  deck-quoits  and  deck-tennis,  and  a  whirring  stone 
ducks-and-drakes,   depend  on  spin  about  the  axis  of  maximui 
inertia  for  their  constancy  in  flight. 

A  Shell,  as  you  know,  is  spun  by  the  rifling,  and  should  preserve 


§96]  ROTATION  66 

its  axial  direction  in  space  unchanged.  It  would  in  vacuo,  but  in 
the  atmosphere  it  doesn't,  it  keeps  its  nose  along  its  trajectory, 
turning  over  quite  a  considerable  angle  during  its  flight.  As  it 
spins,  it  suffers  enormous  air-friction,  it  is  exactly  like  a  very  blunt 
peg-top,  and  therefore  sleeps  determinedly  in  the  position  of  least 
resistance  to  spin,  i.e.  sharp  nose  straight  forward. 

There  is  another  air  action  on  a  shell.  All  its  flight  it  is  subject 
to  the  pull  of  the  earth  downwards  {i.e.,  it  is  '  falling ').  That  means 
that  it  presses  heavier  on  the  air  beneath  it,  as  if  it  lay  on  a  cushion  ; 
therefore,  if  spinning  right-handed,  it  rolls,  or  drifts  off,  to  the  right. 

§  96.  A  teed  golf-ball  is  struck  by  the  sloping  face  of  the  driver ; 
the  blow  gives  it  under- spin  at  from  60  to  120  revolutions  per 
second ;  this  has  been  photographed  as  it  flies  off  at  nearly  200  ft. 
per  second.  If  perfectly  smooth  it  would  fly  forward,  compressing 
a '  pad  '  of  air  straight  in  front  of  it ;  but  it  is  deUberately  roughened, 
and  grips  this  '  air-cushion  '  on  which  its  trifling  weight  is  lying, 
paddles  it  round,  and  packs  it  against  the  under-side  of  the  dense 
forward  pad,  Fig.  25.  The  resultant  air-resistance  is  no  longer  dead 
ahead,  but  inclines  a  trifle  upwards,  and  lifts  the  ball  as  much  as 
gravity  pulls  it  down ;  it  flies  straight  on,  and  not  in  a  falling 
parabola  at  all.  Air-resistance  checks  the  forward  speed  before  the 
rotation,  the  lower  '  pack '  asserts  itself  more  in  comparison,  and 
the  ball  has  to  climb  over  it,  and  '  soars  ' ;  then,  spent,  drops  nearly 
dead.  If  you  pull,  or  slice,  holding  the  driver-face  askew  in  plan, 
the  action  takes  place  in  an  inclined  plane,  and  the  ball  flits  aside 
into  the  rough. 

As  the  bowler  delivers  a  cricket  ball  his  hand  continues  to  move  in  a 
circular  arc  while  the  ball  begins  to  fly  off  tangentially.  That,  aided 
by  their  slight  natural  curvature,  keeps  his  two  flngers  pressed  against 
the  ball,  and  as  this  is  not  moving  much  faster  than  the  hand  still 
follows,  it  has  to  escape  by  rolling  for  an  appreciable  fraction  of  a 
second  on  the  under-side  of  the  fingers,  acquiruig  an  under-spin 
dependent  on  its  roughness  and  their  pressure.  The  axis  of  rotation 
is  oblique,  being  the  direction  in  which  a  thick  stick  would  be  gripped 
across  the  hand  when  in  the  delivery  position.  The  flight  therefore 
resembles  that  of  the  driven  golf  ball,  more  or  less  badly  sliced ; 
the  ball  '  swerves.' 

On  contact  with  the  ground  the  obliquely  spiiming  ball  *  clambers 
along  it,'  getting  a  better  and  longer  grip  on  a  soft  wicket  than  a 
hard,  and  therefore  scrambles  off  to  the  side,  converts  rotary 
momentum  into  cross- wise  component  momentum,  '  breaks.' 

The  roughness  of  a  new  ball  is  mainly  round  the  equatorial  seam, 
and  the  directions  in  which  this  meets  the  fingers,  the  resisting  air, 
and  the  ground,  all  influence  its  path  :  altogether  a  cricket- ball  is  a 
weapon  of  the  most  complex  subtlety. 

Plenty  of  spin  about  various  axes  can  be  put  on  a  Unnia  ball 
also.     Footballs  move  too  sluggishly,  and  are  the  prey  of  the  breeze, 

D 


66  MECHANICS 

though  the  steady  toppling  of  a  rugger  ball  does  sometimes  seem  to 
give  it  a  dim  slow  directness  of  purpose. 

The  strong  persistent  rotatory  individuality  of  whirlpools,  water- 
spouts, hurricanes  and  all  cyclonic  circulations,  can  be  only  mentioned 
here. 


EXAM   QUESTIONS,  CHAPTER  VI 

Follow  the  proofs  in  this  chapter  and  understand  them  :  you  are  very 
unlikely  to  be  called  upon  to  reproduce  them,  but  their  results  are  referred 
to  again  and  again  later  on. 

This  hook  is  never  going  to  tell  you  '  It  can  he  proved  '  when  you  can  perfectly 
well  prove  it  yourself  with  a  little  showing  how. 

The  latter  part  of  the  chapter  is  entirely  for  your  own  edification,  as  a  modem 
mechanically -minded  sportsman. 

1.  A  closed  railway  carriage  moves  uniformly  along  (o)  a  straight  line, 
(6)  a  circle.  How  can  an  observer  inside  determine  anything  as  to  the  motion, 
in  either  case  ? 

2.  Why  can  a  bucket  of  water  be  whirled  over  and  over  at  arm's  length 
without  spilling  ?     Calculate  the  minimum  speed  at  1  m.  radius. 

3.  Calculate  the  difference  in  g  at  the  poles  and  equator  of  a  sphere  6400 
km.  radius  rotating  once  in  24  hr. 

4.  A  stone  on  a  60-cm.  string  whirling  over  twice  per  second,  is  let  go  when 
the  string  is  inclined  at  45° ;   how  long  and  how  far  will  it  fly  ? 

5.  What  is  the  initial  motion  of  the  fragments  when  a  flywheel  bursts  ? 
An  emery  wheel  1  ft.  diameter  bursts  at  2400  r.p.m. ;  what  maximum  distance 
may  fragments  fly  (they  will  be  those  starting  at  45°  up)  ? 

6.  What  speed  would  cause  a  tension  of  2  kg.  (say,  2  million  dynes)  in  a 
hoop  of  wire  weighing  1-2  gm.  per  metre. 

7.  A  motor-cyclist  rides  round  inside  a  sphere  of  effective  diameter  8  m. 
At  what  minimimi  speed  can  he  safely  ride  (a)  upside  down,  (6)  round  the 
horizontal  equator,  if  coefficient  of  friction  is  0-5  ? 

In  the  PRACTICAL  EXAM  various  questions  are  asked  involving  timing,: 
graphing,  and  calculation  of  g  from,  the  simple  pendulmn. 


CHAPTER   VII 


FLUIDS 


§  101.  Matter  that  can  flow  is  fluid.  This  broad  definition 
includes  not  only  liquids  (to  which  the  name  of  fluids  is  popularly 
confined),  but  also  gases,  streaming  masses  of  sand,  grain,  etc., 
crowds  of  people,  pitch  and  candle-wax  in  summer,  even  glacier 
ice  and  metals  plastically  yielding  to  excessive  stresses. 

Every  particle,  every  '  drop,'  say,  of  a  fluid,  of  course  obeys  the 
mechanical  laws  already  described,  but  its  individual  motion  can 
rarely  be  followed ;  it  is  lost  in  the  crowd.  Fluids  are  therefore 
studied  collectively,  their  special  Laws  are  laws  governing  the 
motion  and  equilibrium  of  multitudes  of  particles  in  close  contact ; 
mob  laws,  if  you  like  to  call  them  so. 

The  sand,  ductile  metals,  etc.,  referred  to  above,  differ  from 
typical  fluids  in  one  most  important  respect.  For  they  act  as 
solids  and  do  not  appreciably  respond  to  stress,  until  it  reaches  a 
certain  limiting  value.  For  instance,  at  this  '  yield  point,'  metals 
change  from  springy  solids  and  behave  like  very  viscous  fluids, 
as  by  great  force  they  are  drawn  out  into  wire.  Again,  sand  freely 
trickles  down,  but  stands  solid  at  a  moderate  slope.  The  reason  is 
evident :  solid  friction  among  the  particles — ^whether  held  in  contact 
bj^  molecular  cohesion,  or  merely  by  their  own  weight,  quite  pre- 
vents slipping  under  forces  smaller  than  a  definite  limit,  §  41. 

But  take  well-rounded  sand  grains  lubricated  with  plenty  of  water, 
and  this  '  quicksand  '  notoriously  gives  way  even  to  light  weights. 

From  this  chapter  we  cut  all  these  out,  presenting  them  to  the  civil 
engineers,  and  deal  only  with  the  Typical  Fluid  which  yields 
continuously,  though  it  may  he  slowly,  to  any  force,  however  small. 

An  excessively  viscous  liquid,  like  pitch,  yields  only  year  by 
year  to  its  own  weight — interesting,  but  we  won't  wait  for  it ; 
feathers  and  fluff  give  evidence  of  frictional  drag  as  they  flutter  down 
through  the  air.  The  theoretically  perfect  fluid  would  be  perfectly 
mobile,  its  particles  would  glide  by  one  another  without  friction ; 
it  does  not  exist.  Fortunately,  moderate  fluid  friction  makes  no 
difference  to  the  study  of  fluids  at  rest,  or  in  comparatively  slow 
motion,  for  in  this  friction  there  can  be  no  prelimiruiry  sticking 
stage  whatever  (see  Chapter  XXII). 

§  102.  Noticing  that  Pressure  is  defined  as  the  force  exerted  on  each 
unit  of  area — e.g.  lb.  per  sq.  in.  ;  dynes  per  sq.  cm. — there  flow  from 
the  foregoing  these  Laws  of  Fluids  : 

I.  The  pressure  of  a  fluid  at  rest  on  any  surface  bounding  it  is 
perpendicular  to  that  surface. 

67 


68  MECHANICS  [§  102 

For  whatever  it  may  be,  the  reaction  of  the  surface  is  equal  and 
opposite  to  it ;  resolving  this  into  two  components  perpendicular  and 
parallel  to  the  surface,  the  latter  component  would  urge  the  superficial 
layers  sideways,  and  as  they  are  quite  incapable  of  making  any  stand 
against  it,  they  would  move  until  this  component  had  been  reduced 
to  zero. 

This  principle  is  familiar  to  everyone  in  the  resistance  felt  when 
a  broad  surface  is  slowly  moved  flatwise  against  wind  or  water, 
but  not  when  edge- wise. 

The  free  surface  of  a  liquid  must  consequently  set  itself  at  right 
angles  to  the  resultant  force  acting  on  it  at  the  point.  Usually 
this  is  weight,  vertically  downwards,  and  hence  the  surface  is 
horizontal.  But  if  the  liquid  is  in  rotation,  centrifugal  force  comes 
in,  and  the  surface  banks  up  into  a  wave,  or  the  whirlpool  cone  in 
your  teacup,  or  following  your  paddle. 

II.  The  pressure  at  a  point  in  a  fluid  is  the  same  in  all  directions. 
For  consider  a  minute  equilateral  triangular  volume  in  the  fluid, 

a  prismatic  block  so  small  that  its  weight  is  negligible  compared 
with  the  pressures  on  its  faces.  If  this  remains  at  rest,  there 
can  be  no  resultant  force  acting  on  it,  i.e.  the  three  pressures  per- 
pendicular to  its  three  faces  must  be  all  equal  by  the  parallelogram  law, 
all  sides  being  equal.  As  it  can  be  tilted  about  anyhow,  we  infer  that 
the  pressures  are  the  same  in  all  directions  at  the  point — '  point,'  hydro - 
statically  speaking,  meaning  the  small  region  round  about  a  point. 
The  spirting  of  water  with  equal  violence  in  every  direction 
from  holes  in  a  leaky  hose  illustrates  this  principle.  But  by  far 
the  best  experimental  proof  of  it  is  that  a  well-made  Aneroid  Baro- 
meter reads  the  same  however  it  is  turned  over  and  about  in  the 
hand.  In  this  instrument  (§  117)  the  heavy  pressure  of  the  great 
ocean  of  air,  in  the  depths  of  which  we  live,  is  being  balanced 
against  the  elastic  strength  of  a  spring.  This  of  course  is  unaltered 
by  merely  tilting  the  whole  instrument  about,  hence  the  constant 
reading  means  that  the  air  pressure  on  the  lid  of  the  flat  vacuum- 
box  is  the  same  in  all  directions. 

III.  The  pressure  in  a  fluid  at  rest  the  weight  of  which  can  he 
neglected  is  the  same  throughout.     {Principle  of  Pascal.) 

For  if  different  pressures  acted  on  opposite  faces  of  a  cubical 
volume  in  the  fluid,  it  would  begin,  and  continue,  to  move,  until  the 
pressures  were  equalized.  Porous  partitions  may  slow  the  move- 
ment, but  cannot  make  any  ultimate  difference. 

Of  course  this  law  is  approximate  only  :  no  material  fluid  is 
weightless.  Still,  it  takes  a  good  aneroid  to  measure  the  difference 
of  atmospheric  pressures  on  the  chair  and  on  the  table ;  and  the 
engineer  utterly  disregards  any  variations  of  pressure,  due  to  mere 
weight  of  water,  in  a  hydraulic  cylinder  where  the  average  pressure 
is  a  ton  or  more  to  the  square  inch. 

[The  following  §  103  is  the  supplement  of  this,  it  deals  with  the 
pressures  in  a  heavy  fluid  due  to  its  weight  only.] 


§103] 


FLUIDS 


60 


It  is  on  this  transmissibility  of  fluid  pressure  to  all  parts  that 
steam,  compressed  air,  or  hydraulic  power  distributing  systems 
depend.  The  hydraulic  press  affords  a  good  instance  of  its  adapt- 
ability. In  Fig.  26  a  force  exerted  on  the  small  plunger  P  is  trans- 
mitted by  the  water  and  applied  a  hundred-fold  on  the  plunger  or 
*  ram  '  R  of  100  times  greater  circular  area.  Conversely,  of  course, 
P  moves  100  times  as  fast  as  R,  hence 
it  is  necessary  to  fit  valves  and  make 
it  a  reciprocating  force  pump.  In 
its  smallest  form  this  is  a  hydrauUc 
car- jack  ;  in  its  largest,  a  15,000-ton 
forging-press,  more  efficient  than  any 
steam-hammer. 

Your  Brain  floats  in  cerebro- spinal 
fluid  inside  your  skull,  just  as  you 
float  in  water  in  your  bath  ;  a  blow 
on  the  outside  is  distributed  as  a 
transient  general  rise  of  pressure  over 

its  whole  surface ;  even  after  violent  concussion  the  risk  of  local 
lesions  and  adhesions,  and  consequent  insanity,  is  enormously 
reduced. 

The  foetiis  in  utero  is  likewise  safeguarded  from  external  violence 
1  >  V  the  circumambient  fluid  in  which  it  floats. 


§  103.  Pressure  due  to  weight  of  fluid.  The  pressure  due  to 
gravity  at  a  point  in  a  heavy  fluid  at  rest  is  equal  to  the  weight  of  a 
1  sq.  cm.  vertical  column  standing  on  a  sq.  cm.  horizontal  area  drawn 
round  the  point,  Fig.  27  (left).  For  all  the  pressures  exerted  by 
the  surrounding  fluid  on  the  vertical  walls  of  the  centimetre  cubes 
of  which  the  column  may  be  considered  as  built,  are  perpendicular 
to  these  walls  (Law  I),  and  therefore  strictly  horizontal.  They  are 
drawn  for  the  two  sides,  but  omitted  back  and  front  to  save  con- 
fusion. They  keep  the  column  from  falling  to  pieces  sideways, 
but  we  stipulated  in  the  beginning  that  however  hard  they  squeeze, 
it  can  slip  past  them ;  and  they  have  no  vertical  components 
whatever  capable  of  sustaining  any  part  of  its  weight,  which  there- 
fore rests  wholly  on  the  sq.  cm.  horizontal  base,  and  is  borne  by 
(or  causes)  the  pressure  there. 

Weight  of  column  =  no.  of  c.c.  it  contains  X  weight  of  each 

=  depth  of  base  below  surface  X  density  of 
fluid. 
(Density  being  defined  as  the  mass  of  1  c.c.) 
=  Pressure  p  =  hd  grams-weight  per  sq.  cm. 
=  hdg  dynes/cm.2 

If  there  are  several  fluids  on  top  of  one  another,  e.g.  oil,  water, 
chloroform,  etc.. 


p  =  h^di 


^2^2 


^3^3,  etc. 


70 


MECHANICS 


[§103 


i! 


^.^3 


If,  as  in  the  atmosphere,  the  lower  layers  are  much  compressed  by 
the  weight  of  those  above,  so  that  the  fluid  gradually  increases  in 
density  downwards,  Fig.  27  (right),  p  is  reckoned  by  dividing  the 
total  height  into  small  fractions,  assigning  an  average  density  to 

each,  and  summing  the  product  hd 
throughout ;  a  process  of  integra- 
tion, as  in  §  119. 

Again,  unless  a  vacuum  has  been 
created  over  a  liquid  surface,  there 
is  an  air  or  steam  pressure  P 
which  has  to  be  added  to  all  gravity 
pressures  throughout  the  liquid  (Law 
III),  to  get  the  total  pressure  (the 
'  absolute  '  pressure  of  engineers). 


|P  §  104.  The   pressure  will  be  the 

Fig.  27.  same  everywhere  at  the  same  depth 

below  the  level  surface.  For  hori- 
zontal motion  does  not  involve  the  vertical  force  of  gravity  at 
all,  hence  Law  III  holds  throughout  any  horizontal  plane  in  any 
fluid  at  rest.  The  pressure  at  the  lower  level  in  Fig.  28  (i  to  vi) 
is  the  same  for  all  (and  in  (i,  ii,  and  iii)  the  total  forces  on  the 
equal  bases  are  the  same).  Reciprocally,  of  course,  if  a  number  of 
vessels  communicate  at  one  point,  the  liquid  will  '  find  its  own 
level ' — i.e.  same  height  above  the  common  point — in  all,  whatever 


their  size  and  shape,  and  will  there  remain  at  rest.  Thus  U  bends 
in  the  figure  show  the  same  pressure  at  the  same  level  on  both 
sides,  and  the  greater  or  lesser  pressures  passed  through  on  the 
way  round  the  '  bends  '  need  not  be  reckoned  out. 

Curiously,  in  (iii)  we  see  that  the  pressure  on  the  bottom  of  a 
necked  bottle  may  exceed  the  pressure  of  the  whole  bottle  on  a 
scale-pan.  The  explanation  is  that  the  pressure  of  the  liquid, 
everjrwhere  normal  to  the  glass,  has  a  compensating  upward  com- 
ponent round  the  shoulder ;  if  the  bottle  were  cracked  round, 
the  upper  half  would  be  actually  lifted  until  the  liquid  from  the 
shoulder  had  run  out. 


§  106] 


FLUIDS 


71 


§  105.  The  Siphon.  If  the  limbs  of  a  U  tube  are  filled  to  different 
levels  there  is  an  unbalanced  pressure  on  the  liquid  in  the  bend, 
forcing  it  towards  the  low  side  ;  the  liquid  oscillates,  and  comes 
to  rest  stably  at  the  same  level  both  sides. 

But  in  an  inverted  fl  tube  equihbrium  is  unstable.  At  the 
top.  Fig.  29  (left)— 


Pressure  from  left     =  atmospheric 
,,  ,,     right  =  atmospheric 


hod,  a  smaller  total ;  and 


the  resultant  (^g  —  ^i)^  forces  the  fluid  over  towards  the  right. 
If  now  the  fluid  is  continuously  supplied,  as  in  the  second  figure, 
a  steady  outflow  goes  on.  This  is  the  Siphon,  commonly  employed 
for  drawing  off  liquid  from  any  vessel  without  a  hole  in  the  bottom. 


i/^ 


-^ 


Fig.  29. 


It  may  be  compared  to  an  Atwood's  machine,  the  masses  of  fluid 
hihi  being  the  inert  balancing  masses  and  the  dependent  weight  of 
fluid  in  h2  —  hi  supplying  the  driving  force.  Consequently  the 
shorter  the  arch  and  the  longer  the  long  limb  the  faster  the  outflow. 

But  there  is  a  difference  :  the  Atwood  cord  is  under  tension,  and 
a  liquid  cannot  be  relied  upon  to  endure  a  tension.  The  pressure 
at  the  top  is  less  than  atmospheric,  a  soft  rubber  tube  siphon  often 
tells  you  that,  by  squashing  flat  there,  and  if  there  were  no  atmo- 
spheric pressure  there  would  be  a  minus  pressure  at  the  top,  and  the 
liquid  would  break  and  fall  back  either  side. 

A  siphon  cannot  act  in  vacuo,  nor  if  its  arch  is  higher  than  the 
atmospheric  pressure  can  drive  the  liquid,  i.e.  higher  than  the  barometer 
filled  with  that  liquid. 

The  siphon  is  commonly  started  by '  sucking  the  air  out , '  but  siphons 
arranged  as  in  Fig.  29  (iv  and  v)  start  spontaneously  if  the  cistern 
is  filled  above  their  arch.  Beginning  to  overflow  down  the  pipe, 
as  in  (iv),  the  waterfall  entangles  and  carries  down  the  air,  and 
soon  the  siphon  is  running  full  bore  until  it  has  nearly  emptied 
the  tank,  when  air  gets  in  under  the  edge  and  stops  it.  As  soon  as 
the  tank  is  full  again  the  same  automatic  flush  is  repeated. 

Commonest  of  all  is  Fig.  29  (iii),  where  the  heavy  cast-iron  bell, 


72 


MECHANICS 


[§105 


dropping  suddenly  from  its  lifted  position  shown,  '  wedges '  water 
momentarily  into  its  narrow  upper  part,  and  so  over  the  top  of 
the  pipe  and  as  in  (iv).  ^  „    .  .     r  n- 

When  the  long  waste-pipe  from  a  bath  gets  full  of  water  its  fallmg 
weight  drags  the  water  out  much  faster  than  it  would  run  out  of  a 
mere  hole. 

The  Air  Lock  in  Fig.  28  (right)  is  the  sort  of  thing  that  happens 
in  a  badly  bent  petrol  pipe.  As  it  stands  in  the  figure,  the  liquid  is 
balanced,  and  will  not  run  at  all.  Physical  principles  being  no 
respecters  of  persons,  I  have  known  a  big  air-lock  in  the  new  mains 
hold  up  the  opening  ceremony  of  a  water- works. 

This  is  not  the  air-lock  you  read  about  in  under-water  building 
work  ;  nor  has  the  Siphon  anything  to  do  with  the  comnion  soda- 
water  sjrphon,  which  is  a 
kind  of  wash-bottle  blown 
by  its  own  dissolved  COg- 

§  106.  Pumps.  Pressure  is 
most  commonly  worked  up 
or  down  by  reciprocating 
pumps.  In  Fig.  30  (which 
should  be  intelligible  to  any- 
one who  understands  a  valve) 
are  diagrams  of  the  common 
*  lift  pump  '  at  I,  and  II 
bicycle  pump.  The  *  force 
pump  '  III  when  dealing  with 
liquids  should  have  an  air- 
vessel  on  its  discharge-pipe, 
the  compressed  air  acts  as  a 
spring,  steadying  the  outflow.  With  high  pressures  (Fig.  26)  the 
slightly  compressible  water  itself  affords  elasticity  enough. 

In  the  Heart  the  auricles  dilate  as  they  receive  the  continuous 
influx  from  the  veins,  then,  contracting,  pass  it  through  valves 
into  the  ventricles,  which  in  sudden  systole  force  it  through  other 
valves  into  the  arteries.  The  elastic  arteries  dilate  in  a  '  pulse  ' 
so  that  normally  no  shock  is  felt. 

Air-pumps,  for  exhausting  air,  are  perhaps  of  most  interest  here. 
The  oldest  is  a  '  lift  pump  '  with  oiled  silk  valves  ;  a  pair  of  these, 
driven  opposite  ways,  for  ease  against  the  atmospheric  pressure, 
form  the  time-honoured  machine  for  producing  a  '  vacuum  '— 
rarely  containing  less  than  5%  of  its  original  air.  For  better 
results  one  must  relieve  the  enfeebled  air  of  having  to  lift  a  foot- 
valve,  and  must  do  away  with  all  clearance  spaces  and  all  leakage. 

§  107.  Modern  air  pumps.  In  a  circular  cavity,  about  3  in. 
diameter  by  1  in.  deep,  in  an  aluminium  block,  an  iron  eccentric 
revolves  at  300 — 400  r.p.m.,  as  in  Fig.  31,  left.  A  sliding  gate, 
which  is  kept  pressed  against  the  eccentric  by  a  spring,  divides  the 


108] 


FLUIDS 


73 


crescentic  cavity  into  two  parts.     Everjrthing  is  kept  air-tight  by 

'thick  water-free  oil  in  which  the  whole  pump  is  drowned.     By  the 

I  passage  on  the  left,  air  enters  the  now  expanding  space  to  the 

[left  of  the  gate,  while  that  on  the  right  is  presently  pushed  out,  with 

[oil,  past  the  ball-valve.     Two  of  these  pumps  work  in  tandem  on  the 

j  same  axle,  and  can  pump  down  to  a  vacuum  of  0-001  mm.  of  mercury. 

A  phosphorus  pentoxide  drying-tube  is  included  in  the  system  : 

calcium  chloride  leaves  about  3  mm.  of  water- vapour  pressure  un- 

absorbed,  and  is  quite  useless. 

If  this  vacuum  is  not  good  enough,  a  '  Condensation  '  or  *  Dif- 
fusion '   Pump   is  used  to   extract  the  remainder.     Mercury,   or 
(preferably  an  oil  of  very  much  less  vapour  pressure,  is  kept  boiling 
in  the  little  bulb.  Fig.  31,  right ;  and  a  rapid  stream  of  its  molecules 
rushes  out  and  up  the  wider 
I  tube,    and    along    it    they 
'knock    the    gas    molecules 
(diffusing    over    from    the 
apparatus    under    evacua- 
tion,   giving  them   enough 
additional    momentum    to 
carry  them  along  into   the 
rotary  pump.     The  mercury 
or    oil    molecules    striking 
the  wall,  which  is  kept  cool 
\)y  air  or  water,  stick  to  it, 
and  presently  trickle  back 

into  the  boiler.  Fig.  31. 

i     A  simple  trap,  cooled  by 

i  solid  COg  and  ether,  catches  any  mercury.or  oil  molecules  which  stray 
towards  the  apparatus. 

!  Vacuum  *  getters.'  Charcoal,  cooled  by  liquid  air,  is  occasionally 
a  convenient  absorbent  for  residual  gases  ;  but  commercially,  very 
effective  '  getters  '  save  long  tedious  pumping.  The  wire  filament 
in  a  vacuum  lamp  has  a  speck  of  red  phosphorus  stuck  on  it,  and  the 
I  lamp  is  pumped  out  on  an  oil  rotary  pump  and  sealed  off.  Im- 
;  mediately  on  lighting,  the  phosphorus  is  volatilized,  and  deposits 
I  in  a  perfectly  invisible  film  on  the  glass,  absorbing  then  and  there- 
!  after  all  trace  of  residual  gas. 

A  wireless  valve,  or  radio  tube,  has  a  chip  of  magnesium  left  on 
its  metal  '  plate  '  before  pumping  out  and  sealing  off.  It  is  then 
surrounded  by  a  high-frequency  coil,  which  induces  eddy  currents 
and  raises  all  the  metal  to  red  heat  in  a  few  seconds,  the  magnesmm 
volatilizes,  combines  with  any  oxygen,  hydrogen,  or  nitrogen,  and 
deposits  as  the  familiar  shming  film  on  the  glass,  movable  by  heat, 
perpetually  occluding,  adsorbing,  or  otherwise  sealing,  any  gas  that 
may  happen  to  be  evolved  during  the  working  of  the  tube. 

§  108.  Vacuum  gauges.  Fig.  32,  right,  shows  a  miniature 
barometer,  2  in.  high,  which  has  its  closed  limb  filled  with  boiled- 


74  MECHANICS  [§  108 

out  mercury.     This  leaves  the  top  and  comes  into  action  when  the 
pressure  on  the  right  falls  below  1  in.,  and  can  be  used  down  to 
J  mm.  or  so. 

Beyond  this,  the  McLeod  gauge.  Fig.  32,  left,  is 

kept  connected  through  the  tube  on  the  right  to 

the  apparatus  under  exhaustion.    Mercury  is  just 

being  driven  up  from  below  (by  raising  a  reservoir 

at  the  barometric  height  beneath),  to  drive   any 

gas  from  the  large  bulb  into  its  capillary  tail,  where 

of  course  it  is  comparatively  greatly  compressed. 

If  there  is  none,  the  mercury  stands  at  the  same 

level  in  both  capillaries,  vacuum  above  it  in  each. 

Otherwise  the   elastic   pressure   of   the  gas  com- 

\^/j  pressed  into  the  cul-de-sac  holds  down  the  mercury 

ytuSr  on  that  side,  perhaps  1  or  2  cm.  (as  suggested) ; 

IT  then  the  pressure  in  the  apparatus  is,  by  Boyle's 

Fig.  32.  law, 

volume  V  now  caught  in  capillary       , ,  .  , 

= ^, — „   „  ^ X  this  pressure  p  now  read. 

volume  of  bulb  V  r  jt 

§  109.  Work  done  by  a  pump.  The  work  done  by  a  liquid  lift 
pump  of  any  description  is,  in  gravity  measure,  simply  the  total 
weight  of  liquid  passed  through  x  the  height  it  is  lifted. 

A  pump  which  delivers  volume  v  against  pressure  jp  does  work  pv 
(in  ergs,  if  v  c.c.  and  p  d3nies  per  sq.  cm.). 

For  if  the  discharge -pipe  were  1  sq.  cm.  area,  v  c.c.  forced  into  it 
would  drive  back  resisting  pressure  p  through  v  cm.  =  work  pv, 
§  61,  but  as  fluid  pressure  is  the  same  in  all  directions,  there  is  no 
need  for  this  restriction  as  to  shape. 

But,  in  addition,  if  the  liquid  in  the  pipe  where  p  is  measured  is 
flowing  at  speed  s  cm.  per  second,  the  pump  has  given  it  kinetic 
energy  Jms^  =  J  (volume  X  density  )s^  =  \vds^  ergs. 

e.g.  Work  done  by  heart.  Assuming  the  heart  discharges  per 
beat  75  c.c.  from  right  ventricle  against  pressure  6  cm.  of  mercury, 
and  from  the  left  ventricle  75  c.c.  against  15  cm.  pressure, 

These  =  75  X  6  x  (13-6  x  981)  +  75  x  15  X  (13-6  x  981)  ergs 
=  21  million  ergs 

where  13-6  X  981  =  djrnes  pressure  of  1  c.c.  of  mercury  on  its  base  == 
1  cm.  mercury  pressure  (§111). 

Further,  taking  speed  of  blood  in  both  pulmonary  artery,  and 
aorta,  where  pressures  are  measured,  as  50  cm. /sec,  and  its  density 
as  1-05 

\m  X  speedy  for  both  sides  =  2  X  [J  X  (75  X  1-05)  X  50^] 

=  196,000  ergs 

a  total  of   21-2  piillion  ergs    [2-12   joules,  nearly  2  ft. -lb.,  half  a 
calorie,  per  beat]. 


§  110] 


FLUIDS 


76 


§  110.  Energy  stored  by  fluid.  A  fluid  stores  the  work  quietly 
done  on  it,  as  potential  energy.  That  of  a  mass  m  of  fluid  raised  to 
height  h  is  mh  gm.-cm.  or  mhg  ergs,  just  like  any  solid. 

A  column  of  fluid  of  height  h  has  an  average  height  only  ^h,  and 
therefore  contains  (total  mass  X  ^h  x  g)  ergs. 

Volume  V  under  pressure  p  can  supply  energy  pv  to  a  water-motor 
of  any  sort,     [v  c.c,  p  dynes/cm.^,  pv  ergs.] 

To  those  who  are  '  engine -minded,'  Fig.  33  will  still  further 
illustrate  this  point,  that  pv  represents  the  energy  of  the  fluid.     It 


Hotfe/«f. 


-.  above  atmo. 


^XATMO. 


Fig.  33. 

consists  of  the  Indicator  Diagrams  of  the  engines  of  this  ship,  taken 
to  test  their  working  and  ascertain  their  horse -power,  by  her 
engineer,  J.  Hall  Clark. 

The  horizontal  length  of  the  diagram  is  proportional  to  the 
volume  of  steam  in  the  cylinder,  and  the  vertical  height  to  its 
pressure  at  the  moment.  A  paper  is  clipped  round  the  spring  drum 
on  the  right  of  the  Indicator,  and  then  this  is  pulled  part  way  round 
by  a  string,  hitched,  through  a  simple  reducing  lever,  to  the  piston- 
rod  crosshead,  so  that  a  4-in.  atmospheric  base  line  is  drawn  back- 
wards and  forwards  on  the  paper  by  the  tip  of  the  light -pointer. 

But  this  pointer  copies,  on  an  enlarged  scale,  the  up-and-down 
motion  of  the  little  black  piston  in  the  thumb-sized  steam  cylinder, 
shown  in  section,  which  is  temporarily  screwed  on  to  a  steam-cock 
on  the  engine  cylinder  end,  and  shares  its  steam.  This  piston 
pushes  up  against  a  spiral  spring,  of  which  three  were  employed  in 
obtaining  the  actual  diagrams  shown  on  the  right,  a  120-lb.  per 
sq.  in.  to  the  inch  height  for  the  high-pressure  cylinder,  a  56  for  the 


76 


MECHANICS 


[§110, 


intermediate,  and  a  16  for  the  low-pressure,  of  this  triple-expansion 
engine,  where  the  same  steam  goes  through  all  three  in  succession. 

The  records  start  at  the  top  left ;  steam  is  admitted  to  the  cylinder 
until  about  half-way  along,  when  it  is  cut  off,  and  the  pressure  falls 
towards  *  the  toe  of  the  slipper,'  as  it  expands  to  fill  double  the 
volume  :  during  the  return  stroke  it  is  exhausted  into  the  next 
cylinder,  which  is  much  larger.  The  result  is  that  the  nett  area  of 
the  slipper- shaped  figure  gives  the  product  PV,  representing  the 
work  done  by  the  steam  while  inside  that  cylinder. 

The  three  diagrams  have  been  re-drawn,  on  the  left,  to  a  common 
pressure  scale  of  100  lb.  per  inch,  and  a  common  horizontal  scale 
of  66,000  cu.  in.  cylinder  volume  per  inch ;  you  see  how  they 
fit  together,  and  on  this  diagram  1  sq.  in.  therefore  represents 
100  X  66,000  =  6,600,000  in.-lb.  =  550,000  ft.-lb.  of  Work. 

An  area-measuring  planimeter,  §  153,  shows  that  the  three  areas 
are  205,000,  216,000  and  226,000  ft.-lb. ;  and  as  at  86  r.p.m.  each 
cylinder  is  filled  172  times  per  minute,  multipljdng  by  this,  and 
dividing  by  33,000  (§  66),  give  the  Horse  Powers  as  1066,  1130  and 
1180,  a  total  of  3376  h.p.  for  the  whole  engine. 

A  very  interesting  addition  to  this  engine  is  discussed  further  in 
§294. 

Closely  corresponding  in  medical  practice  to  the  Engine  Indicator 
is  the  Sphygmograph,  which  writes  the  graph  of  the  pulse  at  the 
wrist  on  a  moving  strip  of  paper,  using  the  elastic  wall  of  the  artery 
as  its  piston.  The  elastic  properties  of  this  are  not  known  so 
definitely  as  those  of  a  spiral  spring,  and  the  instrument  is  used  for 
examining  idiosyncrasies  of  the  circulation,  and  not  for  calculation. 


THE   MEASUREMENT   OF  PRESSURE 

§  111.  Manometers.  For  measuring  small  differences  of  gas  pres- 
sure the  U-tube  pressure  gauge  or  Manometer  of  Fig.  34  is  in  common 
use.  Gas  pressure  difference  P  —  P'  is  compensated 
p'  by  an  equivalent  rise  h  of  the  liquid,  so  as  to  maintain 
the  equilibrium  condition  of  equal  total  pressures  on 
each  side  of  an  area  drawn  at  the  bend.  Then  since 
hquid  below  the  lower  level  balances  itself, 


I 
k 

I 


or 


Gas  pressure,  in  grams  per  sq.  cm. 
difference  in  level  X  density  of  liquid 
dynes  per  sq.  cm.  =  hdg. 


By  §  104  there  is  no  obligation  to  have  the  limbs  of 
equal   diameters.     Short    U's    containing    oil,    inky 
water,  etc.,  suf&ce  for  light  pressures,  such  as  furnace 
^^"^^^  draught  (flue,   or  forced),   or   domestic   gas  supply 

Fio.  34.  (usually  4  in.  of  water). 

Long  tubes,  running  up  towers  or  mine  shafts,  and 
filled  with  the  far  denser  mercury,  form  Standard  Manometers  for 
heavy  pressures. 


§112] 


FLUIDS 


77 


The  statement  of  pressure  is  often  left  as  so  many  *  inches  of 
water  *  or  cm.  of  mercury.  [To  convert  X  0-036,  the  weight  of  1  cu. 
in.  water  =  lb.  per  sq.  in. ;  or  X  13-6  =  gm.  per  sq.  cm.  mercury.] 

In  Sphygmo-manometers  for  measuring  blood  pressure,  a  flexible 
rubber  bag  is  strapped  over  the  artery  in  the  arm.  It  is  connected 
with  a  rubber  hand-bulb,  and  is  gradually  blown  up  with  air,  until 
the  pulse  at  the  wrist  ceases,  indicating  the  collapse  of  the  artery 
beneath  the  hard-inflated  bag,  from  which  a  branch  tube  leads  to  a 
mercury  manometer,  or  to  a  dial  gauge  (as  described  later)  also 
graduated  in  cm.  of  mercury. 


Fia.  35. 


§112.  If  one  tube  contains  no  gas  pressure,  but  has  a  vacuum 
libove  the  liquid,  the  instrument  becomes  a  Barometer,  and  measures 
the  absolute  pressure  of  the  Atmosphere,  which  balances  that  of  the 
column  of  liquid  between  the  two  surface  levels. 

The  pattern  in  Fig.  35  (S)  is  called  a  siphon  barometer,  though  the 
open  tube  is  seldom  left  so  long  as  shown.  In  the  domestic  wheel 
barometer,  Fig.  35  (W),  glass  weights  hang  round  a  pulley,  one 
rising  and  falling  with  the  mercury  surface  on  which  it  floats,  and 
a  pointer  conveniently  magnifies  the  motion.  It  lags  a  little  behind 
the  true  reading,  until  sticking  at  the  pulley  pivots,  etc.,  is  relieved 
by  tapping,  which  shows  the  way  the  '  glass '  is  going  (now  called 
the  '  barometric  tendency  '),  and  is  a  most  useful  fault. 


78  MECHANICS  [§112 

For  scientific  accuracy  one  prefers  to  read  the  mercury  column 
direct.  In  a  siphon  this  rises  and  falls  only  half  the  barometric 
change,  for  the  short  limb  moves  equally  and  oppositely,  and  both 
must  be  read.  But  if  the  short  limb  is  broadened,  its  variation  of 
level  is,  of  course,  much  less  (Fig.  S,  as  dotted ;  Figs.  H,  M,  and  F) 
Then  in  the  Kew  or  marine  barometer  M  the  scale  divisions  are 
deliberately  shortened  from  true  inches  sufficiently  to  allow  for 
this  small '  capacity  '.  fall  and  rise  in  the  '  cistern.'  This  instrument 
has  a  constricted  tube  to  hinder  oscillation  of  level  when  the  ship 
rolls,  and  is  hung  on  gimbals  at  the  middle  of  the  protecting  brass 
sheath.  For,  as  in  Fig.  35  (H),  inclining  the  tube  alters  the  reading ; 
it  is  the  vertical  height  that  remains  constant.  The  vernier-head 
resembles  that  of  the  next  pattern. 

In  Standard  Barometers  this  shortening  of  the  divisions  is  in- 
admissible. The  mercury  in  the  glass-walled  cistern  is  always  first 
adjusted  to  touch  a  fixed  pointer,  which  is  the  actual  zero  point  of 
the  vertical  scale.  In  the  Fortin  pattern  (Fig.  35  (F)  shows  one 
graduated  far  enough  down  for  mountain  use,  and  on  a  larger  scale 
its  cistern  and  vernier-head)  this  is  effected  by  moving  the  leather  " 
bottom  of  the  reservoir.  The  scale  is  on  a  protecting  brass  tube 
and  is  read  to  1/500  in.  and  to  1/10  mm.  by  a  vernier  shutter,  borne 
on  an  inner  sleeve  and  racked  down  until  it  just  cuts  off  light  over 
the  middle  of  the  meniscus.  [For  carrying,  the  tube  is  slanted  until 
it  fills  completely,  the  bag  is  screwed  up  until  mercury  exudes  at  the 
little  air-screw  shown  on  the  right,  this  is  screwed  home,  and  the 
instrument  carried  upside-down.] 

§  113.  Readings  on  the  Mercury  Barometer  have  to  be  corrected : 

1.  For  any  errors  of  scale  or  zero  :  in  what  follows  these  are 
assumed  non-existent. 

2.  Mercury  vapour  pressure  in  the  '  Torricellian  space  '  above  the 
column  is  negligibly  small.  But  if  the  air  has  managed  to  stray 
there  enough  to  prevent  the  mercury  hitting  the  top  of  the  tube 
with  a  sharp  snap  when  the  instrument  is  cautiously  inclined,  the  tube 
must  be  refilled  (by  an  instrument -maker,  for  it  requires  boiling  out). 

3.  Capillary  Depression,  §345,  is  negligible  in  a  1-in.  tube,  but  inj 
the  usual  1-cm.  tube  add  0-2  mm.,  or  rather  less  on  '  falling  ' 
readings. 

4.  For  Temperature,  most  important,  see  §  180.  Warmth  makesi 
the  mercury  lighter,  whereas  we  are  relying  upon  it  as  a  substance' 
of  standard  specific  weight.  Therefore  readings  must  be  '  reduced 
to  0°  C  by  the  Rule  (which  is  the  formula  of  §  180). 

Subtract  one  six-thousandth  of  the  observed  height  for  every  degree ^ 
Centigrade  above  zero. 

(The  correction  is  a  trifle  greater  for  scales  other  than  brass, 
but  only  a  trifle.) 

While  nil  outdoors  in  frost,  this  is  usually  by  no  means  small; 
roughly  it  is  1  mm.  in  a  cold  room,  2  mm.  ordinarily,  3  mm.  in  a^ 
hot  room,  and  4  mm.  in  tropical  heat. 


I 


§117] 


FLUIDS 


79 


5  and  6.  Further  corrections,  necessary  on  a  wide  survey  for 
meteorological  purposes,  are  : 

5.  For  Variation  of  Gravity  with  Latitude,  which  afifects  the 
weight  of  the  mercury,  §§  40,  47.  It  is  customary  to  *  reduce  to 
latitude  45°  '  by  a  formula  which  gives — Deduct  J  mm.  (or  J  mb.) 
at  lat.  40° ;  add  J  mm.  (or  J  mb.)  at  lat.  50°,  and  f  mm.  (or  1  mb.) 
at  lat.  55°,  to  the  observed  reading. 

6.  Correct  to  Sea-Level  by  adding  1  mm.  for  every  11  m., 
or  0-1  in.  for  every  90  ft.,  or  1  milhbar  for  25  ft.,  that  the  observing 
station  is  above  the  sea  ;    §  119. 

Many  modern  Barometers  have  contrivances  which  facilitate 
these  corrections  :  see  the  Admiralty  Manual  of  Navigation,  c. 
XXVIII ;   or  the  Meteorological  Observer's  Handbook. 

§  114.  Millibars.  Meteorological  Barometers  are  now  graduated, 
and  Weather  Charts  pubhshed,  in  Millibars,  mb.,  which  are  pressures 
of  1000  dynes  per  sq.  cm.     Roughly,  1  millibar  is  1/32  in.  of  mercury. 

Accurately,  the  Normal  Atmospheric  Pressure,  760  mm.  of  mercury 
(29-91  in.)  at  0°  C,  at  sea-level,  in  latitude  45°,  is  1013-3  millibars  = 
1,013,300  dynes/cm.2. 

§  115.  Since  water  is  13 J  times  less  dense  than  mercury,  the 
Water  Barometer  is  13J  x  2J  =  34  ft.  high.  Glycerine  stands  at 
28  ft.  and  lubricating  oil  at  about  40  ft.,  and  these  are  the  utmost 
heights  to  which  atmospheric  pressure  would  force  up  these  liquids 
into  a  pumped-out  vacuum  ;  see  §  105. 

§  116.  We  now  turn  to  some  Pressure  Gauges 
depending  on  Elasticity,  instead  of  on  dead- 
weight. 

In  the  cottage  weather-glass  of  Fig.  36  water 
is  forced  up  the  neck  of  the  flask,  against  the 
elasticity  of  the  enclosed  air,  as  the  atmospheric 
pressure  outside  increases.  But  the  contrivance 
need  be  kept  in  a  corner  at  a  steady  tempera- 
ture; for  increasing  warmth  expands  the  air  and 
drives  the  water  down,  20°  F.  more  than  com- 
pensating any  ordinary  barometric  change. 

The  gas  in  the  closed  tube  of  the  little  com- 
pressed-air  manometer,  Fig.  37,  halves  its 
volume  every  time  the  pressure  on  the  outer 
end  of  the  mercury  thread  is  doubled,  according 
to  Boyle's  Law,  §  146.  Fio.  36.      Fio.  37. 

§  117.  The  Aneroid  (=  without  liquid)  Barometer  (1848)  is  light 
and  easily  portable.  Fig.  38  shows  the  mechanism  of  a  good 
pocket  aneroid  (an  interesting  travelling  companion  when  provided 
with  an  adjustable  ring  of  altitude  graduations).  Attached  to  the 
base  plate  is  a  flat  vacuum  box  R  of  thin  metal,  corrugated  for 


80 


MECHANICS 


1 

[§  117 1 

or  the  I 
netric  ■ 


Fig.  38. 


flexibility.  The  atmospheric  pressure  would  crush  it  in  but  for  the 
pull  of  a  folded  spring  C  to  which  it  is  hooked.  As  it  is,  barometric 
rise  compels  this  to  yield  a  trifle.  A  long  arm  A  attached  to  C 
magnifies  the  motion  three  or  four  times,  and  is  linked  to  a  shaft 
rocking  on  pivots  PP.  The  distance  of  its  point  of  attachment 
from  the  shaft's  axis  (length  of  lever  arm)  is  variable  by  a  screw 
which  forces  away  the  elastic  free  leg  of  the  forked  rocking  shaft 
from  its  stiff er  pivoted  leg  :  this  modifies  the  total  magnification 
so  that  the  pointer  is  driven  round 
neither  too  fast  nor  too  slowly. 
From  the  rocking  shaft  projects  a 
longer  upright  arm ;  from  this  a 
chain  passes  round  a  pulley  on  the 
pointer  axle  and  is  kept  stretched 
by  hair-spring  H.  The  end  of  the 
pointer  is  thus  made  to  magnify 
the  motion  of  the  box-lid  200  times 
or  more.  [Don't  attempt  to  re- 
member these  details ;  but  do  examine  an  open- front  instrument.] 

Aneroids  must  be  compensated  for  temperature.  Warmth 
weakens  the  spring,  which  gives  way  too  much  and  lets  down  the 
end  of  A,  producing  unduly  high  readings.  This  is  counteracted 
by  making  A  a  compound  bar  (§  175)  of  brass  and  steel  (on  top)  so 
that  its  end  bends  up  as  much  as  the  weakening  would  let  it  drop. 

The  lower  fold  of  C  is  fixed  to  an  L- shaped  bar,  supported  on  the 
base-plate  by  two  steel  posts  at  XY  and  a  '  setting  '  screw  beneath 
Z,  accessible  to  a  small  screwdriver  at  the  back  of  the  instrument. 
Adjusting  Z  rocks  the  bar  on  its  posts,  acting  to  fold  or  unfold  C 
a  very  little,  and  immediately  moving  the  pointer.     The  aneroid 

is  thus  initially  set  to 
agree  with  a  standard 
barometer  corrected  for 
latitude  and  tempera- 
ture, a  zero  adjust- 
ment which  most  ane- 
roids require  every 
year  or  so,  since  the 
spring  slowly  and  per- 
sistently alters  under 
the  constant  strain ; 
though  modern  ane- 
roids are  much  improv- 
ed in  this  respect  : 
thereafter  they  require 
correction  for  Height 
above  sea-level  only. 
Self-recording  instruments  have  frequently  a  stack  of  aneroid 
boxes  as  barograph,  and  a  Bourdon  tube  completely  full  of  alcohol 
as  thermograph,  and  write  on  a  weekly  chart. 


. 

Fig.  39. 


§119] 


FLUIDS 


81 


§  118.  The  Aneroid  mechanism  is  made  for  only  a  very  limited 
range  of  pressure,  but  the  Bourdon  gauge,  Fig.  39  right,  is  used  by 
engineers  for  all  fluid  pressures.  Curled  round  in  nearly  a  circle 
is  a  thin  steel  tube  of  very  flat  elliptic  section.  Increase  of  pressure 
inside  this,  begins  to  fill  out  the  elUptical  shape,  and  this  forces  the 
tube  to  uncurl  to  some  extent,  the  free  end  moves,  and  the  pointer 
geared  to  it  magnifies  its  motion. 

These  are  graduated  by  temporary  attachment  to  a  mercury 
manometer,  §  111,  often  of  great  height,  or  else  to  an  oil  cylinder 
in  which  moves  (easily,  with  rotation)  a  plunger  of  known  sectional 
area  loaded  to  known  weights.  Fig.  39,  left. 

§  119.  Determination  of  heights  by  the  barometer.  As  one  climbs 
above  the  lower  dense  layers  of  the  atmosphere  the  pressure,  of  course, 
diminishes,  by  the  weight  (per  sq. 
cm.)  of  these  layers,  and  the  baro- 
meter falls.  It  was  the  obser- 
vation of  this  fall,  first  made  by 
Pascal  on  the  Puy  de  Dome  in 
1648,  that  established  the  true 
principle  of  the  '  Torricellian  tube.' 

The  calculation  is  this — ^what 
depth  of  air  of  known  density, 
computed  from  its  temperature, 
pressure,  and  humidity,  must 
be  removed  from  above  the  open 
limb,  in  order  that  1  mm.  depth 
of  mercury  may  be  removed  from 
the  closed  limb  ? 

The  depths  are  inversely  as  the 
densities,  §  137. 

Taking  0-0012  average  for  air 
and  13-6  for  mercury  gives  13-6 
-^  0-0012  =  11-3  m.  of  air  per 
mm.  of  mercury,  or  about  an  inch 
fall  of  barometer  for  900  ft.  rise. 

Mining,  surveying,  and  pocket 
aneroids  have  an  Altimeter  Scale 
graduated  on  this  basis,  and  cap- 
able of  rotation  so  that  the  known 
base-level  may  be  set  to  the 
pointer  before  starting  the  climb. 
These  do  fairly  well  for  the  British  Isles,  but  at  greater  heights,  in 
rarer  air,  the  rate  of  fall  of  pressure  with  altitude  diminishes, 
complicating  the  full  calculation.  The  result  is  the  Formula 
employed  in  all  serious  mountain  climbing,  and  in  aviation  : 

Height  in  feet  = 

r     log.  barom.  readings  at  bottoml  ^ 56  200  X  [l  +  0-004  t°  C] 
[_—  log.     do.        do.        at  top       \         * 


cm  10 


82  MECHANICS  [§119 

This,   for   an   average   temperature,   gives   the   Graph   of  Fall  of 
Barometric  Pressure  with  increasing  altitude  of  Fig.  40. 

Up  to  3000  ft.,  the  formula  reduces  to  : 
Height  in  feet  = 

diff.  of  readings  bottom  and  top  ^  ^  ^  ^^^^^,       • 

sum  of  readmgs  bottom  and  top 
or  approximately  the  900  ft.  per  inch  above-mentioned. 

In  variable  weather,  climbers'  readings  are,  of  course,  useless 
unless  afterwards  compared  with  records  simultaneously  made  at 
a  fixed  level. 

FLUIDS   IN  MOTION 

§  120.  Fluids  in  motion.  Fluids  are  set  in  motion  by  differences 
of  pressure  in  different  parts.  The  momentum  gained  per  second 
by  any  portion  =  the  difference  of  the  forces  acting  on  its  opposite 
sides  ;  this  is  the  statement,  for  them,  of  the  Second  Law  of  Motion. 

Evidently,  to  get  into  motion,  the  fluid  has  to  convert  some  of 
its  potential  energy,  due  to  altitude  or  pressure  (§  110),  into  kinetic 
energy  of  motion.  Conversely,  when  the  moving  fluid  is  gradually 
slowed  down,  without  any  wasteful  eddies,  the  energy  returns  to  the 
potential  form,  i.e.  the  pressure  rises  again.  Thus,  if  water  is  flowing 
along  a  pipe  with  gentle  bulges  in  it,  the  pressure  in  the  bulged 
parts,  where  the  motion  is  slower,  is  greater  than  at  the  narrow 
necks ;  wholly  contrary  to  most  people's  expectation,  the  sort  of 
thing  that  used  to  be  called  a  Philosophical  Paradox.  But  why 
should,  and  how  could,  the  fluid  hurl  itself  at  a  narrow  place  with 
increasing  speed,  i.e.  momentum,  unless  it  be  forced,  by  greater 
pressure  from  behind  ? 

The  pressure  at  the  bottom  of  a  water- tank,  or  in  a  steam-boiler, 
may  be  considerable,  but  the  fluid  pressure  in  jets  from  them  is  no 
greater  than  the  atmospheric.  For  if  it  were,  the  unrestricted  jet 
would  burst  and  splutter  in  all  directions.  The  energy  due  to 
pressure  has  gone  into  energy  of  motion,  but  always  their  sum  is 
constant,  PV  -f  Jmv^. 

§  121.  Let  us  calculate  the  relation  between  the  fall  in  pressure 
and  the  speed  of  outflow  of  a  liquid  into  vacuo. 

The  energy  available  is  PV  (§  110),  or  in  1  c.c.  =  P.  The  mass  of 
the  I  c.c.  =  d  the  fluid  density,  therefore  its  energy  of  motion  at 
speed  V  =  ^dv^.     Neglecting  friction,  P  has  entirely  become  ^dv^, 

P  =  ^dv^      or      V  =J—j-. 

Energy  being  in  ergs,  P  in  dynes/cm.2,  v  cm./sec. 
If  P  is  due  to  gravity,  it  =  h  X  d  x  g  (§  103),   where  h  is  the 
'  head  '  of  liquid  above  the  orifice,  and  hence 


V  =  V2hdg  -^  d      or      v  =  V2gh 


I 


§122] 


FLUIDS   IN   MOTION 


83 


This  does  not  involve  d  ;  the  little  fountains  of  mercury  at  Almaden 
play  as  gaily  as  those  of  water  in  any  neighbouring  patio,  if  not  as 
kindly  for  the  vegetation.  The  speed,  which  is  the  same  whichever 
way  the  jet  points,  would,  but  for  friction,  throw  the  jet  up  to  the 
level  of  the  surface  of  the  liquid,  cf.  §  104.  See  also  §  19  ;  we  have 
changed  ato  g  and  s  to  h,  it  is  the  speed  of  free  fall  from  the  surface 
level  :  it  must  be  so,  for  if  the  topmost  particle  cannot  reach  the 
jet,  he  hands  down  his  acceleration  as  a  push  to  the  next,  and  so  on  ; 
and  his  '  proxy,'  the  last  particle,  escapes  with  his  full  authority, 
i.e.  speed. 

§  122.  The  outflow  pipe  from  a  water- works  reservoir  is  coned 
down  to  a  narrow  neck,  and  up  again  to  its  full  size.  This  Venturi 
Meter  does  not  appreciably  obstruct  the  flow,  but  the  square  root  of 
the  fall  of  pressure  at  the  narrow  neck  is  constantly  recorded  by 
the  machinery,  and  meters  the  output. 


1^. 


lf:J 


P'^\ 


R 


Fig.  41. 


Fig.  42. 


In  the  common  jet  pump  or  '  Filter  Pump,'  Fig.  41,  left,  the  high- 
pressure  water  of  the  mains  loses  all  its  pressure  as  speed  as  it 
accelerates  down  the  tapering  jet.  Air  from  around  therefore  pushes 
its  way  into  the  sides  of  the  jet,  and  gets  shot  down  into  the  other 
cone,  whose  gradual  expansion  slows  down  the  speed,  and  brings  it 
up  at  least  to  atmospheric  pressure  again.  With  adequate  '  head  ' 
of  water  it  exhausts  down  to  merely  the  vapour  pressure  of  water ; 
or  it  will  pump  air  for  a  blowpipe,  or  will  pump  water.  Steam-jet 
'  ejectors  '  evacuate  the  pipes  of  the  '  vacuum  brake,'  steam  '  in- 
jectors '  drive  feed-water  into  the  locomotive  boiler,  and  exhaust 
steam  from  the  tapering  blast-pipe  extracts  the  flue  gases  and  blows 
them  up  her  (concealed)  conical  funnel.  Fig.  41,  right,  where  you 
recognize  the  '  Flying  Scotsman's  '  funnel  casing,  and  where  is 
shown  also  the  ring-blower,  used  for  drawing  up  the  fire  when  stand- 
ing, encircling  the  blast-pipe. 

In  Sprayers  for  scent,  disinfectant,  etc.,  a  rapid  air- jet  passing 
over  a  hole  reduces  the  pressure  and  draws  out  liquid,  which  it  then 


84  MECHANICS  [§  122 

blows  away.  Both  here,  and  in  carburetters,  pamt  sprayers,  etc., 
direct  air  pressure  is  often  applied  to  the  liquid,  and  forces  a  more 
copious  supply  of  it  into  the  jet. 

The  Bunsen  Burner  is  the  commonest  laboratory  instance ;  its 
straight  tube  is  wrong,  and  an  expanding  cone  is  used  in  high- 
power  burners  with  much  better  effect. 

§  123.  Fig.  42  shows  a  Centrifugal  Pump;  air  or  water  enters  round 
the  axle  of  a  rapidly  rotating  narrow  paddle-wheel,  which  gives  it  a 
high  speed.  Most  of  this  it  loses  in  the  expanding  outflow  casing, 
gaining  in  pressure  instead.  These  pumps  move  wind  and  water 
for  every  purpose:  domestic  vacuum  cleaners,  forced, draught  to 
boilers  or  cabins,  circulating  condenser  water — a  great  '  all-electric  ' 
ship  reUes  on  them  throughout,  even  for  boiler  feed ;  and  ashore 
they  are  legion.  Do  not  confuse  them  with  screw-propellers,  which 
move  more  fluid  but  at  slower  speed,  and  therefore  much  lower 
pressures. 

§  124.  When  an  obstacle  stands  in  a  stream  of  fluid  one  of  two 
things  happens  :  either  the  fluid  hurries  past  in  Stream  Lines  which 
rejoin  quietly  on  the  lee  side,  or  the  lines  of  flow  break  up  into  rapid 
whirling  Eddies,  which  dissipate  the  energy  and  leave  areas  of 
comparative  quiet.  Sharp  corners  provoke  eddies  :  in  this  way 
you  can  get  shelter  from  the  wind  behind  a  square  telephone-box, 
but  little  behind  a  round  tree -trunk  of  the  same  size  :  the  smooth 
tapering  cone  of  a  filter-pump  jet  discharges  a  surprising  lot  of  water 
for  its  size,  but  the  sharp  edges  inside  an  ordinary  Tap  produce 
obstructive  and  often  noisy  eddies  which  enable  it  to  check  the 
flow  as  desired. 

Any  long  narrow  obstacle  constricting  the  path  of  the  fluid 
induces  accelerated  streaming  :  it  is  always  draughty  under  a  roof. 
Further  narrow  the  way  for  the  wind  by  raising  a  bank,  and  a  long 
building,  and  you  get  a  railway  platform,  draughtiest  of  places. 

Now,  this  increased  speed  to  crowd  past  the  obstruction  must  be 
attained  by  local  reduction  of  air  pressure. 

Ordinary  warm  chimneys  maintain  a  good  draught  and  can  ven- 
tilate a  small  room  effectively,  Chap.  XV,  Q.  10,  but  Ventilators 
for  hospital  wards,  large  halls,  etc.,  are  in  different  plight. 
Slightly  warmer  air,  rising  no  great  height,  weighs  very  little  less 
than  the  colder  air  around ;  its  driving  force  is  very  small  indeed, 
and  yet  we  want  it  to  go  out  into  blustering  winds ;  and  we  want 
them  to  keep  out.  And  the  better  the  ventilation  required,  the  less 
we  can  afford  to  warm  the  great  volume  of  air  ;  so  the  less  chance  it 
has  of  fulfilling  our  wishes. 

If  the  wind  always  blows  steadily  one  way,  as  along  railway- 
carriage  roofs ;  or  if  we  have  a  man  at  the  cowls,  as  aboard  ship, 
it  can  be  managed  ;  but  the  wind  blowing  whither  it  listeth,  curling 
round  buildings  and  romping  over  sloping  roofs,  is  full  of  eddies, 
which  puff  playfully  into  our  feebly  acting  '  holes  in  the  roof,'  until 


f 


126] 


FLUIDS   IN  MOTION 


85 


Fig.  43. 


they  are  cursed  for  their  unendurable  down-draughts,  and  innumer- 
able contrivances  are  sent  for  to  cure  them  ;   and,  mostly,  don't. 

In  Fig.  43,  I,  2,  3,  4  is  an  air  outlet  with  frills  and  a  cap,  to  keep 
out  the  wet.  Adequate  in  a  calm,  but  a  wind  blowing  up  the  slope 
of  the  roof  catches  under  the  cap 
and  puffs  down  :  baffle  that  by  a 
saucer-rim,  5.  Horizontal  wind,  can 
it  be  trusted  not  to  take  a  peep 
down  this  open  thing,  up  which  a 
draught  not  a  tenth  its  strength  is 
trying  to  creep?  Build  a  sheltering 
wall  all  round  it,  6 ;  and  you  have 
a  ventilator  with  the  negative  virtue 
of  no  back-draught. 

But  now,  this  great  round  knob 
standing  up  on  the  ridge  is  an 
obstruction,  which  the  wind  must 
put  on  speed  to  streamline  past ; 
round  about  it,  therefore,  is  reduced 
pressure,  and  out  into  this  partial 
vacuum  the  rising  fumes,  hot  or 
cold,  are  drawn.  This  is  the  Robert- 
son Ventilator,  by  no  means  the  only 
or  the  most  scientific -looking  repre- 
sentative of  such  things,  but  the  most  generally  successful  and 
widely  adopted  one  at  the  time  of  writing. 

§  125.  But  a  stream  of  fluid  exerts  pressure  on  an  obstacle  flatly 
facing  its  motion,  for  this  develops  eddies  to  their  very  greatest. 

The  water-wheel,  beaten  against  by  the  brook,  turns  the  mill. 
Narrow-built  in  steel,  with  its  blades  cupped  like  your  two  hands, 
side  by  side,  as  the  '  Pelton  wheel,'  it  faces  the  fierce  jets  brought 
from  a  mile  high  in  the  mountains,  and  is  spun  with  great  speed  and 
power — Sweden  has  one  of  36,000  h.p. 

As  with  solids  : — 

Force  on  obstacle  =  momentum  destroyed  on  it  per  second, 

=  mass  delivered  per  sec.  X  its  loss  of  forward 
velocity. 

If  the  fluid  be  brought  exactly  to  rest  after  it  strikes  (the  ideal  to 
which  a  wheel-maker  shapes  his  blades),  the  force  pressing  on  the 
surface  =  that  which  originally  set  the  fluid  in  motion,  viz.,  the 
mass  delivered  per  second  X  velocity  of  outflow.  If  it  splashes  back, 
the  force  is  greater :  cricket  ball  and  bat  again. 

§  126.  Reaction  from  jet.  Equal  and  opposite  to  the  force  the 
jet  can  exert  when  stopped  is,  of  course,  by  the  Third  Law  of  Motion, 
a  Reaction  on  whatever  it  started  from. 

A  firework  display  illustrates  this  to  perfection,  from  beginning  to 


86  MECHANICS  [§  126 

end,  for  the  driving  force  of  rockets,  catherine-wheels,  and  all  fly- 
about  fireworks,  is  their  recoil  from  the  outrushing  powder  gases. 

A  rubber  tube  on  the  tap  wriggles  backwards  round  the  sink  : 
a  fireman  has  to  support  a  reaction  of  several  pounds  weight  on  the 
nozzle  of  his  hose,  and  if  that  wrests  it  from  his  hands  the  crowd  soon 
knows  about  it. 

Your  rotary  lawn-sprinkler,  whose  radiating  pipes  continuously 
retreat  from  the  jets  they  deliver  at  a  tangent,  is  a  modern  form  of 
*  Barker's  Mill,'  the  original  '  reaction  turbine  '  ;  while  even  more 
like  that  ancestor  is  the  gaunt  rotary  distributor  down  at  the  sewage 
disposal  works,  whose  four  long  arms  are  driven  slowly  backwards  by 
the  reaction  from  the  streams  of  effluent,  discharged  from  holes 
along  their  following  sides  on  to  the  circular  aerobic  bacteria  bed. 

The  Turbines  of  the  present  day  are  elaborations  and  combinations 
of  this  '  reaction  '  machine,  of  the  '  impulse  '  wheel  of  §  125,  and 
of  the  forward  '  action  '  pressure  wheel  exemplified  in  Windmills, 
old  and  new.  They  vary  from  the  many-thousand-bladed,  thousand- 
revolving,  steam  turbines  of  the  ship,  to  the  cool  slow  simplicity  of 
the  mighty  75,000-h.p.  machines  at  Niagara. 


EXAM   QUESTIONS,   CHAPTER   VII 

Read  steadily  on  to  §  106 ;  read  §§  107,  108  if  you  can  see  modem  apparatus 
inaction.  It  is  often  asked  about,  see  Chap.  IX,  Q.  19.  Thefirstpartsof  §§  109, 
110  are  very  much  to  be  noted;  then  they  borrow  illustrations  from  physio- 
logy and  engineering.  §§  111,  112  are  important,  §  1 1 3  is  for  reference,  and  the 
temperature  correction,  to  which  I  have  given  this  simple  and  acciu-ate  form, 
is  for  use  in  practice.  Note  §  1 14,  and  follow  the  principles  without  the  details 
to  §  119.  §  120  is  of  prime  importance,  it  is  so  wholly  contrary  to  popular 
misconception,  and  is  the  simple  key  to  any  number  of  everyday  mysteries, 
as  you  can  see  by  running  through  the  rest  of  the  chapter. 

You  will  have  noticed  by  this  time  that  these  Exam  Questions 
overlap  quite  a  lot,  so  that  you  need  not  grind  through  them  all,  but 
can  pick  and  choose  on  your  own  plan,  while  there  are  plenty  avail- 
able for  practice  on  particular  points  over  which  you  find  difficulty. 
The  syllabus  is  wide,  and  no  part  of  it  should  be  omitted,  but  of 
course  it  may  happen  in  some  sections — especially  perhaps  these 
mechanical  ones — that  while  you  feel  you  have  a  fair  comprehension, 
you  are  a  bit  of  a  duffer  at  the  problems.  Do  what  you  can  to 
remedy  this — everybody  likes  doing  things — but  without  anxietj^^ : 
get  on  with  the  subject.  \ 

The  following  statement  will  give  you  some  guidance,  and  perhaps  re- 
assurance :  Two  written  papers  are  set,  each  with  ten  or  eleven  questions, 
of  which  you  may  try  seven.  The  morning  paper  usually  contains  two 
'  dry  '  and  two  or  three  '  wet '  mechanics  questions,  two  sound  and  four  or 
five  optics  :  the  other  comprises  three  or  four  heat,  a  magnetic  and  an  elec- 
trostatic, and  the  rest  electrical.  The  marks  of  the  two  papers  are,  of  course, 
added  together,  but  there  is  a  lower  limit,  a  charitable  one,  to  the  marks 


FLUIDS   IN   MOTION  87 

that  can  be  tolerated  in  either  paper  :   you  may  not  neglect  half  the  subject, 
but  you  range  at  will  inside  each  half. 

Two  experiments,  set  at  random  from  the  apparatus  available,  occupy  the 
3-hoiu-  practical  exam;  credit  is  given  for  your  signed  record  of  laboratory 
work,  and  all  marks  add  in  with  the  papers,  often  quite  helpfully. 


1.  Distinguish  between  velocity  and  acceleration,  force  and  pressiu^, 
work  and  energy,  mass  and  weight.  In  what  imits  are  they  measured  (a) 
c.g.s.,  (6)  English? 

2.  A  load  of  700  lb.  rests  directly  on  the  3-in.-diam.  safety-valve.  At 
what  boiler-pressure  will  the  valve  lift  ? 

3.  Wliat  is  the  pressure  due  to  a  '  head  '  of  180  ft.  of  water  ? 

[A  colmnn  of  180  cu.  ft.  of  water,  each  weighing  62-5  lb.,  exerts  180  X  62-5 
lb.  on  the  square  foot  at  its  base  =  1120  lb.  per  sq.  ft.  =  78  lb.  per  sq.  in.] 

4.  Calculate  difference  of  blood  pressure  between  head  and  feet  of  a  man 
1-7  m.  tall;   s.g.  blood  1-05. 

5.  Express  in  gm./cm.^  and  in  dynes /cm.^  the  pressure  due  to  a  76-cm. 
coliunn  of  mercury,  of  density  13-6. 

6.  Calculate  the  height  of  a  column  of  air,  density  0-00125,  which  exerts 
the  same  pressure  on  its  base  as  does  1  cm.  depth  of  mercury. 

7.  Distinguish  between  pressure  and  force. 

Calculate  the  force  on  each  of  the  sides  and  on  the  bottom  of  a  rectangular 
tank  filled  with  water;  the  dimensions  being  80  ft.,  40  ft.,  and  depth  6  ft.; 
given  1  cu.  ft.  =  62-3  lb. 

8.  If  the  atmospheric  pressure  be  that  due  to  76  cm.  of  mercury,  density 
13-6,  at  what  depth  under  sea-water  density  103  will  pressure  be  2  atmos.  ? 

9.  What  do  you  mean  by  the  pressure  at  a  point  ?  An  aneroid  showed 
68  cm.  mercury  pressure  at  1  km.  altitude ;  express  this  in  c.g.s.  imits,  and 
calculate  the  mean  density  of  the  air. 

10.  Prove  that  the  pressure  at  a  point  in  a  liquid  is  proportional  to  the 
depth  below  the  free  surface.  How  would  you  verify  this  result  by  ex- 
periment ? 

A  U  tube  containing  oil  of  s.g.  0-8  is  placed  with  its  limbs  vertical.  One 
end  is  open  and  the  other  is  connected  to  the  gas  supply.  The  difference 
of  level  in  the  two  limbs  is  12  cm.  Calculate,  in  absolute  imits,  the  difference 
of  pressure  between  the  gas  and  the  air. 

11.  Explain  the  mode  of  action  of  a  siphon.  Would  you  expect  its  working 
to  be  impaired  if  a  hole  were  drilled  in  the  long  arm  at  a  point  below  the 
level  of  the  extremity  of  the  .short  arm  ? 

12.  Describe  the  action  of  a  siphon,  indicating  how  it  depends  on  the 
presence  of  an  atmosphere.  How  can  a  siphon  be  arranged  to  start  by  itself 
and  flow  intermittently  ?     (  X  3) 

13.  Draw  a  diagram  of  a  piunp  suitable  for  raising  water  from  a  well  20  ft. 
deep  to  a  tank  30  ft.  above  the  ground. 

What  horse-power  is  necessary  to  piunp  the  water  at  the  rate  of  200  gallons 
per  minute  ?  (A  gallon  of  water  weighs  10  lb.,  and  1  horse-power  =  550 
ft. -lb.  per  sec.) 

14.  What  amoimts  of  energy  are  represented  by  the  joule  and  the  kilogram- 
metre  ? 

A  pump  making  400  strokes  per  minute  delivers  at  each  stroke  2  litres  of 
sea-water,  density  102,  to  a  tank  10  m.  above  sea-level.  Calculate  the  power 
recjuired,  assuming  two-fifths  lost  by  fiiction. 


88  MECHANICS 

15.  In  the  Barcelona  Exposicion  of  1929,  12,000  h.p.  was  devoted  to  the 
fountains.     Calculate  how  much  water  this  would  fling  100  ft.  high. 

16.  The  upper  end  of  a  vertical  glass  tube  of  uniform  cross-section  2  sq. 
cm.  is  connected  to  an  air  pump ;  the  lower  end  dips  into  a  large  reservoir 
of  mercury.  Calculate  the  amount  of  work  done  in  using  the  pump  to  raise 
the  level  of  the  mercury  in  the  tube  to  60  cm. 

17.  Calculate  the  minimum  power  of  a  heart,  making  70  strokes  per  minute 
each  delivering  150  gm.  of  blood,  s.g.  1-05,  against  24  cm.  merciu-y  pressiu-e. 

18.  How  does  an  elastic  tube  differ  from  a  rigid  tube  in  its  manner  of 
conducting  and  delivering  a  pumped-in  liquid  (nature  and  cause  of  pulse- 
wave)  ? 

19.  Describe  an  experiment  to  show  that  the  atmosphere  exerts  a  pressure 
varying  with  height ;   what  is  the  variation  ? 

20.  Describe  some  form  of  standard  Barometer,  and  show  how  to  make 
the  necessary  temperature  corrections  of  the  reading  obtained.  What  is 
the  effect  of  irregular  bore  of  the  tube,  of  narrow  bore,  and  of  change  of  bore 
by  heating  ?     (  x  3) 

21.  Describe  a  Fortin  barometer.  How  is  it  affected  by  temperature 
changes  ?  Calculate  its  difference  in  reading  at  top  and  bottom  of  a  20-m, 
building. 

22.  A  barometric  reading  is  often  given  as  '  corrected  to  sea  level,  in  latitude 
45°,  at  0°  C     Why  are  these  corrections  necessary  ? 

From  which  of  them  should  a  well-made  aneroid  barometer  be  exempt  ? 
Describe  the  construction  of  some  pattern  of  aneroid.     (  X  2) 

23.  A  steel  ball  floats  up  to  the  top  of  a  barometer  ;  what  effect  will  this 
have  on  the  mercury  levels  in  tube  and  cistern,  and  what  difference  will  it 
make  ? 

24.  If  the  atmosphere  sustains  the  barometric  column,  how  is  it  that  a 
barometer  tube  is  heavy  to  lift  from  place  to  place  in  a  basin  of  mercury  ? 
Calculate  the  force  that  must  be  exerted  to  lift  out'  of  the  reservoir  a  glass 
barometer  tube  of  weight  100  gm.  and  0-25  sq.  cm.  area  of  bore,  H  being 
76  cm. 

25.  What  is  meant  by  the  Conservation  of  Energy  ? 

Calculate  the  velocity  with  which  water  will  issue  from  a  hole  at  a  depth 
h  below  the  siu-face  of  a  reservoir.  How  would  your  result  be  modified  if 
the  hole  were  very  small  ? 

26.  Explain  how  the  pressm-e  and  velocity  of  a  fluid  vary  as  it  flows  along 
a  gradually  widening  or  narrowing  pipe,  and  show  how  your  explanation 
accounts  for  the  action  of  a  filter  pump  or  other  jet  pump. 


\ 


CHAPTER  VIII 
FLOTATION  AND  SPECIFIC  GRAVITY 


§  131.  Hiero,  Tyrant  of  Syracuse,  ca.  260  B.C.,  gave  good  weight 
of  gold  to  a  smith,  to  make  him  a  crown.  The  crown  was  made, 
and  returned  its  true  weight,  but  my  lord  entertained  a  suspicion 
that  the  goldsmith  had  abstracted  a  perquisite,  and  had  alloyed  in 
baser  metal.  He  bade  his  court  philosopher,  Archimedes,  discover 
the  truth.  Deeply  pondering,  the  man  of  science  stepped  down  into 
an  already  completely  full  bath ;  its  overflowing  showed  him  a 
possible  solution,  and  he  sprang  out  and  rushed  headlong  to  the 
Presence,  crying  kvp-qKa,  heureka,  I  have  found  it  !  ' 

Do  not  let  this  classic  story  suggest  to  you  scandalized  policemen 
holding  up  the  traflSc,  what  time  a  frantic  greybeard,  in  the  total 
neglige  of  the  bath,  sprints  from  the  Royal  Society  down  St.  James' 
Street  :  Archimedes,  at  the  time,  was  a  man  of  twenty-four,  and 
as  presentable  as  you  are  ;  one's  birthday  suit  is  fitting  wear  under  the 
Sicilian  sun,  among  the  cypresses  and  mjni^les  of  the  gymnasium 
gardens  of  the  palace ;  a  watchful  slave  doubtless  brought  robes 
sufficient  to  obviate  Use  majesU,  while  the  Tjrrant  could  but  be 
immensely  tickled. 

Archimedes  argued  thus  :  I  know  that  it  takes  a  far  bigger  lump 
of  brass  to  lie  as  heavy  in  the  hand  as  a  little  ingot  of  gold  ;  if  this 
crown  is  brazen  it  will  bulk  larger  than  an  equal  weight  of  gold. 
From  the  treasury  he  took  this  gold,  and  lowered  it  by  a  flaxen  thread 
into  a  vase  brim-full  of  water  ;  water,  equal  in  volume  to  the  gold, 
ran  over.  Fishing  out  the  gold,  he  lowered  in  the  crown — and  if 
any  additional  water  overflowed,  the  operations  of  suspension  and 
immersion  were  undoubtedly  repeated  on  the  person  of  the  smith. 

§  132.  But  the  Principle  nowadays  tacked  on  to  the  name  of 
Archimedes  took  more  than  a  sudden  brain- wave. 

Consider  the  fluid  contained  in  a  closed  volume  marked  out  inside 
a  quantity  of  fluid  at  rest,  for  instance  the  water  contained  in  a 
submerged  net.  It  is  acted  on  by  the  pull  of  the  earth,  and  by  the 
pressures  of  the  adjacent  fluid,  and  these  just  balance  each  other, 
for  it  remains  at  rest.  That  is,  the  pressures  of  the  surrounding 
fluid  just  exactly  bear  up  the  weight  of  the  fluid  filling  the  volume. 

Suppose  the  volume  to  be  emptied  of  its  fluid,  and  filled  with  some 
other  material.  The  surrounding  pressures  are  quite  unaltered, 
i.e.  this  foreign  substance  is  borne  up  with  a  force  equal  to  the  weight 
of  fluid  it  has  displaced,  or  apparently  it  loses  that  much  of  its  usual 
weight.    This  is  the  Principle  of  Archimedes. 

89 


90  MECHANICS  [§  133| 

§  133.  If  the  foreign  substance  is  more  massive  (denser)  than  the 
fluid  it  has  displaced,  it  will  still  require  some  other  support ;  but 
if  less  massive,  it  must  be  held  down,  or  it  will  rise  and  float  and 
displace  only  that  fraction  of  its  own  volume  of  fluid  which  has  a 
weight  equal  to  its  own. 

It  is  easy  to  get  muddled  in  any  explanation  of  Floating,  simple 
thing  as  it  is.  As  good  a  way  as  any  is  to  think  of  one's  own  self. 
We  are,  most  of  us,  a  bit  heavier  (denser)  than  fresh  water,  and  a  bit 
lighter  than  sea- water.  Suppose  that,  taking  a  hold  of  the  boat's 
painter,  you  go  over  the  side  into  deep  water.  Hold  on  with  hands 
under  water,  and  gradually  ease  down,  hanging  on  less  and  less  and 
less  ;  hold  your  breath  and  don't  pull  on  the  rope  at  all.  If  you  are 
in  fresh  water,  the  weight  of  all  the  water  your  whole  body  displaces 
is  not  as  much  as  yours,  and  your  2  or  3  lb.  excess  drags  you,  slowly, 
Atwood  machine  fashion,  down  until  your  feet  rest  with  that  much 
force  on  the  bottom  ;  pull  10  lb.  on  the  hanging  rope,  and  the  7  or 
8  lb.  difference  accelerates  you  up,  until  as  much  of  your  head  as 
would  displace  that  weight  of  water  is  now  above  water,  resting  its 
weight  as  usual  on  your  neck.  But  in  sea-water,  you  ease  down  and 
sink  lower  and  lower ;  now  lay  your  head  back  on  the  pillow  of 
water,  and  you  can  let  go  altogether  :  part  of  you  is  displacing 
enough  salt  water  to  weigh  as  much  as  the  whole  of  you,  and  your 
face  and  enough  cubic  inches  of  head  still  remain  unsubmerged 
for  free  breathing.  As  you  breathe  in  and  out,  increasing  and 
decreasing  your  total  bulk  without  change  of  weight,  your  face  rises 
and  falls  a  little,  because  the  immersed  bulk  remains  constant,  that 
of  your  own  weight  of  sea-water. 

The  boat  itself  floats  high,  because  whatever  it  is  made  of  has  been 
split  or  beaten  out  thin,  and  is  put  together  so  as  to  enclose  a  very 
large  bulk,  many  times  more  than  its  weight  of  water  occupies,  so 
that  only  a  fraction  of  it  is  engaged  in  displacing  water ;  the  rest 
bulks  up  above — even  unto  the  six  or  seven  decks  in  a  great  liner. 
It  does  not  sink  completely  when  it  comes  from  salt  to  fresh  water, 
as  you  had  to  with  your  small  projecting  head,  but  a  big  ship  does 
settle  a  few  inches  lower. 

Fig.  44  shows  the  load-line  marking  of  a  British  ship.  S,  on  a 
level  with  the  centre  of  Lloyd's  Register  ring,  is  her  normal  summer  • 
load  line  at  sea,  F  is  for  fresh  water,  and  I 
this  difference  is  deflnitely  due  to  its  l/40th 
less  density.  The  other  marks  involve  the 
more  complex  consideration  of  probable 
weather  ;  T  and  TF  are  for  the  Tropics,  W 
is  for  Winter  and  WNA  for  the  grim  North 
Atlantic  at  that  season,  when  safety  at  sea  is 
Fig.  44.  ^q^  j^g^  a  matter  of  specific  gravity. 

All  American  bath-soap  floats,  but  in 
England  we  still  put  up  with  the  nuisance  of  a  soap  a  little  denser 
than  water,  which  rests  ever  so  lightly  on  the  bottom,  and  therefore 
shps  away  with  almost  frictionless  speed  when  we  fish  for  it. 


§134] 


FLOTATION   AND   SPECIFIC   GRAVITY 


91 


A  body  cannot  rest  midway  in  a  fluid  of  constant  density  not 
precisely  equal  to  its  own.  For  instance,  a  torpedo  cannot  be 
weighted  to  remain  6  ft.  under  water ;  that  depth  must  be  kept  by 
active  mechanical  control.  That  is,  the  thing  must  swim,  even  as 
you  do,  bothering  very  little  as  to  sea-  or  river-water ;  or  as  an 
aeroplane  does  in  the  air,  an  utterly  different  matter  from  the 
flotation  of  a  balloon. 

Fish  almost  manage  the  flotation-in-mid- water  problem,  having 
swim-bladders  full  of  '  air,'  their  volume  under  nerve-control. 
It  was  the  subject  of  an  early  meeting  of  the  new  Royal  Society, 
to  settle  the  much-discussed  question  whether  a  live  fish,  put  into  a 
bucket  of  water,  increased  the  weight  of  it  or  not.  The  fact  being, 
that  since  a  fish  in  swimming  condition  is  almost  exactly  as  dense 
;is  water,  its  own  weight  of  water  will  spill  out  of  a  brim-full  bucket 
as  it  is  slipped  in. 

In  the  Chance  Coal  Washer,  coal  is  separated  from  shale  by  the 
use  of  a  sand-water  mixture — a  very  wet  quicksand — as  a  high- 
density  medium.  An  upward  current  of  water  is  pumped  through 
the  sea-sand  at  a  quite  moderate  speed,  and  maintains  it  in  a  sus- 
pension, the  bulkiness,  and  inversely  density,  of  which  is  readily 
controlled  by  varying  the  water  flow,  equivalent  densities  between 
115  and  1-5  being  attainable.  Consequently  coal,  of  density 
about  1-3,  can  be  made  to  float  out  at  the  top,  while  the  denser 
shale  sinks,  and  is  dredged  out  from  the  bottom. 

§  134.  Archimedes'  Principle  applies  not  only  to  gravity,  but 
e.g.  to  centrifugal  force,  the  employment  of  which  for  separating 
bacteria  from  liquids,  or  cream  from  milk,  is  well  known,  §  87. 

The  Principle   can    be    experimentally  verified  as   in   Fig.    45 


XJ 


TV 


Fig.  45. 


The  ball  and  the  can  of  liquid  are  first  separately  counterpoised,  then 
the  ball  is  lowered  into  the  liquid,  and  to  restore  equilibrium  it  will 
be  found  that  the  same  weight  that  has  to  be  removed  from  the 
first  scale-pan  on  the  right  must  be  put  into  the  last  pan  on  the 
left.     The  liquid  is  bearing  just  exactly  the  missing  weight. 


92 


MECHANICS 


[§136 


SPECIFIC  GRAVITY 

§  135.  The  heaviness  or  *  gravity  '  specific  to  a  particular  sub- 
stance, or  the  specific  gravity  of  a  substance,  is  the  ratio  of  the  weight 
of  a  volume  of  it  to  that  of  the  same  volume  of  water. 

Being  a  mere  ratio,  it  is  the  same  whether  in  cg.s.or  British  measure. 
In  c.g.s.  it  is  equal  to  Density,  for  this  is  the  mass  of  1  c.c,  and  1  c.c. 
of  water  is  1  gm. ;  but  density  if  expressed  in  lb.  per  cu.  ft.  is 
S.g.  X  62-5,  that  being  the  weight  of  the  cubic  foot  of  water. 

[Strictly  speaking,  Specific  Gravity  is  reckoned  from  water  at 
15°  C.  or  60°  F.,  and  neglects  the  fact  that  the  weighings  are  all 
made  in  air.  Hence  small  corrections  to  water  at  4°  C.  and  to  true 
weights  in  vacuo  are  needed  to  make  the  Specific -Gravity  measure- 
ment a  really  exact  one  of  Density.] 

All  specific-gravity  determinations  must  be  made  very  near  the 
standard  temperature,  for  liquids  are  very  expansible. 

Variations  in  the  composition  and  in  useful  properties  of  substances 
are  frequently  accompanied  by  characteristic  slight  changes  in 
their  densities.  Hence  the  accurate  measurement  of  specific  gravity 
or  density  is  of  great  technical  importance,  as  it  very  often  affords 
the  quickest  means  of  discrimination  and  valuation.  It  is  the 
refinement  of  the  familiar  guessing  at  what  a  substance  is  by  its 
'  weight.'  The  mineralogist  uses  it  as  guide  to  the  nature  of 
minerals — gem  stones,  metallic  ores,  etc.  The  apothecary,  the 
analyst,  the  technical  chemist,  the  brewer,  the  exciseman,  all  possess 
tables  drawn  up  to  give  the  concentrations  of  the  particular  solutions 
they  are  dealing  with,  in  terms  of  their  hydrometer  readings,  and 
find  it  vastly  more  convenient,  and  often  more  accurate,  to  make  use 
of  this  instrument,  rather  than  to  undertake  any  chemical  analysis. 

§  136.  A  straightforward  method  of  finding  the  specific  gravity  of 
a  liquid  is  by  means  of  the  Specific-gravity  Bottle  or  Pyknometer. 
This  is  a  bottle  which  can  be  filled  with  always  exactly  the  same' 


Fig.  46. 


volume  of  a  liquid,  either,  Fig.  46  (i)  to  a  flat  plate  (a  scholastic 
contrivance)  or  (ii)  up  to  a  mark  on  a  narrow  neck,  or  (iii)  completely 
up  to  the  stopper  (perforated  for  overflow  when  dropped  in),  or 
(iv)  from  nozzle  to  file-mark  in  the  Sprengel  pattern. 


§137] 


FLOTATION   AND   SPECIFIC   GRAVITY 


93 


The  dry  bottle  is  counterpoised  on  a  balance,  then  the  net  weight 
of  cold  water  filling  it  is  found,  W.  It  is  rinsed  and  filled  with  the 
li(iuid,  the  net  weight  of  which  proves  to  be  L.  Then  sp.  grr.=L  4-  W. 
I' he  English  apothecary  saves  calculation  by  using  a  bottle  with 
\\'  =  1000  grains  at  the  ordinary  temperature  :  he  buys  his  bottle 
with  its  counterpoise  weight  complete,  and  often  omits  any  decimal 
})oint  in  his  s.g. 

For  insoluble  solids  (powders)  M  gm.  are  weighed  into  the  previously 
counterpoised  bottle,  which  is  then  filled  with  water.  Its 
contents  now,  of  course,  weigh  less  than  M  +  W  by  the  weight  of 
water  which  cannot  get  in  on  account  of  the  presence  of  the  solid, 
i.r.  which  would  occupy  the  same  volume  as  the  solid.  Then 
M  -^  this  shortage  =  sp.  gr.  of  solid. 

With  soluble  solids  any  limpid  oil  would  be  used,  and  then  M  -f- 
skortage  =  ratio  sp.  gr.  of  solid  to  sp.  gr.  of  liquid,  this  last  being  found 
as  before  described. 

[These  are  most  favourite  practical  examination  exercises.] 


§  137.  *  Hare's  apparatus  '  of  balancing  columns  of  fluid.    If 

two  non-miscible  fluids  are  poured  into  a  U  tube.  Fig.  47,  they  will 

come  to  rest  at  different  levels.     Omitting 

the  changes  of  pressure  below  their  contact 

level  (§  104),  the  pressures  on  either  end 

of  the  portion  of  (denser)  liquid  in  the 

f)end  below  this  common  level  must  be 
I  equal.     The  air  pressure  is  the  same  in    ^ 
I  both  open  tubes  and  can  be  left  out  of      | 

account.   Then  by  §  103,  ^j  X  c^i  =  /ig  X  <?2       1 


d, 
or  V 


^2 


and  therefore  if  liquid  (2)  is  water,  d^  the 
Specific  Gravity  of  the  first  liquid  =  height 
of  water  ^height  of  liquid,  both  from  the 
common  level. 

For  miscible  liquids  the  form  of  appa- 
ratus shown  on  the  right  is  preferable. 
The  atmospheric  pressure— the  reduced 
air  pressure  in  the  bend  (sucked  out)  = 
the  pressure  due  to  either  column  of 
liquid,  hence,  as  before,  heights  Hi  Hg  above 
the  levels  in  their  respective  reservoirs  are 
inversely  as  densities. 

It  is  often  a  little  more  convenient  to 
draw  the  liquids  just  up  into  the  tubes  by 
a  slight  suction,  and  to  use  their  two 
levels  in  the  tubes  as  the  starting  points. 


L 


D 


"^^^ 


Fio 


Notice  that  the  sizes  of  the  tubes  are  quite  immaterial,  §  104. 


94 


MECHANICS 


[§  137 


The  Barometer  is  a  '  Hare's  apparatus  '  with  one  column  miles  | 
high  ;   Hg  versus  air. 

§  138.  The  Hydrostatic  Balance  method,  introduced  by  Galileo 
in  1588,  of  weighing  a  body  in  air  and  then  in  water,  applies  Archi- 
medes' principle  directly.  A  balance  is  arranged  as  in  Fig.  48  with 
a  '  stirrup  '  of  thick  wire  (flexible  and  heavy  to  enfold  and  sink 
things  that  want  to  float,  when  W  exceeds  M)  hanging,  scrupulously 
touching  nothing,  by  a  thin  silk  thread,  under  cold  London  tap- 
water  ;   and  this  is  counterpoised  (s). 

The  body  is  laid  on  the  left-hand  pan  and  weights  =  its  M  placed 
on  the  other  pan  until  equilibrium  is  restored.  The  body  is  removed 
from  pan  to  stirrup  ;  it  is  found  that  a  portion  W  of  the  weights  must 
be  removed  from  the  weights  pan 

M       weight  of  body       „      -n    n      -^      j^  v  j 
W  =  losa  in  water   =  ^^'^'fi'  ^""'^y  "^  '"^^^ 

For  soluble  solids,  weigh  in  oil,  and  Sp.  Gr.  = sp.  gr.  oj  oi 

°  '  ^  loss  m  oil 


Fig.  48. 


The  apparatus  can  be  used  to  find  the  specific  gravity  of  liquids, 
such  as  this  oil,  for  this  is  the  loss  of  weight  in  the  liquid,  of  a  '  sinker  ' 
or  'plummet,  divided  by  its  loss  of  weight  in  water,  these  being  the 
weights  of  the  two  liquids  that  the  same  bulk  displaces. 

The  plummet  may  be  any  old  glass  stopper,  but  very  convenient 
ones  displacing  exactly  10  c.c.  (or  less,  to  1),  i.e.  10  gm.  water,  are 
obtainable.  The  weight  that  has  to  be  hung  on  the  same  side  as 
the  plummet  (previously  counterpoised  in  air)  to  keep  it  under  the 
liquid,  divided  by  10  (or  less,  to  1)    =  sp.  gr.  of  liquid. 

Nicholson's  hydrometer  is  a  variant  of  the  hydrostatic  balance  now 
long  past  all  usefulness,  even  educational,  a  crank  old  craft  always 
plunging  to  a  watery  grave.  There  let  it  be  ;  there  are  numerous 
*  spring-balance  '  hydrostatic  balances  which  are  far  quicker  and 
more  serviceable. 


§139] 


FLOTATION  AND   SPECIFIC   GRAVITY 


96 


§  139.  The  common  Hydrometer,  used  for  liquid  specific  gravities, 
consists  of  a  glass  buoy  ballasted  by  a  load  of  shot  or  mercury  at  the 
bottom,  and  having  a  thin  stem  projecting  above  the  liquid,  Fig. 
4:9.  It  floats,  therefore,  always  displacing  a  weight  of  liquid  equal 
to  its  own  constant  weight,  or 

volume  displaced  x  density  of  liquid 
=  mass  of  hydrometer 

hence  it  displaces  less,  i.e.  floats  higher,  in  a  denser  liquid. 

A  scale  of  specific  gravities  is  therefore  graduated  on  the  stem, 
with  the  largest  readings  at  the  lower  end,  and  in  such  a  way  that 
the  volumes  of  the  instrument  up  to  the  scale  divisions  are  inversely 
as  the  specific  gravities  marked  on  them  :  the  divisions  get  rather 
wider  apart  towards  the  top.  [The  liquid's  specific 
gravity  is  the  reading  at  which  the  stem  cuts  the 
surface.]  For  a  given  size  of  bulb,  their  length  {i.e. 
possible  delicacy  of  reading)  is  greater  on  a  thinner 
stem,  being  inversely  as  its  cross -section. 

Hydrometers  are  commercially  obtainable  of  various 
degrees  of  sensitiveness,  over  various  ranges  of  specific 
gravity  {e.g.  1-0 — 0-9  ;  1-2 — 1-4,  etc.)  under  different 
names — lactometer,  salinometer,  alcoholometer,  etc. — 
and  to  various  arbitrary  scales,  e.g.  Twaddell's  or 
Beaume's,  or  even  directly  graduated  in  concentrations 
of  cream,  salt,  etc. 

[Specific  gravity  =  1  +  degrees  Twaddell  -^  200, 

=  144-3  -^  (144-3  —  degrees  Beaume).] 

In  using  a  hydrometer  give  it  a  spin,  and  tap  the  jar 
to  eliminate  friction  ;   don't  forget  to  rinse  the  instru-  . 
ment  after  use. 

Sykes's  hydrometer,  a  pretty  little  gilt  brass  thing, 
which  the  exciseman  uses  in  gauging  the  alcohol  content  of  wines 
and  spirits,  works  like  the  common  hydrometer,  but  can  be  loaded 
with  collar  weights  which  make  the  same  instrument  available 
for  several  ranges  of  specific  gravities. 

Hydrometer  readings  are  adversely  affected  by  the  Surface  Ten- 
sion of  the  liquids.  Chap.  XXIII.  Not  only  does  this  lift  up  the 
liquid  in  a  clinging  ring  round  the  stem,  obscuring  the  mark  (but 
this  is  usually  best  seen  from  below  the  surface),  but  this  clinging 
actually  drags  the  float  down  too  low.  This  could  be  allowed  for 
if  the  surface  tension  were  the  same  for  all  liquids  examined,  but  it 
varies  a  good  deal,  e.g.  it  is  only  J  as  much  for  soapy  water  or  for 
oils.  Fortunately,  the  makers  do  not  graduate  their  hydrometers 
by  dead  reckoning,  as  in  Ex.  11  below,  but  by  trial  in  a  series  of 
common  test  liquids  ;  and  the  discrepancies  are  seldom  as  great  as 
0-001  s.g.,  an  error  comparable  with  that  caused  by  2°  or  3"^  variation 
of  temperature.  But  it  prevents  them  attaining  the  same  accuracy 
/  as  the  bottle  and  balance. 


Fig.  49. 


96  MECHANICS  [§  HO] 

§  140.  The  specific  gravity  of  substances  of  which  only  small  chips 
or  drops  are  available  is  found  by  preparing  a  jar  full  of  a  mixture  of 
liquids  of  the  same  density  as  the  substance,  as  determined  by  placing 
a  fragment  or  drop  of  it  in  the  midst,  when  it  must  show  no  appreci- 
able intention  of  either  rising  or  sinking.  (Just  as  the  specific 
gravity  of  ice  can  be  found  as  that  of  a  mixture  made  by  pouring 
alcohol  into  water  until  the  ice  just  ceases  to  float.) 

A  mixture  of  chloroform   (sp.  gr.   1-526)  and  benzole   (sp.  gr, 
0-889)  is  made  up  until  a  drop  of  human  blood  floats  undecided 
then  a  small  s.g.  bottle  is  filled  with  the  mixture. 

In  finding  the  specific  gravity  of  mineral  fragments,  or  separating 
the  constituent  minerals  in  a  powdered  rock,  very  dense  liquids 
are  used,  such  as  mercury  biniodide  in  potassium  iodide  solution 
(max.  3-2),  cadmium  borotungstate  solution  (3-6)  or  methylene 
iodide  (3-3) ;   diluted  with  water  or  alcohol. 


EXAM    QUESTIONS,    CHAPTER   VIII 
A  chapter  of  value,  theoretically  and  practically. 

1.  Equal  masses  of  alcohol  (s.g.  0-80)  and  water  are  mixed,  and  after  cooling 
to  the  original  temperature  the  mixture  is  found  to  have  contracted  in  volume 
by  2  % .     What  is  its  specific  gravity  ? 

2.  The  densities  of  three  liquids  are  as  1:2:3;  what  are  the  relative 
densities  of  mixtures  containing  (a)  equal  volumes,  (6)  equal  weights,  of  all 
three  ? 

3.  Distinguish  between  density  and  specific  gravity.  What  convenience  is 
there  in  keeping  to  c.g.s.  measure  ? 

How  would  you  determine  experimentally  the  specific  gravity  of  a  liquid 
of  which  only  a  few  drops  are  available  ?     (  X  2) 

4.  Given  a  hydrometer,  a  measuring  microscope  and  a  quantity  of  bacilli, 
how  could  you  find  the  mass  of  a  bacillus  ? 

5.  State  the  Principle  of  Archimedes. 

A  silk  balloon  weighing  150  kg.  contains  1000  cu.  m.  of  hydrogen  (density 
0-00009  gm.  per  c.c.)  and  is  surrounded  by  air  of  density  0-00129.  Calculate 
the  additional  weight  it  can  lift,  and  explain  why  the  fabric  of  the  upper 
part  is  tightly  pressed  out  by  the  contained  gas,  although  the  neck  is  open 
below.  Explain  also  exactly  why  the  balloon  will  float  in  stable  equilibrium 
at  a  constant  altitude. 

6.  Calculate  the  lifting  power  of  a  3-kgm.,  4-m.-diam.  spherical  balloon, 
containing  hydrogen.  Compare  the  volumes  of  hydrogen-  and  helium- 
filled  balloons  necessary  for  the  same  lift,  neglecting  difference  in  weight  of 
fabric. 

7.  Compare  the  lifting  power  of  a  non-expanding  (open  below)  balloon 
of  10,000  cu.  m.  capacity  originally  full  of  hydrogen,  at  sea-level,  and  at 
2000  m.  higher.     Barometer  falls  8-3  cm.  per  1000  m. 

8.  A  straight  tube  10  cm.  long,  of  glass  of  sp.  gr.  2-5,  its  lower  end  closed 
by  very  thin  glass,  is  upright  in  water.  The  cross-sectional  area  of  the  in- 
terior is  four-fifths  that  of  the  exterior.  At  what  level  will  it  float  if  empty, 
and  what  depth  of  water  inside  will  just  sink  it  ? 


I 


i 


FLOTATION   AND   SPECIFIC   GRAVITY  97 

;).  state  the  Principle  of  Archimedes  and  show  how  it  applies  to  the  common 
h\  (Irometer  of  constant  weight. 

Why  are  the  specific  gravity  graduations  on  these  instruments  not  equally 
spaced,  at  which  end  are  they  closer  together,  and  how  would  you  check 
ilioir  accuracy  ? 

10.  Explain  the  principle  of  the  common  hydrometer,  and  state  precisely 
w  hat  experiments  you  would  perform  to  check  the  accuracy  of  its  specific 
mavity  graduations. 

A  floating  hydrometer  indicates  s.g.  1-200.  Its  weight  is  30  gm.  What 
;i<l(litional  weight  will  depress  it  to  float  at  the  1-000  mark  ? 

11.  On  a  certain  hydrometer  stem  the  I-O  and  1-3  s.g.  marks  are  9  cm. 
apart;  where  are  the  1-1  and  1-2  marks  ? 

12.  A  hydrometer  to  measure  specific  gravities  from  1-2  to  1-4  has  stem 
r*  in.  long.     Find  length  of  stem  which  has  voliune  equal  to  bulb. 

13.  Give  the  principles  of  hydrometers  of  constant  volume,  and  of  constant 
w  light,  showing  how  to  calculate  the  graduations  of  the  latter.     (  x  2) 

14.  A  U  tube  contains  mercury  sp.  gr.  13-5  in  the  bend;  on  it  in  one  limb 
stands  20  cm.  salt  water  sp.  gr.  1-1,  in  the  other  10  cm.  of  ether  sp.  gr.  0-73. 
\\  hat  is  difference  in  merciu-y  levels  and  what  height  of  ether  added  would 
make  them  the  same  ? 

15.  Explain  how  the  '  gas-pressure  '  in  the  pipes  is  greatest  at  the  top  of 
the  house. 

16.  A  solid  weighed  in  air  14-86,  in  water  8-67,  in  a  liquid  9-85.  Find 
densities  of  solid  and  liquid,  explaining  why  they  are  densities. 

17.  A  can  of  water  stands  on  a  balance  pan.  Into  it  is  lowered  a  glass 
ball  of  s.g.  2-5  counterpoised  on  a  second  balance  by  200  gm.  What  alterations 
in  the  weights  restore  equilibriiun  ? 

18.  Prove  that  the  loss  in  weight  of  a  body  partly  immersed  in  a  liquid 
IS  (iqual  to  the  weight  of  the  liquid  displaced. 

A  uniform  solid  cylinder,  of  volume  2  cu.  ft.  and  s.  g.  0-5,  is  held  in  a  liquid 
of  s.g.  0-8,  axis  vertical,  by  a  vertical  string  attached  to  the  centre  of  the  top, 
with  J  of  its  length  immersed.     Find  the  tension  of  the  string. 

[A  cubic  foot  of  water  weighs  62-5  lb.] 

19.  A  piece  of  wood  4  ft.  long,  4  in.  wide  and  6  in.  deep,  weighs  42  lb.  Will 
it  float  in  (a)  water,  (6)  sea-water,  s.g.  1-03  ?  If  it  floats,  what  fraction  of  it 
projects  above  the  surface  ? 

20.  Explain  the  floating  of  a  body  which  is  partially  immersed.  Taking 
the  specific  gravity  of  ice  as  0-917  and  that  of  sea- water  as  1-025,  what  fraction 
of  the  total  volume  of  an  iceberg  is  not  immersed  ? 

21.  How  would  you  measure  the  volume  of  the  human  body  ?  If  a  150-lb. 
man  has  a  volume  of  2-4  cu.  ft.,  compare  his  buoyancy  in  river  and  sea. 

22.  A  body  floats  in  water  5/6  immersed.  A  3  cu.  in.  cavity  is  made  in 
it  and  it  now  floats  with  3/4  its  apparent  volume  inmiersed.  What  was  its 
volume  ? 

23.  A  metal  cube  weighed  42-5  gm.  in  air  and  37-5  in  water;  what  fraction 
of  it  is  immersed  in  mercury  in  which  it  is  floating  ? 

24.  State  the  Principle  of  Archimedes.  What  is  the  combined  weight  in 
water  of  15  gm.  wood  s.g.  0-6  and  57  gm.  lead  s.g.  11-4  ? 

25.  1  oz.  of  wood  sp.  gr.  0-5  is  just  sunk  in  water  by  a  stone  of  sp.  gr.  2-5. 
Find  weight  of  stone. 

26.  A  2-cm.  cube  floats  in  water  5/6ths  immersed  ;  a  O-S-cm.  cube  is  now 
attached  to  it,  and  both  are  just  submerged,  find  their  s.g.'s. 

27.  Oil  of  s.g.  0-90  floats  on  brine  s.g.  1-15,  a  golf  ball  dropped  in  rests 
with  2/3  its  bulk  in  the  brine,  what  is  its  s.g.  ? 

E 


98  MECHANICS 

28.  A  piece  of  cork  floats  on  water  with  1  /4  of  its  volume  immersed.  What 
effect  on  the  volume  immersed  in  the  water  would  be  produced  by  pouring 
petroleum  (density  =  0-8)  on  the  water,  so  as  to  cover  the  cork  completely  ? 

29.  A  bucket  half  full  of  water  is  filled  to  the  brim  with  petrol.  Several 
pieces  of  wood  are  now  put  in,  and  sink  below  the  surface  ;  discuss  the  effect 
on  the  total  weight  of  the  bucket. 

30.  A  block  of  non-porous  material  floats  exactly  awash  in  oil  of  s.g.  0-8 
inside  the  receiver  of  an  air  pump.  The  air  pressure  is  now  reduced,  what 
happens  ? 

The  oil  is  now  replaced  by  oil  of  vitriol  s.g.  1-6,  and  the  experiment  repeated  ; 
with  what  effect  ? 


See  also  under  BALANCE,  p.  121. 


PRACTICAL  QUESTIONS 

Measure  s.g.  and  deduce  diameter  of  wire,  or  length  by  diameter  and  vol. 
(in  water).     Find  the  density  of  mercury  by  a  tube  method. 

Measure  the  density  of  wax  by  the  hydrostatic  balance,  also  of  copper 
sulphate  crystal. 

Find  the  densities  of  insoluble  and  soluble  powders  by  s.g.  bottle. 

Make  a  straight-tube  hydrometer  and  test  it  at  two  points. 

Plot  the  density  of  a  salt  solution  against  its  concentration. 


I 


CHAPTER    IX 
ELASTICITY 


§  141.  In  elementary  mechanics  one  thinks  of  solids  as  rigid, 
retaining  their  shape  perfectly,  and  of  liquids  as  incompressible, 
retaining  their  bulk  perfectly,  whatever  forces  act.  Such  substances 
do  not  exist.  For  if  two  perfectly  rigid  bodies  ever  met,  the  abso- 
lutely instantaneous  change  of  momentum  on  contact  caused  an 
infinite  force,  which  broke  them,  as  they  would  not  otherwise  yield. 
Thus  the  sea  rounds  its  hard  pebbles,  and  grinds  their  debris  to  sand, 
which  the  wind  whirls  in  sand-devils,  and  chases  over  the  dunes, 
rounding  and  polishing  it  to  perfection.  Thus  sharper  sand  splinters 
off  the  glass  surface  at  which  it  is  blown  in  '  sand-blasting,'  but  falls 
harmless  from  the  gelatine  covering  parts  to  be  clear  in  the  pattern. 

All  solids  and  fluids  yield  more  or  less  to  force,  all  elastically 
regain  their  bulk  when  the  force  is  removed.  And  those  that  have 
a  shape  of  their  own  (solids)  either  elastically  regain  also  that  shape 
or  have  been  plastically  moulded  into  another. 

§  142.  Hooke's  Law.  Elasticity  was  studied,  in  what  at  first 
sight  seems  its  simplest  form,  by  Hooke.  He  hung  weights  on  a 
wire  and  measured  its  elongation,  and  summed  up  his  results  in  the 
law — Ut  tensio  sic  vis — '  as  the  stretching  so  is  the  force,'  i.e.  stretch 
and  force  causing  it  are  proportional  to  each  other. 

Robert  Hooke  was  born  at  Freshwater,  I.o.W.,  in  1635.  The 
great  practical  outcome  of  these  elastic  researches  was  his  invention 
of  the  Balance  Spring,  which  has  been  the  very  soul  of  all  watches 
ever  since.  In  clocks,  he  replaced  the  Verge  by  the  Anchor  Escape- 
ment, still  used  in  most  of  them.  He  made  the  first  great  improve- 
ments in  the  Compound  Microscope,  Fig.  245,  pubUshing  his 
Micrographia  in  1665.  In  the  following  year,  after  the  Great  Fire, 
he  became  City  Surveyor,  but  is  reputed  to  have  stored  the  wealth 
acquired  from  the  emoluments  of  this  post  in  an  iron  chest ;  while  he 
pursued  his  customary  studies  until  2  or  3  a.m.  He  was  one  of  the 
founders  of  the  Royal  Society  ;  d.  1702. 

Two  long  wires  of  the  same  metal  hang  from  the  same  hook,  thus 
eliminating  thermal  expansion  and  yielding  of  support ;  one  is 
stretched  by  a  constant  load  and  bears  a  scale  ;  alongside  this  moves 
a  vernier  attached  to  the  second  wire,  and  reads  its  elongation  as  its 
load  is  progressively  increased. 

Plotting  load  against  extension  gives  a  straight  line,  retraced  as  the 
load  is  reduced.     See  the  pecked  line  OE  for  wrought  iron,  Fig.  50. 

A  long  thin  heavily  loaded  wire  of  course  stretches  more  than  a 
short  thick  lightly  loaded  one.     To  get  a  number  depending  on  the 

99 


100  MECHANICS  [§  142 

nature  of  the  substance  alone,  one  must  adopt  a  standard  size  and 
force,  viz.  1  cm.  long  and  1  sq.  cm.  cross-section  (a  1-cm.  cube)  and 
1  dyne.  The  coefficient  of  linear  elastic  extensibility  is  the  fraction 
of  a  cm.  that  the  cm.  length  of  the  cube  stretches,  in  response  to  a 
pull  of  1  dyne,  applied  over  the  sq.  cm.  base.  It  is  very  small,  and 
the  smaller  the  less  yielding  and  '  stronger  '  is  the  substance.  This 
is  inconvenient,  and  one  inverts  it  and  defines  instead  Young's 
modulus  of  elasticity,  Y,  as  the  ratio  of  the  force  per  sq.  cm.  (the  tension, 
or  pressure)  to  the  elongation  or  compression  of  1  cm.  which  it  produces. 
Then  stretching  force  per  sq.  cm.  =  Y  x  elongation  of  1  cm., 

stretching  force  _  ^         total  elongation  of  whole  length' 

**^'  area  in  sq.  cm.  ~  whole  length  at  start         ' 

stretching  force 

^^       area  in  sq.  cm.        stress  ,  , 

or  X  =        ;         ..  =  — — — ,  see  below. 

elongation  strain 

initial  length 

Ex.  1.  A  wire  3  m,  long  and  0-8  mm.  diameter  is  stretched  1-5  mm.  by 
a  weight  of  5  kgm. ;  find  Y. 

Stretching  force  =  5  kgm.  =  5000  gm.  =  5000  X  981  dynes 

Area  of  cross  section  =  ttt"^  =  0-042  ^  22/7  =  0-0050  sq.  cm. 
Elongation  =  1*5  mm.  =  0*15  cm. 

Initial  length  =  3  m.  =  300  cm. 

5000x981/0-005       0-98x109        ,  ^^        ,„,„  , 

•••  Y  =      0-15/300 —  =    0-0005    =  2ii2ii21!  dy^««  p«r  «q-  ^^' 

Note. — ^The  common  words  stress  and  strain  have  acquired 
specialised  meanings  in  Elasticity  : — 

Stress  =  force  per  unit  area. 

Strain  =  change  of  length  per  unit  length,  or  of  volume  per  unit 

volume,  as  the  case  may  be. 
Then  a  Modulus  =  stress  ~  strain.     Strictly  speaking,  a  modulus 

should  be  called  a  Modulus  of  Resilience. 
And  Hooke's  Law  can  be  generalized  to.  Strain  oc  Stress. 

§  143.  This  sort  of  elasticity,  called  into  play  by  direct  push  or 
pull,  is,  however,  not  the  simplest.  For,  as  anyone  can  see  with 
india-rubber,  the  substance  contracts  sideways  as  it  is  stretched 
lengthways,  or  bulges  when  compressed,  changing  shape,  but  evading 
much  change  of  bulk.  A  body  undergoes  a  more  simple  elastic 
change  when  subjected  to  uniform  (fluid)  pressure  from  all  sides, 
it  contracts  in  bulk  without  change  of  shape  (except  crystals,  which 
are  unequally  elastic  in  different  directions),  the  diminution  in 
volume  per  c.c.  per  dyne/cm.^  pressure  =  its  coefficient  of  com- 
pressibility ;  the  reciprocal  of  this  is  its  bulk  Modulus,  B. 

For  a  good  many  solids,  bulk  modulus  is  roughly  equal  to  Young's 
modulus,  but  this  is  far  from  being  the  case  with  rubber,  which  is  a 
fairly  solid  substance  when  prevented  from  free  change  of  shape. 


I: 


§  144] 


ELASTICITY 


101 


Bulk  elasticity  is  the  only  possibility  in  liquids,  in  fact,  all  bulk 
moduli  are  measured  in  hydraulic  apparatus,  but  is  practically 
unimportant  m  solids,  as  it  cannot  break  them.  Young's  modulus 
oontrols  the  bending  of  beams,  carriage  and  clock  springs,  etc.,  for  the 
inner  side  is  directly  compressed  and  the  outer  stretched. 

Twisting  brings  in  a  third  species  of  elasticity  the  Modulus  of  which, 
that  of  Shear,  or  Rigidity,  is  less  than  Young's.  Bones  are  fractured 
most  easily  by  a  twisting  blow.  It  controls  the  strength  of  rotating 
shafting,  of  helical  springs  directly  pulled  (which  purely  twists  the 
wire),  of  resistance  to  shearing,  etc.  The  history  of  a  specimen 
twisted  to  destruction  resembles  that  of  one  broken  in  tension, 
which  follows  : — 

§  144.  Solids  acted  on  in  one  direction  by  great  forces  presently 
reach  an  Elastic  Limit.     Thereafter  their  modulus  has  little  or  no 


£5 

^ 

X.O 

"njinnlog 

«.p.j 

«^ 

y 

y 

WROUGH- 
Y 

■     IRON. 

io 

/ 

4 

''"^ElKren* 

orj   cria<)i3i{i« 

ilOOr.tem 

f 

s 

/ 

/ 

CAST 

IRON.^ 

- 

/ 

^-' 

■s- 

_J[ 

Exrentxoa 

o'                1 

o' 

STRAIN  -  EXTENSION     OF  10"    SPECIMEN. 

Fig.  50. 

meaning.  They  are  over- strained,  they  do  not  immediately  return 
to  shape,  they  retain  a  deformation  or  '  set,'  the  extent  and  perman- 
ence of  which  depend  largely  on  how  long  time  the  excessive  stress 
was  acting.  The  solid  has  begun  to  show  the  Plasticity  of  a  very 
viscous  fluid.  Watch  and  compare  the  yielding,  and  the  efforts  to 
return  to  shape,  of  a  stick  cut  from  the  hedge,  after  you  have  bent 
it  a  little,  for  however  long  ;  and  more  severely,  for  different  times. 

Wrought  iron  and  Steel  show  this  remarkably  well.  A  stress- 
strain  diagram  for  an  ordinary  10-in.  specimen  on  the  testing  machine 
is  given  in  Fig.  50.  The  specimen  stretches,  though  hardly  appreci- 
ably, with  perfect  elasticity  up  to  a  high  limit  E,  the  weigh-beam 
bouncing  under  the  hand.  Beyond  E  begins  a  permanent  set,  very 
slight  at  first,  but  at  the  *  yield-point '  Y  the  beam  drops,  and  the 
specimen  stretches  and  stretches  visibly,  though  slowly.  Presently, 
however,  it  recovers  itself  and  picks  up  the  load  again.  [Recollect 
that  steel  wire,  once  coiled,  never  comes  quite  straight  again ;  or 
that  rolled-up  papers  refuse  to  flatten  out.]     Increases  of  load  now 


102  MECHANICS  [§  144 

cause  very  large  plastic  extension,  but  slowly  coming  to  a  standstill 
for  each  load.  This  forms  a  new  elastic  limit  for  the  now  altered  and 
hardened  specimen  (as  at  17,  18,  21  tons).  Ultimately  the  specimen 
terminates  the  experiment  by  pulling  out  a  neck  and  breaking  there.  ^ 

The  elastic  limit  is  sometimes  poorly  marked,  e.g.  the  cast  iron  in  1| 
Fig.  50  gives  a  line  falling  away  from  the  direction  it  started  in 
almost  from  the  very  beginning,  i.e.  it  fails  to  obey  Hooke's  law  as 
soon  as  any  serious  stress  is  put  upon  it ;  it  has  no  definite  modulus. 
And  in  most  malleable  metals  the  plastic  stage  starts  gradually, 
without  any  remarkable  yield-point  (travelling  along  the  round 
dotted  line  in  Fig.  50).  The  proof  stress  required  to  produce  a  half 
per  cent,  permanent  elongation  is  a  more  useful  constant  for  struc- 
tural material  than  either  the  elastic  limit  or  the  yield  point. 

On  quasi-fluid  behaviour  depends,  of  course,  the  possibility  of 
drawing  into  wire  (ductility)  or  extruding  by  steady  pressure  or 
by  hammering  (malleability).  Under  the  microscope  it  is  observed 
that  the  constituent  crystals  of  the  mass  develop  layers,  which  glide 
over  one  another  without  loss  of  cohesion,  §  145.  Continued,  this 
process  develops  a  stream -line  structure,  recognized  as  the  grain  of 
wrought  iron,  and  as  a  fictitious  stratification  in  gneiss,  etc. — ^rocks 
crystallized  from  fusion  with  granite  structure  and  then  distorted 
by  earth  movements,  while  confined  under  pressure  too  great  to 
permit  their  losing  coherence  and  crumbling  up. 

Alternating  or  repeated  Stress  {e.g.  in  bicycle  forks)  has  been 
shown  to  be  perfectly  harmless  if  within  the  natural  elastic  limit,  but 
rapidly  destructive  if  a  trifle  beyond  it. 

A  true  elastic  limit  in  many  materials,  e.g.  glass  and  rubber — 
colloid  materials — is  very  low,  if  indeed  it  exists.  They  return 
quickly  nearly  to  shape,  but  not  quite,  the  latter  stages  of  the 
return  may  lag  for  minutes  or  hours. 

A  material  which  has  a  high  modulus  and  yields  but  little,  and 
when  over- stressed  cannot  save  itself  plastically,  is  brittle. 

Hard  substances  must  have  a  high  elastic  limit  and  an  enormous 
breaking  stress,  but  their  very  inflexibility  exposes  them  to  the  fate 
of  the  two  rigid  bodies.  A  mixture  of  hard  and  soft — glittering 
carborundum  set  in  concrete  ;  or  the  natural  mixture  constituting  a 
high-speed  tool-steel,  or  the  structure  of  your  teeth — often  displays 
their  good  qualities  to  best  advantage.  Hardness  varies  enormously. 
Think  of  it,  what  comparison  can  you  possibly  draw  between  the 
softness  of  your  own  body,  of  leaves,  and  living  things ;  and  the 
hardness  of  all  inanimate  mineral  matter  ? 

The  special  interest  of  this  paragraph  to  us  lies  in  this,  that  we 
ourselves  are  built  of  good  constructional  materials  :  bone  and  i 
muscle,  and  brain  and  nerve ;  of  necessity  they  obey  these  simple 
physical  elastic  rules.  They  have  their  elastic  limits,  vague  no 
doubt,  and  hard  to  fix,  but  existent  somewhere.  Within  these 
elastic  limits  they  serve  us  faithfully ;  but  we  want  to  grow  in 
strength  and  wisdom ;  how  set  about  it  ?  Plainly,  not  by  over- 
valiant  feats  in  the  field — ^they  will  only  land  us  in  bed  and  bandages. 

fl 


§  145]  ELASTICITY  103 

Nor  by  sudden,  intense,  and  excessive  *  burning '  of  the  midnight 
current — that  way  lie  disappointment  and  breakdown.  Up  to  the 
elastic  limit,  and  a  little  more,  and  then  '  sleep  on  it,'  day-by-day, 
patiently  :  in  Fig.  50  lies  the  physical  basis  of  all  sound  '  training.* 

§  145.  The  Hardness  of  metal  is  usually  tested  by  the  Brinell 
method  :  a  hard  chrome-steel  ball  is  pressed  by  a  heavy  load, 
30  kg.  X  (mm.  diam.)^  of  ball,  into  the  surface,  and  the  diameter 
of  the  shallow  depression  produced  ismeasuredby  a  small  microscope. 
For  heavy  sections  a  10-mm.  ball  is  standard,  but  for  light  articles 
with  polished  surfaces,  such  as  surgical  cutlery,  and  for  the  hardest 
materials,  a  1-mm.  diamond  ball  is  used. 

Sometimes  the  indentation  made  by  a  pjrramidal  point  of  diamond 
is  used  instead  ;  and  in  another  method  this  is  drawn  over  the  surface, 
under  a  load  increased  until  it  makes  a  visible  (0-01  mm.)  scratch. 

This  at  once  calls  to  mind  glass-cutting  :  brittle  as  glass  is,  the 
microscope  will  show  that  the  diamond  crystal,  the  faces  of  which 
meet  in  a  very  obtuse  angle,  actually  drags  up  the  surface  on  both 
sides  of  the  break,  just  as  your  finger-nail  will  drag  up  a  wet  rag 
spread  on  the  table,  as  you  draw  it  across. 

Also  the  questions  arise,  why  does  a  razor  or  a  scalpel  ever  lose  its 
edge,  when  it  is  never  asked  to  meet  obstacles  of  anjrthing  like  the 
hardness  of  steel,  and  why  should  stropping  it  on  leather,  with  no 
harder  dressing  than  rouge  (ferric  oxide),  restore  its  edge  ? — which 
is  actually  a  hemi-cylinder  of  about  0-2  micron  radius. 

The  answer  is  rather  a  shock  to  one's  notion  of  hard  steel :  it  is 
that  the  surface  layers  get  brushed  away  from  the  edge  by  the  work, 
and  are  coaxed  up  into  position  again  by  the  adhesive  drag  of  the 
strop,  much  as  you  can  wipe  melted  solder  about  on  the  surface  of 
the  copper  bit.  Of  course  this  is  a  small-scale  process,  a  polishing, 
with  atoms  as  the  moving  particles ;  and  if,  like  a  good  many  safety- 
razor  blades,  metal  has  never  been  ground  away  to  anything  near 
an  edge  to  start  with,  you  strop  in  vain. 

Metallic  bismuth  polishes  by  simply  rubbing  with  the  finger ;  a 
feather  edge  forms  on  the  edge  of  the  block  and  breaks  into  dust, 
and  this  dust  is  seen  under  the  microscope  to  consist  of  spherical 
droplets  0-01  mm.  diameter. 

In  crystals,  the  constituent  atoms  are  packed  in  perfectly  definite 
ways,  of  which  there  are  many,  forming  layer  on  layer,  as  all  packing 
does  ;  and  these  layers  are  bound  together  by  the  electrical  attrac- 
tions between  the  atomic  systems  which  constitute  Ck)hesion. 

One  can  imagine  these  layers  like  a  nobbly  board  lying  on  a  cobble- 
Btoned  roadway,  and  held  down  by  a  load,  the  rounded  projections 
fitting,  and  so  locking  together.  But  a  hard  enough  pull  will  jump  the 
board  along,  one  cobble,  and  it  settles  down  in  the  next  lot  of  hollows, 
and  holds  just  as  hard  as  ever,  and  so  on,  all  along  the  road  ;  yet  if 
you  get  it  on  the  run  it  will  glide  along  over  the  tops  quite  easily. 

'  Gliding  planes  '  like  this  can  be  developed  in  most  crystals ; 
some  felspars  are  as  full  of  them  under  the  polarizing  microscope  as  a 


104  MECHANICS  [§  145 

book  is  full  of  leaves ;  the  crystal  endures,  without  splitting,  a 
shearing  distortion  just  as  a  book  does.  This  is  how  a  metal  can 
yield  '  plastically,'  beyond  the  elastic  limit  :  the  gliding  planes 
actually  show  up  on  the  machined  surface  of  the  test-piece  as  a  fine 
parallel  striation ;  and  this  is  mainly  how  a  solid  glacier  manages 
to  flow  as  a  river  of  crystal  ice.  The  crystal  remains  just  as  solid  as 
ever,  the  electrical  forces  go  on  holding,  for  the  layers  of  atoms  never 
move  far  enough  apart  to  weaken  them,  during  the  motion.  If 
they  do,  the  crystal  'cleaves'  along  these  'cleavage  planes,'  as  do 
mica,  calcspar,  etc. 

Near  the  surface,  atoms  are  tied  in  place  from  below  only,  and  are 
consequently  by  no  means  so  tightly  held  as  in  the  mass  :  like  the 
gliding  board,  the  superficial  layers  can  run  as  liquid,  under  sufficient 
stress,  but  set  perfectly  solid  between- whiles  ;  the  surface  is  plastic. 
Fig.  50,  as  a  normal  state  of  affairs. 

On  the  crudest  scale  of  all,  the  farmer  harrows  his  ploughed  fields, 
dragging  down  the  ridges  to  fill  the  furrows,  and  they  soon  set  solid. 
Common  metal-work,  brass  taps,  etc.,  are  '  burnished  '  by  pressing 
against  a  rapidly  running  belt  faced  with  hard  material :  sectioned 
at  right  angles,  polished,  and  etched,  the  surface  layers  are  seen 
under  the  microscope  to  be  '  flowed  '  over  the  irregular  outlines  of  the 
crystalhne  mass  beneath.  The  flow  shows  little  structure,  but  is 
actually  harder,  more  resistant  to  both  mechanical  and  chemical 
interference  ;  it  is  '  packed  '  differently  from  the  regular  crystals  of 
the  metal. 

On  the  finest  scale  there  is  the  edge  of  razors  and  surgical  knives  : 
it  is  also  the  state  of  the  surface  of  Optical  Glass,  which  is  polished 
by  identically  the  same  treatment,  §  544  :  all  roughnesses  are  flowed 
full  of  material  which  has  never  let  go  and  been  crumbled  off,  as 
most  was  during  the  emery  grinding,  but  holds  exactly  as 
firmly  as  if  it  settled  there  from  fusion  or  from  solution.  And  why 
shouldn't  it  ? 

This  local  mobility  under  stress  is  paralleled  by  what  goes  on  inside 
the  tool  steel  under  warming.  Naturally,  a  1-3  per  cent,  carbon  tool 
steel  is  crystallized  in  a  condition  one  may  compare  with  wet  sand, 
the  binding  '  wet '  being  Cementite,  FcgC,  and  the  granules,  consti- 
tuting the  great  bulk,  PearUte,  which  is  pure  iron  interleaved  with 
a  little  cementite  in  a  laminated  mother-of-pearl  structure.  Heated 
to  700°  C,  the  cementite  attacks  the  pearlite,  and  by  bright  redness 
between  800°  and  900°  the  steel  has  become  a  uniform  '  solid  solu- 
tion '  called  Austenite — which  is  not  exactly  a  mobile  liquid,  as  you 
find  when  you  start  to  hammer  it  into  shape.  Quenched  suddenly 
in  cold  water,  this  shows  an  exceedingly  fine  acicular  marking — 
something  moved,  but  not  much — and  is  now*Martensite.  Reheated 
merely  to  230°  C. — little  more  than  the  temperature  of  the  water 
in  a  locomotive  boiler — it  begins  to  change  ;  the  steel  of  the  razor 
edge  tempered  at  this  heat  is  not  quite  the  brittle  martensite ;  by 
100°  hotter  (the  melting  point  of  lead),  the  temper  has  been  let  down 
right  through  the  whole  range  of  cutting  tools  and  springs  ;  at  400? 


I 


I" 


§146] 


ELASTICITY 


105 


it  enters  a  rather  useless  stage,  called  Troostite,  and  at  500°,  the 
dull  red  at  which  the  blacksmith  leaves  off  hammering,  a  scattering 
of  fine  microscopic  droplets  of  cementite  shows  throughout  the  mass, 
constituting  Sorbite,  extremely  strong, 
but  quite  ductile. 

Even  without  heating,  motion  is 
going  on  in  a  solid.  The  crystal '  grain ' 
of  lead  increases  rapidly  month  by 
month,  etching  entirely  differently  six 
months  after  manufacture,  indicating 
an  atomic  repacking.  Watch-main- 
springs '  crystalHze,'  and  snap  '  un- 
i  accountably ' ;  like  elm  boughs.  At  the 
'Mint,  a  gold  coin  blank  was  welded  to 
the  end  of  a  round  rod  of  lead,  which  lay 
'  on  the  shelf  for  ten  years,  and  was  then 
ishced  up  and  assayed,  and  gold  was 
1  found  to  have  diffused  even  six  coin- 
,  thicknesses  down  the  bar.  These  two 
I  metals  were  chosen  merely  for  chemical 
convenience,  there  is  nothing  excep- 
tional about  them ;  zinc-aluminium 
I  die-castings  are  notorious  for  growing 
; larger  ;  etc.,  etc. 

It  would  be  a  hasty  Hamlet  who 
would  complain  about  this  '  too  too 
sohd '  anything  nowadays. 

I  §  146.  The  Elasticity  of  Gases  (volume 
compressibility)  can  be  investigated 
throughout  far  greater  change  of  bulk 
than  can  that  of  solids  or  liquids. 
Hooke's  law  still  holds  for  small 
changes  {e.g.  §  395,  compression  in 
sound  waves),  but  fails  for  greater, 
and  is  superseded  by 

Boyle's  Law.  At  constant  temperature, 

tite  volume  V  of  any  particular  mass  of 

\<iny  gas  varies  inversely  a^  its  pressure  P. 

P  oc  ^  or,  alternatively,  PV  oc  1, 

i.e.  PV  is  constant  for  a  constant  mass 
of  any  particular  gas  at  a  constant 
temperature. 

[When  you  are  asked  for  Boyle's 
Law,  one  of  these  alternative  state- 
ments must  be  given  in  full.] 

This  relation  between  P  and  V  is  graphically  expressed  by  the 
hyperbola  of  Fig.  51. 


77    \ 

6o       ] 

PV- 

'90G 

v*^. 

V 

\ 

Q. 

V 

O         I 

f        .3. 

V 

0        4 

AB 

5        6 

0         7 

'    .* 

Fig.  51. 


106 


MECHANICS 


[§  1461 


Robert  Boyle  was  the  seventh  son  of  the  first  Earl  of  Cork ;  h 
resided  in  Dorset,  in  Oxford,  and  ultimately  in  London,  devotin[ 
himself  to  philosophical  pursuits,  and  being  one  of  the  founders  o 
the  Royal  Society.  In  1662,  in  a  research  '  Touching  the  Spri: 
of  the  Air,*  he  established  this  law  by  aid  of  a  U-tube  containing 
air  in  its  short  sealed  limb,  shut  in  by  an  increasing  height  of  quick- 
silver in  its  long  open  limb.  He  found  that  an  extra  29  in.,  which 
amounted  to  doubling  the  barometric  pressure,  halved  the  volume 
of  the  imprisoned  air. 

The  laboratory  apparatus  of  Fig.  51  consists  of  two  tubes  a  foot 
long  connected  by  bicycle-pump  tubing.  Air  or  other  gas  ia 
enclosed  in  the  left-hand  tube  between  mercury  and  the  flat  sealed-in 
stopper  ;  its  volume  is  proportional  to  its  length  AB. 

The  pressure  it  is  sustaining  is  that  due  to  the  extra  height  ol 
mercury  BC,  plus  the  barometric  height  H  representing  the  atmos 
pheric  pressure  on  the  top  of  that  mercury  in  the  open  tube.  With 
this  apparatus,  taking  care  not  to  warm  the  air  by  sudden  compression) 
or  by  handling,  one  can  prove  that 

PV  =  (H  db  height  BC)  X  (length  AB)  =  constant 

[-[-  in  (i)  above  atmospheric  pressure  and  —  in  (ii)  below  atmo- 
spheric pressure  H].  The  two  positions  shown  lie  on  the  hyperbolic 
curve  beneath. 


8o             iSo 

Zi,o 

PRESSURE        IN 

ATMOS 

Fig.  52. 

320 


§  147.  Boyle's  Law  has  been  tested  to  high  pressures  by  using  verj 
tall  mercury  columns.     The  product  PV  decreases  at  first  for  al 


§149]  ELASTICITY  107 

gases  except  hydrogen,  but  at  higher  pressures  it  increases  gradually 
for  all,  Fig.  52.  Anywhere  near  their  liquefying  points,  gases 
col  lapse  with  undue  ease  ;  compare  COg  at  40°  and  at  200°  ;  hydro- 
izcii,  of  course,  is  very  far  from  liquefaction. 

§  148.  Work  absorbed  in  elastic  stretching.  If  a  specimen  steadily 
stretches  a  distance  e  under  a  force  which  has  steadily  increased 
from  zero  to  F,  averaging  therefore  JF,  the  work  done  is  the 
product  JeF. 

Thus  elastic  materials  give  under  a  blow,  absorbing  its  energy 
without  fracture  or  permanent  deformation  ;  hence  their  great  value. 
Most  of  the  energy  is  returned  as  the  stress  passes,  and  the  strain 
relieves  itself  ;  herein  lies  the  use  of  Springs  of  every  sort. 

India-rubber  by  reason  of  its  enormous  extensibility  can  store, 
per  pound,  10  times  as  much  elastic  energy  as  spring-steel  (instance 
its  use  in  toy  aeroplanes,  etc.),  but  on  account  of  its  elastic  lag  does 
not  restore  it  all,  losing  a  little  in  that  internal  friction  which  accounts 
for  the  lack  of  '  life  '  in  a  slack  bicycle  tyre^  and  for  the  heat  deve- 
loped in  motor  tjrres,  especially  when  under- inflated,  and  therefore 
flexing  unduly  under  load.  Most  of  the  loss,  however,  is  in  the 
'  fabric' 

Weight  for  weight,  no  solid  can  compete,  in  storage  of  elastic 
energy,  and  its  restoration,  with  a  compressed  gas  :  the  pneumatic 
tyre  and  the  inflated  ball  are  supreme. 

Sometimes,  of  course,  one  wants  the  energy  of  the  blow  smothered 
and  not  flung  back ;  the  soft  answer  that  tumeth  away  wrath. 
Ordinary  car- springs  effect  this ;  their  leaves  slide  on  one  another 
with  much  friction,  the  rebound  is  less  than  the  blow,  and  the  car 
does  not  go  on  bouncing,  even  without  the  further  aid  of  accessory 
shock-absorbers . 


ELASTIC  DATA 

§  149.  Some  values  of  Young's  modulus  are,  in  millions  of  millions 
of  djrnes  per  sq.  cm. :  Steel  2-0  ;  copper,  brass,  and  bronze  0-75  to 
1-0  ;  quartz,  glass,  and  rocks  1-5  ;  wood  0-2  to  0-1  ;  catgut  and  silk 
0-03  ;  vulcanised  rubber  10  million  only. 

Rigidity— steel  0-8  ;  copper,  etc.  0-3  to  0-4  ;  glass  0-17  to  0-24. 

Some  liquid  compressibilities  are,  in  milUonths  of  millionths 
of  the  original  volume  for  I-  dyne  per  sq.  cm.  :  Water  50, 
glycerine  25,  various  oils  48,  alcohol  90,  ether  140,  at  ordinary 
temperatures. 

Any  gas  at  a  million  dynes  pressure,  which  is  nearly  1  atmo., 
compresses,  of  course,  a  millionth  for  1  additional  dyne,  and  is 
therefore  20,000  times  more  compressible  than  water. 


108  MECHANICS 


EXAM   QUESTIONS,  CHAPTER   IX 

Study  §  142,  it  is  troublesome  to  recollect,  and  do  it  in  the  lab.  for  wire  or^ 
for  rubber.  §  145  is  for  those  who  feel  interested,  §  146  is  the  first  thing  an 
examiner  would  ask  about  if  suddenly  roused  from  sleep — it  controls  all  gas 
calculations;  §  147  you  will  read  with  §  296;  §  148  is  the  antithesis  of  §  141, 
and  is  a  phase  in  the  all-important  Avoidance  of  Shock  dealt  with  further 
in  §  295. 

1.  State  Hooke's  Law  of  Elasticity. 
Define  Young's  modulus  and  explain  how  you  would  determine  it  for  a 

steel  wire.     What  is  its  value  if  a  wire  3-5  m.  long  and  1-5  mm.  diam.  if 
stretched  2-5  mm.  by  a  load  of  5  kgm.  ?     (  X  4) 

2.  Define  a  Modulus  of  Elasticity.  Show  that  the  work  done  in  stretching 
a  cord  or  spring  is  half  the  product  of  the  stretching  force  and  the  extension. 
Calculate  the  energy  stored  in  a  spring,  which  1  gm.  wt.  stretches  1  cm.,  when 
stretched  10  cm. 

3.  Define  Work  and  Power,  and  their  units. 
If  a  pull  of  20  lb.  increases  the  length  of  a  chest-developer  from  18  to  32  in.,( 

calculate  the  work  done,  assuming  Hooke's  law  obeyed.     Calculate  also  tho^ 
horse-power,  if  the  developer  can  be  stretched  forty-five  times  a  minute. 

4.  Into  a  vertical  cylinder,  area  12  sq.  in.,  length  8  in.,  closed  below,  a 
4-lb.  piston  is  inserted.  Where  will  it  rest,  atmospheric  pressure  being 
15  Ib./sq.  in.? 

5.  State  carefully  Boyle's  Law,  and  how  you  would  verify  it.  A  1-m. 
glass  tube,  closed  at  the  top,  is  forced  vertically  half  under  mercury.  What 
is  the  pressure  inside  it,  the  barometric  height  being  75  cm.  ? 

6.  Find  the  internal  volume  of  an  oxygen  cylinder  to  hold,  at  120  atmos., 
20  cu.  ft.  of  oxygen  under  normal  conditions. 

7.  A  long  narrow  vertical  tube,  closed  at  the  lower  end,  contains  2  ft. 
length  of  air  shut  in  by  2  ft.  length  of  oil.  When  inverted  so  that  the  open 
end  is  down,  the  air  expands  to  2  ft.  2 J  in.  Calculate  the  height  of  the  oil 
barometer. 

8.  Air  at  atmospheric  pressure  (32  ft.  water  barometer)  is  taken  down  to 
30  ft.  under  water  and  liberated  to  form  a  spherical  bubble.  Show  diameter 
of  this  has  increased  one-fourth  when  it  reaches  sm-face. 

9.  Give  a  careful  statement  of  the  law  relating  the  pressure  and  volume 
of  a  mass  of  gas  at  a  fixed  temperatm-e.     A  litre  fiask  has  a  neck  1*4  cm. 
diameter;   what  force  will  be  pressing  on  the  cork  at  a  depth  of  10  m.  in  a* 
lake,  and  how  much  water  will  enter  if  it  gives  way  ? 

10.  What  is  meant  by  the  pressure  at  a  point  in  a  fluid?     A  cylindrical i 
vessel  is  lowered  into  the  sea,  open  end  downwards.     What  would  be  the" 
reading  of  a  suitable  aneroid  inside,  when  the  water  had  risen  a  third  way  up  ? 
At  what  depth  would  this  occur  ?     Atmospheric  pressiire  =  76  cm.     Density 
of  sea  water  =  1-025. 

11.  A  Kelvin  sea-sounding  tube,  60  cm.  long  (closed  at  the  top),  leaving] 
the  surface  full  of  air,  is  sunk  in  sea-water  density  1-03.  When  raised,  there' 
are  indications  that  the  water  has  risen  up  45  cm.  along  the  tube  :  what  isn 
the  sounding  ?     Barometer  750  mm. 

12.  Show  in  a  diagram  the  relation  between  pressure  and  volume  of  a  mass 
of  gas  at  constant  temperatm-e.     A  Boyle's  law  tube  contained  20  c.c.  of  airw 
at  9'5  cm.  apparent  mercm-y  pressure,  and  12  c.c.  at  65-5  cm.;   what  was  the ^ 
barometric  pressure  ? 

13.  State  Boyle's  law  and  describe  how  to  verify  it  for  pressures  less  thaiu 
atmospheric. 


ELASTICITY  109 

When  10  c.c.  of  air  at  N.T.P.  are  introduced  above  the  merciuy  in  a  baro- 
meter tube,  this  is  depressed,  leaving  a  volume  of  15  c.c.  at  the  top.  What 
is  its  final  height  ? 

14.  The  Torricellian  space  in  a  barometer  at  30  in.  being  2-5  cu.  in.  and 
(  r  ss-section  of  tube  0*5  sq.  in.,  an  air  bubble  measiu'ing  0-1  cu.  in.  at  atmo- 
s[-horic  pressiu-e  is  admitted.     How  far  will  the  mercury  fall  ? 

]  5.  A  faulty  barometer,  with  tube  40  in.  long,  contains  air  and  reads  29  in. 
iiLstead  of  30.     What  will  it  read  when  true  barometer  reads  29  ? 

16.  A  breathing-bag,  with  face-mask,  oxygen,  and  caustic  soda,  is  issued 
to  crews  of  submarines.  Show  how  a  man  wearing  this  may  escape  from  a 
grounded  submarine,  and  rise  with  increasing  speed  to  the  surface. 

17.  Why  can  a  balloon  remain  at  rest  in  equilibriiun  with  the  fluid  surroimd- 
ing  it,  while  a  submarine  cannot  ?  What  happens  to  a  balloon  when  the  sun 
shines  on  it,  and  to  a  submarine  when  it  comes  into  fresh  water  ? 

18.  Explain  by  a  diagram  the  action  of  a  piston  pump  in  exhausting  air. 
Kthe  vessel  is  of  1000  c.c,  and  the  barrel  of  the  pump  100  c.c,  what  are  the 
theoretical  values  of  the  pressure  of  the  air  remaining  after  one,  two,  and 
three  complete  strokes,  neglecting  any  change  of  temperature  ?     (  X  2) 

19.  Describe  some  form  of  pump  suitable  for  the  production  of  high  vacua, 
and  explain  its  action.     (  X  6) 

How  would  you  measure  the  remaining  pressure  ? 

20.  The  air  in  a  bulb  of  capacity  30  c.c  is  compressed  into  a  capillary 
tube  5  cm.  long  and  1  mm.  in  diameter,  and  the  pressiu-e  of  the  g&a  is  found 
to  be  2  cm.  of  mercury.     What  was  the  original  pressure  ? 


PRACTICAL    QUESTIONS 

Measure  Young's  modulus  for  rubber;   and  ditto  for  wire. 

Measure  the  work  done  in  stretching  a  rubber  band. 

Find  how  the  period  of  oscillation  of  a  spring  depends  on  the  suspended 
load. 

Investigate  Boyle's  Law  and  plot  graphs,  or  deduce  bfiurometric  height. 

No.  7  above,  often  with  mercury. 


CHAPTER   X 


THE   PRECISE   MEASUREMENT   OF  LENGTH,   TIME, 

AND  MASS 


THE   MEASUREMENT   OF   LENGTH 

§  151.  The  difficulty  in  using  subdivided  scales  to  read  ordinary 
lengths  very  closely  is  that  the  subdivisions  soon  become  too  smal 
to  see.  Without  a  magnifying-glass,  it  is  better  to  guess  at  th( 
decimals  of  a  tenth-of-an-inch  division  rather  than  attempt  to  re? 
a  hundredth-inch  scale. 

To  a  soldier  of  fortune,  Pierre  Vernier  {ca.  1620),  is  due  the  con^ 
trivance  most  widely  used  in  reading  scales  on  all  sorts  of  instruments. 
The  main  scale  is  graduated  throughout  into  equal  divisions 
small   as   can   be   distinguished   comfortably.      Attached   to   thej 
moving  part,  and  sliding  alongside  the  main  scale,  as  in  Fig.  53," 


I,  U,  1,1,1  ,1 


I    I    I    I    I 


1 1  iV 


I   I  I 


M  I  I  I  I 


0 

Li 


rr 


U 


10 

jl 


10 


I.  I  I  I  1  M 


^ 


zo 


Fig.  53. 

is  another,  the  vernier,  each  of  the  divisions  of  which  is  one  nth 
part  less  than  those  of  the  main  scale.  Evidently  n  divisions  of 
this  fall  short  n  nths  =  1  whole  scale  division  ;  or  n  vernier  divisional 
=  71—1  scale  divisions  (in  the  simplest  form  10  and  9,  Fig.  53,1 
top  scale  ;   in  the  lower,  30  and  29). 

Now,  if  the  index  mark  on  the  vernier  (either  its  edge  or  elsei 
marked  with  an  arrow  or  O)  lies  in  line  with  a  scale  mark,  them 
the  mark  1  on  it  falls  short  of  a  scale  mark  by  1/nth  scale  division, » 
mark  2  by  2/nihs,  etc.  Pushing  the  vernier  forward  2/?iths  willl 
therefore  bring  the  2  into  line  with  a  scale  mark,  and  so  on,  coin- 
cidence at  the  mth  vernier  mark  meaning  that  it  has  been  bodily 
pushed  m/7iths  of  a  scale  division  beyond  the  last  scale  mark  pre- 
ceding its  index. 

Thus  the  vernier  always  reads  nths  of  the  smallest  scale  division, 
say  tenths  of  the  tenth  of  an  inch,  tenths  of  a  milHmetre,  thirtieths 
of  half-a-degree,  etc.,  e.g.  the  upper  verniers  in  Fig.  53  are  indicating 

110 


152] 


PRECISE  MEASUREMENTS 


111 


0-0  and  17-4 ;  and  the  lower  vernier  is  reading  12/30th8  beyond 
50-5°  on  the  scale,  i.e.  12'  of  arc  plus  50°  30',  which  is  50°  42'. 

Sometimes  the  vernier's  10  divisions  =  19  of  the  scale,  as  on  the 
Fortin  barometer.  Fig.  35,  left  scale  ;    this  is  merely  to  get  more 

*  open  '  divisions  and  avoid  dazzling  the  eye  ;  it  is  a  vernier  to 
twentieths  with  alternate  marks  invisible,  only  even  twentieths  can 
be  read,  i.e.  tenths. 

[The  original  vernier,  to  be  found  on  old  instruments,  had  its 
divisions  1/nth  greater  than  the  scale  divisions,  and  read  backwards.] 
Grct  perfectly  used  to  vernier  calUpers  in  the  laboratory. 

§  152.  Better  than  a  long  vernier  is  the  Micrometer  Screw. 
A  true  screw  will  advance  through  a  perfectly  fitting  nut  1/wth 
its  Pitch,  i.e.  its  distance  from  thread  to  thread,  for  each  1/nth  of  a 
revolution.  In  practice,  a  well-made  screw  and  nut  are  carefully 
ground  together  to  smooth  away  irregularities  ;  there  must  be  some 
clearance  between  screw  and  nut,  and  this  gives  rise  to  shake  and 

*  back-lash'  —  travel 
of  the  screw  without 
turning,  or  vice  versa. 
To  avoid  these,  lubri- 
cate well  with  grease, 
take  readings  with  the 
screw  going  always  in 
the  same  direction, 
and  if  possible  have  a 
spring  to  press  screw 
and  nut  together,  always  one  way,  with  a  steady  pressure. 

The  head  of  the  screw  is  enlarged  and  graduated  into  a  large 
number  of  equal  parts.  A  very  common  arrangement  has  a 
J-mm.  pitch  screw  and  50  divisions  on  the  head.,  and  therefore  reads 
to  hundredths  of  a  millimetre.  A  scale  to  read  whole  turns  of  the 
screw  is  provided,  but  be  careful,  it  marks  only  every  second  turn ; 
watch  out  for  odd  half -mm. 

In  the  screw-gauge.  Fig.  54,  the  flat  end  of  the  screw  works  up 
to  a  flat  anvil  formed  on  an  extension  of  the  nut,  the  object  the 
thickness  of  which  is  required  being  put  between,  and  very  genUy 
gripped.  The  large  micrometer  benches  used  by  engineers  are  m 
principle  screw-gauges  with  long  adjustable  gaps,  which  are  first 
standardized  by  using  *  end  measure  bars '  of  known  length.  Of 
these,  those  most  in  use  are  steel  bars  of  cross-section  about  the  size  of 
a  postage-stamp,  and  with  their  ends  so  true  and  flat  that  when 
»noistened  with  paraffin  oil  they  can  be  wrung  together  end  to  end, 
and  stick  firmly.  From  a  set  of  these,  of  lengths  arranged  like  the 
weights  in  a  weight-box,  any  required  length  can  be  built  up,  a 
milUonth  of  an  inch  being  allowed  for  each  oil-film. 

In  all  such  length  measurements  great  care  must  be  taken  as  to 
temperature,  §  174. 


Fio.  54. 


112 


MECHANICS 


[§152 


Most  micrometer  screws  nowadays,  in  instruments  of  all  sorts, 
are  of  half -millimetre  pitch,  makers  wisely  concentrating  on  the 
manufacture  of  one  good  screw.  (English  screw-gauges,  however, 
employ  a  1/40-inch  pitch  screw,  and  graduate  to  25ths,  reading 
thousandths  of  an  inch,  '  mils  '.) 

In  the  screw  spherometer  the  screw  is  mounted  so  that  it  can 
measure  small  heights,  especially  the  height  CZ  =  h,  Fig.  55,  of 
the  arc  of  a  curved  surface  above  its  chord  AB  joining  rigid  pro- 
jections, or  feet  on  its  nut  (for  the  moment,  imagine  one  at  B). 
This  measures  the  Curvature  of  the  surface,  1/R,  the  reciprocal  of 
the  radius  with  which  it  was  struck. 

By  a  property  of  the  circle  (Euclid  III.  35) 

AC  X  CB  =  CZ  X  the  large  remaining 
part  of  its  diameter 


r^  =  h  X  (2R 
h^  =  2nh. 


h) 


It  is  seldom  that  one  has  to  measure 
a  lens  or  mirror  so  strongly  curved  that 
h^  is  as  much  as  1/400  of  r^ ;  so  that 
within  ordinary  limits  of  experimental 
error 

r2  =  2Rh 


Curvature  ^^ 

IS) 


2h 


55,  of  lens- 
laboratories 


Fig.  55. 


The  Spherometer,  Fig. 
makers  and  physical 
usually  has  three  feet  (points,  or  better, 
small  steel  balls)  fixed  at  the  corners  of 
an  equilateral  triangle,  and  each  distant 
r  cm.,  to  be  measured  carefully  with 
vernier  calHpers,  from  the  screw-point, 
when  standing  on  a  plane.  This  enables 
it  to  stand  alone,  and  makes  no  differ- 
ence to  the  reading  on  a  sphere  ;  the  foot  B  has  virtually  split  into 
two,  and  one  has  travelled  to  60°  E.  and  the  other  to  60°  W.  longitude 
round  the  '  small  circle  of  latitude  '  in  which  the  plane  ACB  cuts 
the  sphere.  On  a  cylinder  the  pattern  with  feet  in  line  (see  lens- 
gauge.  Fig.  201)  reads  full  curvature  one  way  and  zero  the  other,  the 
tripod  reads  half  curvature  in  any  position. 

Contact  of  the  screw-point  is  indicated  by  the  instrument  beginning 
to  be  able  to  spin  round,  or  just  perceptibly  totter.  Find  the  zero 
on  a  piece  of  flat  plate-glass,  and  then  keep  cautious  tally  of  all 
graduations  passing,  usually  200  to  the  mm.,  until  it  stands  true  on 
the  curved  surface. 

In  the  much  more  convenient,  and  moderately  accurate,  spring- 
spherometer  or  '  lens -gauge,'  used  by  spectacle -makers.  Fig.  201, 
three  points  in  line  are  pressed  on  the  surface.     The  middle  spring 


165] 


PRECISE   MEASUREMENTS 


113 


point  gives  way,  measuring  h  ;  and  its  motion,  magnified  by  multi- 
plying gearing,  moves  a  pointer  round  a  dial  equidistantly  graduated 
in  Curvatures  1/R,  in  units  to  be  described  in  §  509. 

In  dividing- engines,  travelling  micrometer-microscopes,  catheto- 
meters,  etc.,  the  end  of  the  screw  presses  on  a  carriage  sliding  on 
'  geometrical  ways,'  and  carrying  the  cutting  tool,  or  a  cross- wire 
microscope  or  telescope.     See  Fig.  61  and  §  622. 

Most  Microscopes  depend  for  their  fine-focussing  adjustment  on  a 
micrometer  screw,  working  against  a  rather  strong  spring  to  prevent 
back-lash. 

§  153.  Area.     Areas  of  irregular  shape  are  measured  : 

(1)  By  tracing  on  squared  paper  and  counting  squares. 

(2)  By  tracing  on  ordinary  paper  of  uniform  thickness,  cutting 
out,  and  weighing. 

(3)  By  Planimeters,  instruments  which  in  one  way  or  another 
'  integrate  '  the  area  as  their  tracing  point  is  carried  once  round  it. 

The  Area  of  a  Circle  is  nr^  =  3-1416  x  radius^,  of  a  Sphere  is  ^ttt^. 

§  154.  Volume.  The  volume  of  a  parallel-sided  block,  or  cylinder, 
whether  rectangular  or  obUque,  is  area  of  base  X  height  perpen- 
dicular to  it.     Of  a  pyramid  or  cone,  one-third  this. 

The  Volume  of  a  sphere  is  (4/3) rr^. 

The  volumes  of  irregular  solids  are  easily  measured  by  dropping 
them,  like  the  thirsty  crow  in  the  fable,  into  water  or  any  other 
liquid  partly  filling  a  jar  graduated  in  cubic  centimetres. 

Or  they  =  loss  of  weight  in  water,  §§  132,  138. 

§  155.  The  measurement  of  angle.  Angles  are  measured  in 
degrees  (360  to  the  ckcle),  minutes  (60'  =  1°),  and  seconds  (60"  = 
1').  Practically,  their  measurement  is  that  of  distances  round  a 
circular  scale,  verniers,  micrometers,  etc.,  being  employed.  At 
Greenwich  angles  are  quoted  to  a  hundredth  of  a  second,  about  the 
thickness  of  this  paper  at  a  distance  of  a  mile. 


Fig.  56. 

But  the  Natural  Measure  of  Angle  is  (Length  of  arc-fradius),  and 
is  in  Radians,  largely  used  in  theory,  although  awkward  to  graduate 
on  circles,  the  circumference  being  2-k  radians. 

Thus  in  the  four  figures  the  angle  A  is  measured  by  stepping 
along  the  curve,  and  then  expressing  its  length  as  a  fraction  of  its 
radius  r. 


114  MECHANICS  [§  155 

When  the  angle  is  small  you  see  that  another  common  everyday 
way  of  measuring  it,  as  a  Gradient,  of  height  reached  in  distance 
along,  comes  to  the  same  value,  for  the  difference  in  length  between 
the  little  curve  and  its  straight  chord  is  insignificant.  What  is 
more,  it  does  not  matter  whether  you  measure  distance  along  by 
pacing  it  out  up  the  slope,  or  whether  you  stay  at  home  and  measure 
it  on  the  flat  map  of  the  district. 

In  the  second  figure  it  visibly  does  begin  to  matter :  the  arc  is 
longer  than  CB,  i.e.  the  angle  is  really  larger  than  the  gradient 
expressed  as  CB  in  r.  At  the  same  time,  it  is  smaller  than  the  '  map  ' 
gradient  CB/AB,  for  by  similar  triangles  this  is  the  same  as  DE/AE, 
and  DE  is  longer  than  CE. 

In  the  third  figure,  A  is  half  a  right  angle,  the  mountain  slope  is 
1  in  1-4  actual  climbing  distance,  but  1  in  1  on  the  map  ;  and  in 
the  fourth  figure  A  is  1  radian  (57-3°),  CB/r  is  0-84,  and  CB/AB  = 
1-56. 

The  expression  of  the  slope  as  perpendicular  -^  actual  climbing 
distance,  CB/r,  is  called  the  Sine  of  angle  A,  sin  A 

and  the  gradient  perpendicular  -:-  plan  distance,  CB/AB,  is  the 
Tangent  of  A,  tan  A  (being,  as  in  the  second  and  third  figures,  the 
length  of  the  actual  tangent  -^  radius  of  circle). 

So,  when  you  meet  with  Sines  and  Tangents  later  in  this  book, 
don't  call  them  incomprehensible  trigonometrical  functions. 

Here  also,  for  reference,  are  some  simple  and  handy  '  results  of  the 
binomial  theorem  '  : — 

Approximately,  when  a  and  b  are  small 

N  (1  ±  a)  (1  ±  6)      =^{l±a±b) 

N  (1  ±  «)  ^  (1  ±  6)  =  N  (1  ±  a  T  6) 
N  (1  zfc  g)^  =l^{l±ma) 

N  X  VI  ±  «  =  N  (1  ±  a/m) 

Check  them  for  yourself,  if  you  feel  doubtful. 


THE   MEASUREMENT   OF  TIME 

§  166.  How  Time  is  defined  you  read  in  §  3,  how  the  pendulum 
ticks  it  out  in  §§  84,  85,  or  the  balance-wheel  in  §  90 ;  and  how 
these  are  preserved  from  a  major  source  of  irregularity  you  will  find 
in  the  next  chapter,  §  175.  Other  troubles  arise  from  the  fact  that 
the  arc  of  swing  of  a  pendulum  cannot  be  infinitesimal,  but  must 
be  kept  constant  §  85 ;  that  barometric  changes  alter  the  mass  of 
air  clinging  to  the  pendulum,  the  air- resistance  to  its  swing,  and  the 
buoyancy,  §  163,  which  opposes  gravity ;  that  clock-oil  thickens, 
and  that  every  impulse  given  it,  except  exactly  at  mid-swing, 
upsets  all  timekeeping.     For  plainly  enough,  if  a  pendulum  bumps 


^'\ 


156] 


PRECISE   MEASUREMENTS 


115 


into  anything  at  either  end,  it  gets  bounced  back  sooner,  so,  since 
you  must  hit  it  to  keep  it  going,  hit  it  only  at  mid-swing,  so  avoiding 
all  buffering  action. 

This  is  not  a  treatise  on  clockmaking,  but  here  is  the  action  of  the 
Shortt  Clock  : 

In  the  deepest  dungeon  of  the  old  castle  of  Duke  Humphrey  of 
Gloucester,  underlying  Greenwich  Observatory,  on  pillars  of  masonry 
4  ft.  thick,  hang  two  seconds  pendulums,  the  one  swinging  N.  and 
S.,  the  other  E.  and  W.  They  swing  in  vacuo,  at  only  2  cm.  mercury 
pressure  ;  they  are  of  Invar,  §  174,  but  as  a  further  precaution  the 
temperature  of  the  clock-room  is  kept  constant  by  thermostat, 
§  204. 

Seven  inches  down,  the  pendulum-rod.  Fig.  57,  M,  carries,  in 
jewels,  a  little  wheel.  Once  every  30  sec.  the  Slave  Clock  S  (shown 
on  a  much  smaller  scale)  drops — electrically  of  course — the  light 
lever,  shown  pivoted  on  the  right-hand  end,  so  that  the  flat  face  of 


m 


O 


M 


Fig.  57. 


its  black  jewel  falls  on  the  wheel,  which  rolls  under  it  unaffected,  until, 
as  the  pendulum  moves  through  mid- swing  to  the  left,  the  edge  of 
the  jewel  runs  down  the  right  side  of  the  wheel,  giving  a  push 
equivalent  to  0-4  gm.  falling  2  mm.,  and  restoring  the  1 /700th 
loss  of  arc  (of  34  mm.)  which  the  pendulum  has  suffered  in  the  half- 
minute.  The  inevitable  residual  frictions  which  cause  this  loss, 
and  this  tiny  restoring  impulse,  are  the  only  interferences  with  the 
free  motion  of  this  14-lb.  Pendulum. 

That  the  slave  dropped  the  jewel  at  the  wrong  time  does  not 
matter,  it  only  means  0-01  mm.  more  or  less  run  under  the  flat  face, 
but  now  its  sharp  edge  falls  clear  of  the  wheel  at  the  exact  time  oj  the 
master,  and  it  continues  its  fall  until  in  the  dotted  position  it  releases 
all  the  electrical  relay  gear.  This  (a)  sends  out  a  signal  to  Ime,  (6) 
lifts  back  the  lever  to  its  place,  and  (c)  pulls  down  a  thin  buffer  mto 
the  dotted  position  to  the  left  of  S,  the  slave  pendulum-ro<l. 

If  the  slave  pendulum  is  up  to  time,  the  end  of  the  long  U-shaped 
sprmg  attached  to  it  (shown  much  too  short  and  thick)  has  already 
escaped  to  the  left,  under  this  buffer  ;  but  it  is  deliberately  given  a 
losing  rate,  and  every  now  and  again  the  buffer  is  there  first,  and 


H6  MECHANICS  [§  166 

the  slave,  1/200  sec.  late,  is  bounced  back  1/200  sec.  early  again. 
Thus  the  slave,  which  is  an  ordinary  *  Synchronome  '  pendulum, 
can  never  run  away  from  the  master,  nor  ever  lag  far  behind. 

These  clocks  are  admittedly  not  perfect,  for  their  arc  of  swing, 
which  is  inspected  at  intervals  through  a  microscope,  sometimes 
unaccountably  changes,  and  a  maintained  addition  of  0-02  mm. 
would  increase  the  '  circular  error,'  §  85,  enough  to  make  the  clock 
lose  half  a  second  a  year.  But,  practically,  the  two  pendulums 
swing  together  within  0-01  sec.  for  months  together,  when  circular 
corrections  from  the  micro- observations  are  applied  their  differences 
disappear,  and  these  British  contrivances,  keeping  Time  some  hun- 
dred times  better  than  any  mechanism  ever  before,  are  a  superlative 
instance  of  the  benefit  of  simplifying  experimental  conditions  and 
getting  down  close  to  first  principles. 

Inevitably,  these  pendulums  slow  down  as  the  moon  passes  over- 
head, '  tidally  '  lifting  the  bob  against  the  earth's  controlling  gravity, 
and  further,  they  have  actually  succeeded  in  throwing  doubt  on  the 
regular  rotation  of  the  earth.  For  this  there  may  be  various  reasons, 
e.g.  annually  Arctic  masses  of  snow  and  cold  air  are  released,  and 
travel  south  to  greater  distances  from  the  earth's  axis,  increasing 
its  moment  of  inertia  and  correspondingly  slowing  its  rotation,  for  it 
is  the  moment  of  momentum  that  remains  constant,  §  88,  and  the 
Antarctic  does  not  exactly  compensate. 

These  clocks  are  never  regulated,  their  probable  error  is  always 
known  well  within  0-01  sec,  but  the  work  of  the  world  demands 
good  Greenwich  time  daily,  or  oftener.  A  third  clock,  with  a  losing 
rate  (at  present)  of  0-03  sec.  a  day,  is  corrected  daily  at  10  hr.,  13  hr., 
18  hr.,  and  21  hr.,  by  bringing  up  a  magnet  under  an  iron  bar  on  the 
end  of  its  pendulum,  so  as  to  increase  the  controlling  force  just  long 
enough  ;  from  this  the  '  six  pips  '  are  sent  out,  with  an  error  unlikely 
to  exceed  0-01  sec.  Making  all  allowances,  you  may  rely  on  these, 
as  you  receive  them,  to  within  l/20th  sec. 

§  157.  Another  timekeeper,  largely  used  in  alternating-current 
supplies  and  wireless  controls,  is  the  electrically-maintained  tuning- 
fork,  Fig.  155.  This  is  independent  of  gravity,  but  must  be  kept 
in  a  thermostat,  §  204,  or  else  it  would  keep  no  better  time  than  an 
uncompensated  watch,  §  175. 

The  modern  A.C.-mains  clock  is  a  little  *  synchronous  '  motor,  and 
borrows  its  time  from  the  supply  station  tuning-fork. 

Better  than  a  steel  fork  is  the  electro -elasticity,  §  802,  of  a  little 
plate  of  rock-crystal,  controlling  an  oscillating  circuit  :  it  is  a 
first-rate  timekeeper  quite  independent  of  gravity,  see  §  451  for  a 
calculation  of  its  speed,  and  Fig.  380,  §  837,  for  the  circuit. 

§  158.  For  continuous  time  records  involving  fractions  of  a 
second,  recourse  is  had  to  a  chronograph,  a  rotating  paper-covered 
drum.  As  it  would  require  elaborate  mechanism  to  secure  quite 
uniform  rotation,  this  is  not  looked  for,  but  the  paper  is  electrically 


§159]  PRECISE  MEASUREMENTS  117 

marked  off  in  seconds  by  a  pendulum  as  in  Fig.  152.  The  ingenious 
chronograph  used  with  the  Shortt  clock  punctures  the  paper,  by 
spark,  with  an  easily  readable  accuracy  of  0-001  sec. 

Other  similar  electrical  markers,  mounted  in  line  above  the  first, 
are  connected  direct  to  the  experimental  apparatus,  or  else  to  the 
observer's  press-button.  Direct  connection  is  preferred  where 
possible,  for  there  is  always  a  fraction  of  a  second  interval,  called 
the  observer's  personal  equation,  between  his  seeing  or  hearing  a 
signal  and  his  pressing  the  button.  Between  one  observer  and 
another,  personal  equations  vary,  and  must  be  measured  and  allowed 
for. 

Small  calculable  intervals  of  time,  often  required  in  physiological 
experiments,  are  easily  obtained  by  letting  a  heavy  pendulum  knock 
down  two  triggers  placed  a  definite  distance  apart  in  its  path,  near 
mid-swing. 

Technically  the  Error  of  a  clock  is  the  amount  it  is  slow  at  noon, 
its  Rate  is  its  loss  per  day. 


THE  MEASUREMENT   OF   MASS 

§  159.  The  common  Balance  is  a  Lever  with  two  equal  arms : 
the  Forces  which  it  balances  are  the  pulls  of  gravity  on  the  masses 
placed  in  the  pans  ;  when  these  forces  are  equal  the  masses  are  equal, 
for  pendulum  experiments  from  the  days  of  Newton  onwards  have 
persuaded  everybody  that  gravity  acts  on  matter  of  every  kind  with 
total  indifference  as  to  its  nature. 

Gravity  varies  from  one  locality  to  another,  §  40,  but,  balance 
pans  being  close  together,  the  variations  affect  both  equally.  In 
a  spring  balance,  however,  where  the  pull  of  the  earth  comes  against 
a  fixed-strength  spring,  variations  of  gravity  interfere  if  an  accuracy 
exceeding  1  in  1000  is  aimed  at,  whereas  a  fine  beam-balance  works 
to  1000  times  this. 

The  stiff  balance  beam,  Fig.  48,  bears  three  '  knife  edges  ' — 
sharp-edged  prisms  of  steel  for  heavy  weights,  agate  or  rock  crystal 
for  light.  The  middle  inverted  one  rests  on  a  flat  plate  of  the  same 
material  on  the  supporting  pillar,  the  others  carry  flat  plates  from 
which  hang  the  pans,  etc .  To  preserve  the  delicate  edges  from  crush- 
ing, mechanism  (not  shown  in  the  figure)  is  provided  for  lifting 
edges  and  plates  out  of  contact,  except  when  actually  testing  the 
equiUbrium.  Use  this  when  it  is  provided,  don't  alter  weights 
except  with  beam  fixed,  don't  exceed  the  specified  load,  don't  use 
weights  in  mixed  order  or  from  different  sets,  don't  weigh  hot  things 
which  set  up  draughts,  nor  damp  things  which  lose  weight  by^ 
evaporation — in  short,  don't  *  weigh  like  a  damned  apothecary,* 
the  somewhat  libellous  dictum  of  an  eminent  Scot. 

The  maker  sets  the  three  knife-edges  parallel  and  all  touching  a 
straight  line,  the  theoretical  lever,  dotted  in  Fig.  48.     For  if  not, 


118  MECHANICS  [§  159 

let  them  be  ACB,  when  tilted  they  move  to  A'CB',  Fig.  58,  and  now 
horizontal  a'c  is  not  equal  to  c6',  and  these  are  the  arms  perpendicular 
to  the  weights,  §  73.  Such  a  balance  alters  its  readings  according 
to  the  zero  to  which  one  adjusts  it  to  work,  unless  levelled  with 
extraordinary  care. 

The  beam  is  made  stiff,  an  open  lattice  deeper  in  the  middle,  so 
that  this  still  remains  essentially  a  straight  line,  under  full  load. 

The  balance  arms  should  be  of 
equal  length,  or  equal  weights  will 
not  balance.  In  most  of  our  labora- 
tory experiments  in  physics  and 
chemistry  it  does  not  matter  if  we  are 
working  to  a  false  standard  weight,  so 
long  as  we  keep  to  it  throughout  the 
experiment  :  this  means  that  we  may 
use  a  balance  with  unequal  arms,  provided  that  the  body  is  always 
placed  on  the  left,  and  the  weights  in  the  right-hand  pan — ^the 
natural  place,  unless  you  are  left-handed — and  keep  the  weight- 
box  near  by,  and  use  and  keep  the  weights  in  regular  sequence,  it 

SAVES    TIME. 

Pans,  stirrups,  etc.,  are  never  exactly  paired,  but  should  be 
marked  with  .  or  :  so  that  left-hands  and  right-hands  are  each  kept 
all  together ;  then  if  the  two  aggregates  are  not  exactly  equal 
(a/c  corrosion,  etc.)  little  weights  running  on  fine  screws  along  the 
beam  facilitate  final  adjustment. 

§  160.  If  the  imperfectly  equal  lengths  of  the  balance  arms 
be  r  and  I,  the  body  in  the  left  pan  will  be  balanced  by  Ijr  times  its 
weight  placed  in  the  right.  For  in  a  lever  the  forces  are  inversely 
as  their  distances  from  the  fulcrum.  Now,  changing  the  body  to 
the  right  pan  it  is  counterpoised  by  r/Z  times  its  weight  in  the  left. 
Multiplying  the  two  weights  together  and  taking  the  square  root, 

this,  a/  -w  X  -,  W  =  w.     Or,  what  comes  to  the  same  thing  in  practice 

in  any  balance  not  too  glaringly  lopsided  to  be  used,  add  them 
together  and  divide  by  2. 

This  is  Double  Weighing,  and  was  understood  and  practised 
between  buyer  and  seller  centuries  before  weighbeams  were  made 
true. 

Weighing  by  substitution  of  weights  for  body  is  another  means  of 
wringing  the  truth  from  an  unjust  balance,  only  a  temporary 
counterpoise  being  used  in  the  other  pan. 

You  may  be  asked  in  an  examination  to  find  the  Ratio  rjl  of  the 
Arms ;  to  do  this,  double-weigh  a  body,  divide  one  weight  by  the 
other  (don't  mix  up  which  is  which),  and  take  the  square  root 

J  .        ,      ,  X-      XI  •  .    .     ,    ,    1    difference  in  weight 

and  m  actual  practice  this  amounts  to  1  +  A  . r-^rr ^—  • 

^  '    2  weight 


I 


162] 


PRECISE  MEASUREMENTS 


119 


§  161.  A  Rider  of  aluminium  wire  weighing  10  mg.  can  be  placed 
anywhere  on  one  arm,  which  is  divided  into  10  equal  parts  (some- 
times 12  mg.  and  12  parts).  Placed  directly  over  the  end  knife 
edge,  this  has  a  moment  10  mg.  x  full  distance  1  at  which  the 
weights  in  the  pan  act.  Placed  say  at  the  third  division  from 
centre  of  beam,  its  moment  =  10  mg.  x  distance  0-3,  which  is 
equivalent  to  3  mg.  x  distance  1 .  Thus  it  now  turns  the  balance 
just  as  much  as  a  weight  of  3  mg.  in  the  pan.  The  one  rider  saves 
fiddling  with  weights  from  10  mg.  down  to  0*1  mg.  The  thing  is 
a  common  steelyard  in  miniature,  §  75  G. 


§  162.  Now,  suppose  a  perfect  balance  has  been  reached,  and  1 
mg.  excess  is  put  in  one  pan.  There  is  no  force  to  oppose  this,  and 
the  system  gradually  accelerates  until  it  '  crashes,'  that  pan  resting 
on  the  floor.  But  we  want  the  balance  to  point  out  the  heavier 
side  to  us  quietly  and  stably,  and  to  give  us  an  idea  how  much 
heavier  ;  not  to  flop.  To  this  end,  the  centre  of  mass  Q  of  the  beam 
and  pointer  (not  pans)  is  adjusted  to  lie  a  little  below  the  centre 
knife-edge,  and  therefore  swings  out  on  the  high  side  of  the  inclined 
beam,  and  the  weight  B  of  the  beam  now  gains  a  leverage  and  helps 
a  lighter  weight  w  to  balance  a  heavier  W. 

In  Fig.  59,  equating  turning- moments  about  the  point  of  support  C. 

{w  X  LC)  -f  Beam  weight  x  Q^'  =  CR  X  W 

LC  =  CR 
.-.  Beam  weight  X  Qg  =  CR  X  (W  —  t^), 

i.e.  the  beam  tilts  and  opens  out  the  lever  distance  Qg  until  it  comes 
to  rest  in  the  inclined  position  obtainable  from  this  equation,  the 
arm  slant  being  about  proportional 
to  the  excess  weight.  Everybody,  of 
course,  is  familiar  with  this  action 
of  a  '  pair  of  scales.' 

From  the  figure  it  is  plain  that 
the  smaller  h  is  made,  the  more  the 
beam  must  kick  to  open  out  Qg,  so 
if  ^  be  altered  by  fixing  a  small 
weight  higher  or  lower  on  the  pointer 
of  the  balance— if  this  is  I /100th  of 
the  beam  weight,  h  is  altered  by 
1/100  its  movement— the  sensitive- 
ness of  the  balance  is  controlled, 
this  being  defined  as  the  number  of 
scale  divisions  the  pointer  is  displaced  for  a 
{often  1  milligram). 

And,  from  the  equation,  if  B  is  lessened,    ^^         -r  r  u*. 
light  beam  tilts  more  for  a  given  overload.     But  if  lightness  is 
attained  by  shortening  the  arm  CR,  the  sensitivity  decreases. 

In  §  90  it  was  shown  that  the  balance  beam  is  a  compound 
pendulum  with  a  time  of  swing  which  increases  as  h  dimmisnes, 
but  decreases  as  mass  and  size  of  beam  are  reduced. 


Fio.  59. 


definite  small  overload 
Qq  must  increase  :    a 


120  MECHANICS  [§  162 

Between  these  you  can  puzzle  out  why  it  is  that  modem  balances 
are  made  with  short  light  beams,  but  provided  with  magnifiers  for 
reading  deflexions.  You  can  also  see  what  a  Waste  of  Time  it  is  to 
use  a  balance  more  sensitive  than  suits  your  job.  Balances  are 
very  accurate  machines,  and  it  is  a  Lack  of  Judgment — which 
learn  to  rectify— to  save  ten-thousandths,  while  letting  hundredths 
leak  away  somewhere  else  in  an  experiment. 

§  163.  Correction  for  weighing  in  air.  Finally,  it  is  necessary 
to  correct  for  both  body  and  weights  being  buoyed  up  by  the  air 
around  them  to  an  extent  equal  to  the  weight  of  air  they  displace, 
§  132.  This  prevents  the  weights  exerting  their  full  face  value  W, 
while  the  body  M  appears  lighter  than  it  should. 

1  c.c.  of  dry  air  at  0°  C.  and  760  mm.  barometer  weighs  0-001293 
gm.,  and  this  requires  reduction  for  temperature,  pressure,  and 
humidity  :   commonly  1  c.c.  of  air  may  be  taken  as  1/800  gm. 

The  volume  of  the  Body  =  M  ^  its  specific  gravity,  which  must 
be  known  approximately;  it  .*.  displaces  (M/its  s.g.)  X  1/800  gm. 
of  air,  and  presses  on  its  scale-pan  with  force 

^-body-ss.g.^800g"^- 
Likewise  the  Weights  press  on  their  scale-pan  with  force 

^       weights'  s.g.  ^  800  ^^' 
Weights  are  usually  brass,  s.g.  8-4 ;    and  this  becomes  nearly 
enough  W  —  W/7000.     The  balance  equates  these  two  forces 

M M_       J__  JW 

body's  s.g.       800  7000' 

•M  =  W  M  W 


True  weight  M  =  W  + 


800  X  body's  s.g.       7000' 

}  it  will  make  n< 
the 

W  W 


The  right-hand  M  being  divided  by  800  it  will  make  no  appreciable 
difference  if  we  write  W^  instead ;  hence  the 


800  X  body's  s.g.       7000' 

a  formula  which  shows  you  plainly  how  far  out  you  are  without  it, 
and  is  quite  near  enough  in  practice. 

With  very  light  and  bulky  bodies  it  may  become  necessary  to 
alter  the  1/800  to  allow  for  barometric  variations  (see  example  10)  : 
actual  Weighing  in  Vacuo  demands  very  special  apparatus. 

§  164.  The  Balance,  as  described  above,  has  been  made  probably 
the  most  sensitive  and  accurate  of  all  ordinary  physical  apparatus 
(0*1  mg.  in  100  gm.  corresponds  to  a  second  in  a  fortnight),  and  for 
many  practical  purposes  a  great  deal  of  this  can  be  sacrificed  for 
speed  and  facility  in  use,  mainly  by  reducing  the  handling  of  weights. 


^^  1()4]  PRECISE  MEASUREMENTS  121 

i  tactions  of  the  pound,  for  instance,  are  taken  care  of  by  a  spring, 
or,  better,  by  a  counterpoise  which  swings  out  to  a  greater  leverage, 
moving  a  pointer  over  a  graduated  scale,  whole  pounds  only  having 
to  be  put  on  the  weights  pan.  Damageable  and  shifting  knife- 
edges  are  replaced  by  thin  strips  of  springy  metal,  secured  to  both 
])arts,  like  that  from  which  a  clock-pendulum  hangs.  Excellent 
{)alances  of  this  description  are  now  familiar  to  us  all,  in  shops. 


EXAM   QUESTIONS,   CHAPTER   X 
Very  much  a  laboratory  chapter  :  clocks  are  not  asked  about,  Balances  are. 


1.  You  are  given  a  rod  about  10  cm.  long  and  3  nun.  diameter.  How 
would  you  measure  its  dimensions  accurately  ?  Describe  the  instrtunents 
you  would  use  and  explain  how  they  act. 

2.  What  conditions  determine  the  sensitiveness  of  a  balance  ?  A  balance 
has  unequal  arms ;  a  piece  of  lead  weighs  apparently  452  gm.  in  the  left  pan, 
but  in  the  right  pan  454  gm.  Calculate  the  true  weight  of  the  lead,  and  the 
ratio  of  the  balance  arms.     (  X  3) 

3.  Describe  an  accurate  balance,  explain  the  principle  of  the  rider  and 
divided  beam,  and  state  precautions  in  use  of  the  balance. 

4.  A  balance  with  10  gm.  in  each  pan  rests  several  divisions  out  of  centre. 
How  would  you  find  whether  this  is  due  to  defects  of  balance  or  inticcurticy 
of  weights  ? 

5.  Explain  the  principle  of  the  lever. 

A  common  balance  rests  inclined  when  the  weights  in  the  pans  are  slightly 
unequal;   show  in  a  diagram  why  this  is. 

If  the  beam.  20  cm.  long,  weighs  100  gm.,  where  must  be  its  centre  of  mass 
if  0-05  gm.  overload  in  one  pan  causes  an  inclination  of  1  in  50  ?     (  X  2) 

6.  Against  brass  weights  in  air  a  litre  flask,  of  glass  sp.  gr.  2-4,  weighs 
150  gm.     Find  its  true  weight  in  vacuo. 

From  §  163,  w  =  150(1  -  1/7000  +  00012/2-4) 
=  150(1  +  000036)  =  150054. 

7.  The  flask  is  then  filled  with  liquid  and  appears  to  weigh  960  gm.  Find 
true  weight  of  liquid. 

8.  What  are  the  important  qualities  of  a  good  balance,  and  how  are  they 
secured  in  practice  ?  What  is  the  mass,  and  the  volume,  of  a  glass  stopper, 
8.g.  2-5,  coimterpoised  in  air,  s.g.  000125,  by  25  gm.  brass  weights  s.g.  8-4  ? 
(X  2) 

9.  A  hollow  sphere  of  thin  copper  is  balanced  on  a  light  rod,  in  a  box  of 
air,  against  a  solid  brass  knob.  What  would  be  the  effect  of  displacing  the 
air,  (o)  by  hydrogen,  (6)  by  COj  ?  How  could  this  be  used  to  determine 
relative  densities  ? 

10.  An  electric  lamp  bulb  is  balanced  against  brass  weights;  on  which 
side  must  additional  weights  be  placed  to  restore  equilibrium  if  the  barometric 
height  increases  ?  How  may  the  true  weight  of  the  bulb  be  determined  ? 
(X  2) 

11.  Describe  a  balance  for  loads  of  only  a  few  milligrams,  and  show  how 
it  is  affected  by  temperatm'e  and  pressure. 


HEAT 


CHAPTER   XI 


THERMAL  EXPANSION 


That  most  familiar  of  scientific  instruments,  the  Thermometer] 
measures  the  hotness  or  temperature  or  coldness  of  its  surroundings^ 
by  means  of  the  expansion  or  contraction  of  its  contents  past 
scale.     Let  us  therefore  study  the  expansion  of  things  in  general  to' 
start   with ;    merely,   for  the   present,  using  a   bought  centigrade 
thermometer  of  certified  accuracy. 

§  171.  The  Expansion  caused  by  heating  things.  Illustrations 
of  this  abound  on  every  hand,  though  the  expansion  is  too  small 
to  be  directly  visible  except  on  long  lengths  ;  for  the  great  swelling 
of  a  red-hot  poker,  or  of  a  lamp  filament,  is  an  optical  illusion  due  to 
local  dazzling  of  the  eye.  We  warm  the  neck  of  a  bottle  to  ease  out 
a  stuck  stopper.  Telegraph  wires  sag  noticeably  more  in  summer. 
On  a  hot  day  the  '  distant  signal,'  pulled  down  through  1000  yd.  of  ' 
wire,  only  languidly  indicates  a  clear  line.  The  rods  to  distant 
'  points,'  where  half-way  motion  cannot  be  tolerated,  have  to  be 
contrived  so  that  half  their  length  pulls  and  half  pushes.  The  gaps 
at  the  ends  of  rails  visibly  close  up,  and  very  exceptionally  more 
than  do  so  ;  I  have  known  the  traffic  delayed  while  eighty  yards  of 
line  was  being  persuaded  to  lie  down  flat  again. 

Conversely,  the  reduction  of  temperature  is  accompanied  by 
contraction  :  the  tyre  grips  the  wagon-wheel  tightly  when  quenched 
from  the  blacksmith's  bucket. 

Liquid  expansion  is  instanced  by  the  thermometer  itself,  or  byi 
the  overflowing  of  a  saucepan,  quite  full  of  cold  water,  long  before « 
it  boils.  Smoke  rises,  for  its  Gases  have  expanded  until  less  denser 
than  the  air  around,  and  solid  sparks  are  borne  upward  by  thei 
little  invisible  uprushes  of  hot  air  which  they  themselves  produce. 

§  172.  Forces  involved  in  thermal  expansion.  All  substances^ 
yield  to  force  ;  and  the  stretching  of  a  wire  due  to  a  pull  may  be-* 
compared  in  the  laboratory  with  that  due  to  heating.  Small  as  the 
latter  is,  it  will  be  found  to  equal  that  produced  by  large  forces. 
Red-hot  rivets  hammered  up  tight  draw  the  plates  together,  as  they 
cool,  with  a  pressure  of  many  tons.  If  the  engineer  cannot  allow 
for  free  expansion  in  his  structures — 6  in.  is  allowed  in  the  Forth 
bridge — ^he  must  design  them  to  meet  heavy  stresses.     Tram  rails 

122 


§173]  THERMAL   EXPANSION  123 

are  welded  solidly  together  and  prevented  by  weight  of  surrounding 
road  metal  from  the  lifting  that  expansion  would  otherwise  cause ; 
they  are  in  compression  on  a  hot  day,  and  in  tension  on  a  frosty  one. 

The  Assouan  dam  endures  stresses  from  temperature  not  less 
severe  than  those  from  the  weight  of  Nile  water  :  it  has  been 
heightened,  but  the  strengthening  buttresses  are  not  built  into  it, 
they  press  against  stainless  steel  slipping  plates  on  the  vast  wall. 

Ghostly  crackings  in  the  house  at  night  are  often  due  to  such 
sUpping,  as  things  cool  down  and  contract. 

Glassware  has  to  be  cooled  slowly  (annealed)  or  else  it  contains 
strains  that  are  released  in  cracking  under  a  trifling  blow  (as  is  so 
common  in  china-ware),  or  immersion  in  hot  water. 

The  builder  was  discovered  setting  the  fronts  of  the  tanks  into 
their  metal  frames,  in  the  New  York  Aquarium,  with  hard  unyielding 
cement ;  remonstrated  with,  what  cared  he  for  the  laws  of  expansion 
and  contraction  ?  the  laws  of  Tammany  Hall  were  good  enough  for 
him  :    so  the  great  panes  split. 

Wires  to  be  sealed  into  glass  must  be  no  more  expansible  than  the 
frlass,  or  they  break  loose  in  cooling  :  platinum,  or  nowadaj's  a 
special  nickel- steel,  copper  plated,  leads  into  electric  bulbs. 

Pyrex  and  other  modern  glasses  are  very  much  less  expansible, 
beakers  are  made  thicker  without  risk  of  fracture  from  flame  outside 
and  cold  water  within,  you  are  spared  many  minor  catastrophes  of 
the  chemical  lab,  and  you  even  find  dishes  of  them  in  use  at  home  in 
the  oven.  Pure  silica  glass  is  the  limit,  it  is  so  inexpansible  that 
patches  can  be  fused  in  anyhow,  and  quenched  with  water. 

§  173.  The  expansion  of  solids.  The  expansion  of  solid  rods  can 
be  quickly  measured  with  the  apparatus  of  Fig.  60.  The  rod  is 
geometrically  gripped  at  one  end,  and  the  other  flat  end  presses 
on  the  spring  point  of  an  ordinary  optician's  spherometer  clamped 
on  the  framework.  In  this,  by  multiplying  gear,  the  motion  of  the 
end  of  the  rod  is  magnified  about  200  times,  one  dial  division  corre- 
sponding to  0-001  cm.     The  rod  is  50  cm.  long. 

Ice-cold  water  is  run  through  a  jacket  corked  on  the  rod,  and  in 
a  minute  or  two  the  dial  reads  steadily.  The  water  is  run  out  and 
steam  blown  through,  the  dial  read  when  steady,  and  the  temperature 
obtained  from  the  barometric  height.  Fig.  82. 

If  the  wooden  framework  increased  in  length  during  the  experi- 
ment, this  alteration  would  subtract  itself  from  the  rod's  expansion. 
But  the  teak  frame  shown  is  well  away  from  the  changing  temper- 
atures, and  secured  by  its  form  against  warping.  Hot  and  cold 
readings  can  be  alternated  at  three-minute  intervals. 

Dividing  the  difference  in  dial  readings  by  1000  gives  the  elonga- 
tion of  the  rod  in  centimetres.  Now,  2  it  is  merely  left  on  record 
that  a  certain  rod  expanded  so  much  for  such  a  rise  of  temperature, 
it  will  involve  a  compound  proportion  sum  when  we  want  to  reckon 
out  the  extension  of  some  other  piece  of  the  same  material  for  a 
different  rise.     As  the  whole  rod  is  equally  heated,  each  centimetre 


124 


HEAT 


[§  173 


is  contributing  an  equal  share  to  the  whole  extension,  i.e.  in  our  caseJ 
each  contributes  one-fiftieth  part.     Again,  if  we  divide  the  observed! 
expansion  by  the  rise  of  temperature  that  caused  it,  we  get  the! 
average  expansion  caused  by  1°  rise.     That  is,  the  observed  exten- 
sion expressed  in  centimetres  (or  whatever  units  the  length  of  the 
rod  is  measured  in)  divided  by  the  length  of  the  rod,  and  divided  by  ji 
the  rise  of  temperature  which  caused  it,  gives  us  the  expansion  of  one  ll 
unit  length  when  heated  one  degree,  which  is  the  *  coefficient  of  ^' 
expansion  in  length  '  or  the  <  linear  expansibility/  a,  of  the  material 
of  the  rod. 


Fig.  61.  V?| 

§  174.  In  the  great  Comparator  of  Fig.  61  the  test  bar  U'  under  i 
examination  by  the  micrometer  microscopes  can  be  kept  at  any 
desired  temperature.  Before  and  after  every  measurement  of  it, 
the  bar  U  is  pushed  back  into  place,  and  the  micrometers  are  reset 
upon  that.  Being  kept  packed  with  melting  ice  all  the  time,  it  forms 
an  unchanging  standard,  and  this  procedure  does  away  with  any 
uncertainties  due  to  the  possible  expansion  of  the  long  iron  bed  B. 

Thus,  to  calculate  the  expansion  of  a  length  of  material,  we  rtmsti 


f; 


§175] 


THERMAL   EXPANSION 


12i 


multiply  its  a  by  its  length  and  by  the  rise  of  temperature  ;  a  length 

Lq  in  ice  expands  Lp  x  a  X  100°,  increasing  in  total  length  to 
Lq  +  Lq  X  a  X  100°  in  boiling  water. 

Lo  at  0°  becomes  Lq  +  LoaT  at  T° 

or,  from  some  intermediate  temperature,  very  nearly  indeed 

L  at  t°  becomes  L  +  La(T  —  t)  at  T° 


Linear  Expansibility  per  °C.,  in  parts  per  million. 


Fused  silica 
'  Invar  '  steel    . 
'  Pyrex  '  glass,  \ 

Porcelain        / 
Pine,  along  grain 

„    across    „ 
Common  glass, 

granite     . 


0-5 
0-9 
3 

4-5 
40 

8-5 


Platinum     .      . 

■pui  Its  pt 

9 

steel       .      .      . 

11 

Iron        .      .      . 

12 

Copper,  eureka . 

17 

Brass      .      .      . 

18-5 

Aluminium 

23 

Zinc,  Lead  .      . 

28 

Ebonite 

70 

Paraffin  wax 
Calcspar,  along 
axis 

„       contracts 
across  axis 
Stretched    india- 
rubber        con- 
tracts strongly. 


110 


-5 


Per  °F  the  expansibilities  are  ten -eighteenths  of  the  above. 

'  Invar  '  is  a  nickel-steel  which  has  undergone  a  special  heat- 
treatment.  It  is  invaluable  to  surveyors,  and  in  pendulums,  but 
not  for  primary  standard  measure  bars  ;  for  some  specimens  '  grow  ' 
minutely,  year  by  year.     '  Pyrex  '  is  a  highly  siUceous  glass. 

Ex.  1.  A  half-metre  aluminium  rod  expands  1-15  mm.  between  0°  and 
lOO*".     Find  a. 

60  cm.  become  50-115  cm. 


L  +  L  .  a  .  100° 
.'.    5000  a  =  0-115. 


a  =  0000023. 


What  correction  will  be  necessary 
15)  =  12-0112  in. 


Ex.  2.     A  steel  foot-rule  is  correct  at  15°. 
ill  boiling  water? 

12  in.  at  15°  becomes  12  +  12  X  0-000011  X  (100 
.*.  the  rule  is  0-011  in.  too  long. 

Ex.  3.  What  length  of  lead  bob  compensates  a  pendulum  made  of  44  in. 
of  pine  and  a  2-in.  suspending  spring  ? 

Centre  of  lead  must  remain  nearly  fixed,  i.e.  its  lower  half  expands  as  much 
as  the  wood  and  steel,  i.e. 

2  X  0000011  +  44  X  0000004  =  0000198  in.  per  degree. 
.*.  half  length  X  0000028  =  0000198  .'.  length  =  14  in. 

Ex.  4.  A  tram-rail  40  ft.  long  is  heated.  Its  normal  expansibility  is 
0000011,  but  expansion  is  prevented  by  the  end-pressure  of  adjoining  rails. 
Find  the  increase  in  this  pressure  between  40°  F.  and  90°  F.,  given  that  10 
itons  shortens  the  rail  0-04  in. 

i  Free  rail  would  expand  40  X  0-000011  X  50°  X  12  =  0-264  in. 
;  If  10  tons  shortens  it  004  in.,  66  tons  would  shorten  it  0-264  in. 
j 

§  175.  Compensation  contrivances.  Fig.  62  shows  various  devices 
I  by  which  timekeepers  are  freed  from  thermal  errors. 

The  problem  in  a  Pendulum  is  to  prevent  the  middle  of  the  bob 

(practically  the  centre  of  mass)  being  lowered  when  the  rod  expands, 

which  would  make  the  clock  lose  (half  a  second  a  day  per  1°  C. 

warmer,  for  an  iron  rod).     The  expansion  upward  of  (the  lower 

ihalf  of)  a  long  lead  or  mercury  bob  raises  the  centre,  as  much  as 


120 


HEAT 


[§  1751 


ffl 


PINE  WOOD 

•ooooo  4- 
OR 

PYREX 

•oooooS 


STEEL 

■oooo  11 


LEAD 

•oooo2^ 


I 


ZiNCTUBE 
•0000^9 


OUTER 
STEELTUBE 


c=; 


fflZ] 


ra 


STEEL 


^Bf 


MERCURY 
■000  16^3  u.f> 


INVAR 

■ooooooj 


MODERN      COMPENSATED      PENDULUMS 


Scale  =  /7X  '• 


CHRONOMET 


^Compensating  metals  in  blackj 

Fig.  62. 

the  expansion  of  the  wooden  or  steel  rod  lowers  the  bottom,  of  th<i 
bob,  which  rests  on  the  regulating  nut  at  the  end  of  the  rod.  In  th<t 
modern  representative  of  the  '  gridiron '  pendulum  (long  sincn 
obsolete  in  England),  a  zinc  tube  rests  on  the  nut,  and  expands  u]( 


§  176]  THERMAL   EXPANSION  127 

as  much  as  the  inner  steel  rod  and  outer  iron  tube  (hanging  from  the 
top  of  the  zinc)  together  expand  down,  and  the  bob  is  unmoved ; 
being  supported  at  its  centre,  its  own  expansion  does  not  count. 
This  inexpensive  construction  is  employed  in  most  tower  clocks, 
including  '  Big  Ben.'  With  invar  steel  a  few  brass  washers  provide 
sufficient  upward  expansion  ;  it  is  used  now  in  all  the  best  clocks. 

In  a  compound  bar,  two  thin  strips  of  very  differently  expansible 
metals,  such  as  iron  and  brass,  or  nickel-steel  and  nickel-copper,  are 
w  elded  together  throughout  their  lengths.  When  heated,  the  bar 
must  bend,  the  higher  expansive  strip  taking  the  longer  outer  curve. 

Compound  bars  the  size  of  a  micro-slide  are  used  for  Fire  Alarms, 
distributed  over  the  ceiling  :  any  one  becoming  overheated  bends, 
makes  an  electric  contact,  and  starts  the  fire  bell.  Long  thin  ones, 
curled  up  in  spirals,  work  the  self-recording  Thermographs  of  the 
Meteorological  Office,  or  oven  pyrometers,  or  little  dial-room  thermo- 
meters, made  up  to  look  like  something  off  a  car  dashboard. 

By  this  means  also  the  masses  on  the  balance  wheel  of  a  Chrono- 
meter, or  '  compensated  '  Watch,  are  brought  in  nearer  the  centre, 
as  the  temperature  rises.  This  reduces  the  moment  of  inertia,  and 
K.mpensates  not  only  the  expansion  of  the  wheel  as  a  whole,  but 
iilso  the  enfeebled  elasticity  of  the  balance  spring,  which  has  a  twenty 
times  worse  effect  on  timekeeping  than  the  mere  expansion  of  an 
iron  pendulum. 

A  very  great  advance  on  this  costly  and  only  partially  satisfactory 

:itrivance  has  been  made  by  Guillaume,  the  inventor  of  Invar,  who 

;i  years  later  produced  Elinvar,  a  spring  steel  almost  unaffected 
in  elasticity  by  temperature  :  a  plain  white  Invar  balance  wheel 
and  Elinvar  hair  spring  have  superseded  these  complications  :  look 
in  your  wrist -watch. 

One  proposal  for  eliminating  residual  error  sounds  amazingly  cheap 
and  simple  ;  ordinary  hard-rolled  sheet  zinc  is  about  six  times  as 
expansible  one  way  as  the  other,  balance  wheels  are  to  be  stamped 
out  of  it  with  their  one  cross-spoke  along  the  highly  expansible 
direction  ;  and  then  their  rims  are  annealed,  by  being  gripped  for  a 
second  in  hot  jaws,  and  lose  this  property.  Now,  when  temperature 
rises,  the  cross  spoke,  expanding  excessively,  forces  the  rim  into 
a  long  0  shape,  the  sides  pulling  in  so  much  that  the  moment  of 
inertia  is  actually  reduced. 

§  176.  Expansion  of  Area  and  of  Volume.    An  expanding  Square 

increases  towards  the  east,  and  also  towards  the  north  ;  it  increases 

i  twice  as  fast  as  its  length  of  side  ;  any  Area  can  be  built  up  of  little 

;  squares  ;   ureal  expansibility  is  twice  linear  expansibility. 

I      A  cube  expands  towards  the  east,  and  towards  the  north,  and 

I  upwards  ;    any  volume  can  be  packed  with  cubes  ;    the  *  volume  * 

or  *  bulk '  or  «  cubical '  expansibility  is  three  times  the  linear. 

The  appearance  of  greater  exactness  which  might  be  given  here 

t  by  using  algebraic  formulae,  or  geometrical  figures,  is  illusory.     For 

there  are  few  substances  so  isotropic  (same  turned  any  way)  that 


.128  HEAT  [§  176 

they  expand  quite  equally  in  all  directions  :  wood  is  5  to  10  times 
as  expansible  across  the  grain,  while  some  crystals,  and  stretched 
indiarubber,  actually  contract  lengthwise  when  heated,  although 
expanding  in  volume. 

The  internal  volume  of  a  hollow  vessel  has  the  same  volume 
expansibiUty  as  the  material  of  its  walls.  For  it  might  be  filled  with 
a  solid  mass  of  their  material,  which  would  then  expand  with  them 
and  always  exactly  fill  the  cavity. 

Since  Volume  X  Density  =  constant  Mass,  it  follows  that  the 
Volume  Expansibility  is  also  the  Coefficient  of  Diminution  of  Density ; 
when  Vq  increases  to  Vq  +  V^e^,  D^  decreases  to  D^  —  h^et,  as  is 
seen  upon  multiplying  out,  e^  being  quite  negligible. 

§  177.  Expansion  of  liquids.  The  expansion  of  fluids  is,  of  course 
volume  expansion  :  nothing  else  is  possible. 

Taking  that  quantity  of  liquid  which  occupied  1  c.c.  at  0°  C,  its 
expansion  in  c.c.  when  heated  l""  is  its  expansibility  E. 

Or  The  expansibility  or  coefficient  of  thermal  expansion  of  a 
liquid  is  its  increase  in  volume,  when  heated  1°,  expressed  as  a 
fraction  of  its  volume  at  0*  C. 

Notice  carefully  that  it  now  has  to  be  specified  at  what  temperature 
the  original  volume  is  measured.  This  is  because  liquids  expand  so 
much  more  than  solids,  e.g.  1  c.c.  alcohol  at  0°  becomes  1-015  c.c.  at  15°, 
and  if  only  1  c.c.  at  15°  were  taken,  E  would  work  out  1|%  too  small. 

As  in  §  176,  Vq  at  0°  becomes  Vq  +  VqET  at  T°, 

But  now  V  at  t°  does  not  become  V  +  VE(T  —  t)  at  T°, 

It  has  to  be  dealt  with  in  two  steps  : 

First,  V  at  ^  =  Vo  +  VqE^  =  Vo(l  +  E^ 

from  this  calculate  what  its  volume  Vq  at  zero  would  be ; 

then,  V  at  T^  -  Vq  +  VqET 

The  expansibilities  of  most  liquids  increase  rather  fast  at  higher 
temperatures,  and  E  usually  given  is  only  an  Average  Value  over 
some  ordinary  range  of  temperature,  which  ought  to  be  specified. 
See  Water,  at  end  of  §  179. 

§  178.  *  Apparent '  Expansion.  The  vessel  containing  a  liquid 
complicates  measurements  by  expanding  and  leaving  more  room 
for  the  contents.  [If  a  flask,  filled  with  water  to  somewhere  in  its 
long  narrow  neck,  be  suddenly  plunged  into  hot  water,  the  liquid  in 
the  neck  goes  down  for  an  instant — the  glass  has  got  heated  first.] 

The  Apparent  Expansibility  e  of  a  fluid  in  glass  is  therefore  less 
than  its  true  or  '  absolute  '  expansibility  E. 

It  is  fairly  plain  that  the  difference  between  them  is  the  volume 
expansibility  g  of  the  glass. 

Apparent  expansion  =  true  expansion  —  volume  expansion  of  vessel, 
e  =  E  -  gr.  i 

Here  again,  the  introduction  of  algebra  might  give  a  more  precisf  < 


f, 


§  178]  THERMAL   EXPANSION  129 

formula,  but  the  difference  is  so  small  that  it  would  be  immediately 
neglected  in  practice,  even  with  mercury,  which  is  only  seven  times 
;is  expansible  as  glass  :  see  table  below. 

The  Measurement  of  '  net '  or  <  apparent '  expansibility  in  glass 

vessels  is  really  what  concerns  us  most  in  practice  : — 

I.  By  the  weight  or  overflowing  dilatometer.  This  is  neither 
more  nor  less  than  a  specific -gravity  bottle,  §  136.  Its  weight 
( iiipty  (or  containing  a  few  bits  of  glass  to  act  as  stirrers)  is  6.  It 
is  stood  in  ice,  filled  with  ice-cold  liquid,  wiped  with  a  cold  cloth, 
and  weighed  quickly,  b  +  ^o-  ^^  ^^  warmed  in  a  water-bath  and 
kept  for  several  minutes  at  a  steady  t°  until  no  more  exudes,  wiped, 
and  weighed  (after  partial  cooling,  §  159),  b  +  Wf  This  may  be 
repeated  at  several  rising  temperatures  ;  then  for  each  : 

Wt  is  the  weight  of  liquid  at  t°  filling  the  bottle  of  volume  v,  which 
//•„  filled  at  0°.  {We  are,  of  course,  agreeing  to  neglect  the  change  of  v 
with  temperature.)  If  liquid  density  at  0°  he  d,  v  =  wjd,  since 
volume  =  wt. /density. 

If  cooled  to  0°  again,  Wt  would  have  a  lesser  volume  =  Wt/d,  the 
part  of  the  bottle  {Wq  —  Wt)ld  being  now  left  empty. 

Heated  to  t°  again,  volume  Wtjd  would  again  expand  and  fill  this 
part ;  Wtid  expands  {Wq  —  Wt)ld  .  t  per  degree. 

.*.  unit  volume  expands  — S — ^ -r  =  -^ r-  =  e. 

^  a .  t  a  Wt .  t 

Notice  it  is  w^  in  the  denominator,  and  not  Wq,  for  part  of  the 
expansion  of  the  whole  mass  occurred  in  the  thrown-away  overflow 
after  it  got  outside. 

Ex.  5.  A  sp.-gr.  bottle  contained  40  gm.  of  a  liquid  at  0^;  after  keeping  at 
3')"  until  no  more  exuded  it  contained  only  39-5  gm. 

39-5  expands  (40  -  39-5)  =  0-5  for  35°. 
.-.  1  expands  0-5  ^  (39-5  x  35)  =  0-000362  for  1°. 

II.  By  the  hydrometer  or  the  hydrostatic  balance. 

Since  mass  =  volume  X  density,  when  a  liquid  volume  1  expands 

1  -f-  et  its  density  or   specific    gravity  decreases  in  the  ratio 

1/(1  +  et).     Hence,    measuring    its    specific    gravity    at    different 

temperatures  with  a  common  glass  Hydrometer  [or  a  glass  ball  hung 

from  a  Hydrostatic  Balance]  enables  e  to  be  calculated. 

Ex.  6.  A  hydrometer  in  terebene  at  0°  read  0-870  and  at  61°,  0-820. 

1/(1 +  61e)  =  0-820/0-870. 
.-.  0-820  X  61e  =  0-05  e  =  0-00100. 

Note.— If  a  fluid  is  enclosed  in  a  long  tube  its  increase  in  length 
denotes,  not  a  linear,  but  a  volume  expansion.  For  the  tube  being 
unyielding,  all  three-ways  expansions  are  squeezed  into  one  way. 
In*^speaking  of  the  linear  expansion  of  solids  then:  sideways  swelUng 
is  ignored. 


130 


HEAT 


[§178 


E  —  e,  the  reduction  in  expansion  caused  by  the  vessel,  is,  per 
cent.  : 


I 


Common 

glass 

(g  =  0-000025), 


Pyrex 

glass 

(gr  =  0-000009). 


Fused  silica 
(gr  =  0-0000016). 


Mercury 
Alcohol  . 
Air   . 


0-00018 
000110 
0-00367 


14% 
2-5 

0-7 


5% 
0-9 
0-25 


1% 
0-15 
005 


Thus  nowadays  one  must  know  what  kind  of  glass  is  in  use  ;  and 
an  expansion  bulb,  or  in  II  a  ball,  of  fused  silica  glass  all  but  abolishes 
the  vessel  correction. 

This  is  the  practical  way  of  doing  it  nowadays,  but  the  following 
method  is  of  theoretical  interest. 

§  179.  True  or  *  absolute  '  expansibility  experimentally.    A  Hare's 

apparatus  of  balancing  columns  is  used,  the  legs  being  filled  with  the 

same  liquid,  cold  and  hot.  As  ex- 
plained in  §§  104  and  137,  this  is 
quite  independent  of  the  sizes  of  the 
tubes,  therefore  the  swelhng  of  the 
hot  glass  does  not  affect  it  at  all 
(provided  the  scales  are  not  on  the 
tubes).  In  the  apparatus  of  Fig.  63, 
designed  as  a  laboratory  illustration 
of  the  method,  two  lengths  of  glass 
joined  by  a  short  narrow  rubber 
tube  form  a  U  tube,  kept  at  100° 
on  one  side  by  a  steam-jacket  and 
cooled  on  the  other  by  ice -water. 
The  liquid,  which  at  the  start  is  at 
the  same  level  on  both  sides,  finally 
stands  at  69-6  cm.  on  the  cold  and 
73-1  on  the  hot. 

Since  each  represents  the  same 
hydrostatic  pressure,  i.e.  the  same 
weight  per  square  centimetre  cross- 
section  of  tube,  a  volume  equal  to  the 
69-6  at  0°  has  expanded  3-5  for  100°, 
or  0-035  per  degree.  Therefore  1  at 
0  expands  per  degree  0-035  -f  69-6 
=  0-00053  =  E. 

The  absolute  expansibility  of  mercury  was  determined  by  Regnault 
with  an  elaborate  form  of  this  apparatus,  but  an  excellent  plan  is 
simply  to  '  boil  the  barometer,'  utilizing  the  atmosphere  as  the  cold 
balancing  column.  A  siphon  barometer  of  the  shape  shown  in 
Fig.  35  (S)  is  enclosed  in  a  jacket  (out  of  which  only  the  end  of  its  i 


§  180]  THERMAL  EXPANSION  131 

open  tube  protrudes),  and  is  read,  at  various  temperatures  of  circulat- 
ing fluid,  by  a  cathetometer  and  scale  kept  at  the  constant  room 
temperature.  Calculation  as  above,  *  cold  '  =  initial  height  of 
barometer.  Any  small  variation  of  the  atmospheric  pressure  during 
the  experiment  must  be  observed  on  the  laboratory  barometer  and 
allowed  for,  and  there  must  be  added  to  the  height  at  each  tem- 
perature the  small  depression  due  to  the  increasing  pressure  of 
mercury  vapour  in  the  TorricelUan  space  (0-03  cm.  at  100°  1-83  at 
200°,  etc.,  see  §  282). 

Note. — There  is  no  need  to  measure  the  absolute  expansibility 
of  any  other  liquid  by  this  method,  for  E  of  mercury  once  known, 
a  glass  dilatometer  can  be  filled  with  it  and  g  of  the  glass  =  E  —  e, 
the  observed  falling-ofE  in  expansibility.  Then  g  is  added  on  to 
other  liquids  examined  in  the  same  dilatometer. 

Some  average  values  of  E  are,  in  parts  per  10,000  : 

Glycerine  5 ;  strong  sulphuric  acid  6 ;  olive  oil  7 ;  parafl&n  oil 
9 — 10  ;  xylol  10  ;  alcohol  and  acetic  acid  1 1  ;  methyl  alcohol, 
benzene,  CSg,  chloroform,  petrol  12 ;   ether,  pentane  16. 

Water:  5—10°,  0-53;  10—20°  1-5;  20—40°  30;  40—60°  4-6; 
60—80°  5-9  ;   average  0—100°  4-5. 

§  180.  In  §  113  the  Rule  for  correcting  Mercury  Barometer  readings 
for  temperature  was  quoted.     It  is  arrived  at  as  follows  : 

•  Ex.  7.  The  Barometric  Column  stands  at  76  cm.  at  O*',  what  will  be  its 
true  height  at  25°  ? 

The  problem  is  to  keep  the  hydrostatic  pressure,  i.e.  the  weight  of  a  square 
centimetre  column,  the  same.  The  expansion  of  the  glass  has  nothing  to 
do  with  it. 

I  c.c.  of  mercury  at  0°  becomes  1  +  1  X  0000182  X  25  =  1-00455  c.c. 
ai  25°. 

.-.  1  c.c.  at  25°  weighs  only  1/100455  of  the  c.c.  at  0°,  and  .-.  1-00455 
times  as  many  c.c.  must  be  piled  up  on  the  1  sq.  cm.  btise. 

.*.  True  height  =  76  x  1  00455  =  76-346  cm. 

In  the  foregoing,  if  the  height  were  being  measured  in  a  Brass  Scale,  correct 
at  0°,  and  of  linear  expansibility  000001 8,  what  woidd  the  reading  be  ? 

Pretty  plainly,  the  scale  is  engaged  in  a  hopeless  attempt  to  overtake  the 
mercury,  and  we  must  deduct  its  expansion,  by  way  of  discount,  from  that 
of  the  mercury.  In  actual  practice,  we  have  to  reduce  the  reading  on  a  warm 
barometer,  to  its  true  value  at  0° ;   thus  : 

Ex.  7a.  A  Barometer  reads  on  its  Brass  Scale  76-50  cm.  at  t°,  what  is  the 
reading  at  0°  ? 

Strictly  Ho  +  Hp  x  (0000182  -  0-000018)  X  t  =  76-50 

which  becomes,  for  all  practical  purposes. 

Ho  =  76-50  -  76-50  X  (0-000182  -  0-000018)  X  i°. 

Hence  the  Practical  Rule  for  correcting  the  Barometer  for  tem- 
perature :  From  the  observed  height  deduct  (observed  height  X 
0000164  X  f  C),  i.e.  deduct  1/6000  the  observed  height  for  every  degree 
of  temperature  Centigrade. 


7 


132  HEAT  [§  181 

§  181.  Water.  Water  expands  increasingly  faster  at  high  tem- 
peratures and  contracts  increasingly  slower  at  low,  as  do  most 
liquids,  but  it  gradually  ceases  to  change  at  all,  and  thereafter  begins 
to  expand  on  the  way  down  to  its  freezing  point,  becoming  1/8000 
part  bulkier  at  0°.  Thus  there  is  a  temperature  at  which  its  volume 
is  least,  and  therefore  its  density  a  maximum.  This  is  4°  C.  or  39- 1  °  F. 
Altered  either  way  it  very  slowly  expands ;  it  is  because  the  change 
for  1°  is  hardly  measurable  that  this  temperature  was  taken  in 
defining  the  gramme.  Conversely,  the  slow  change  makes  it 
•difficult  to  find  this  maximum  density  temperature  accurately. 

Joule  used  a  large  absolute -expansion  apparatus,  Fig.  63,  but, 
instead  of  attempting  to  observe  difference  in  level,  he  opened  a 
cross-channel  at  the  top  and  watched  which  way  a  floating  bulb 
drifted  {i.e.  towards  the  denser  column,  down  which  the  water  sank). 
Arguing  that  at  equal  distances  either  side  of  the  maximum,  the 
water  would  be  equally  lightened,  he  found  that  with  one  column 
at  2°  and  the  other  at  6°,  the  float  did  not  move,  and  the  mean  of  all 
such  pairs  of  temperatures  was  4°. 

We  all  see  a  similar  apparatus  in  action  at  home,  producing  the 
sulky  fire  of  muggy  weather,  and  the  bright  clear  one  of  frost,  when 
the  colder  denser  outside  air  sinks  heavier,  and  pushes  a  sharper 
draught  up  the  chimney,  blowing  up  the  fire,  and  so  heating  the 
chimney  hotter  and  intensifying  the  effect. 

This  idiosyncrasy  of  water  has  an  effect  in  nature  which  can  be 
illustrated  by  a  tall  jar  of  water  and  floating  ice,  with  a  ther- 
mometer dropped  to  the  bottom  and  another  held  near  the  top. 
Both  run  down,  but  the  bottom  one  slows  up,  and  stops  a  little 
above  4°,  while  the  top  continues  to  0°,  the  water  getting  lighter. 

When  the  ice  is  fished  out,  the  top  rises  to  4°  before  the  bottom 
begins,  for  although  the  surrounding  air  warms  all  parts  of  the  jar, 
yet  as  long  as  there  is  water  at  4°  anywhere  it  sinks  to  the  bottom. 
Provided  with  a  waist-band  for  ice,  this  jar  is  called  Hope's  apparatus, 
but  this  is  a  needless  elaboration,    try  this  experiment. 

In  consequence,  fresh-water  fish  can  lie  quiet  in  4°  C.  beneath  the 
ice-shield,  while  fish  in  the  salt  marsh  must  endure  —  2°  C,  for  sea- 
water  then  begins  to  freeze,  before  having  reached  its  maximum 
density,  -  3°  C. 

Some  oily  liquids  can  be  seen,  in  the  polarizing  microscope,  to 
form  swarms  of  minute  spherular  '  liquid  crystals,'  a  few  degrees 
above  their  freezing  to  crystalline  solids.  It  may  fairly  be  inferred 
that  some  such  molecular  re-arrangement  is  taking  place  here,  though 
on  an  entirely  ultra-microscopic  scale,  and  the  ice  structure  being  so 
much  bulkier  than  water,  the  increasing  proportion  of  it,  as  the 
temperature  falls,  swells  the  mixture. 

Indeed,  there  is  a  good  deal  of  evidence  that  while  steam,  from  its 
vapour  density,  is  HgO,  liquid  water  is  mainly  '  dihydrol '  (H20)2, 
but  contains  from  16%  at  the  boiling  point,  to  37%  at  the  freezing 
point,  of  '  trihydrol '  {H^^)^,  of  which  ice  mostly  consists.  At  the 
same  time,  it  contains  a  proportion  of  HgO  increasing  towards  the 


s^  182]  THERMAL   EXPANSION  133 

l)oiling  point,  and  accounting  for  its  rapidly  increasing  rate  of 
expansion,  for  which  see  §  179,  end. 

'  Heavy  water  '  has  now  arrived  to  compUcate  matters,  being 
})resent  in  ordinary  water  to  about  1 /6000th  part.  It  has  a  maxi- 
mum density  1-106  at  11-8°. 

§  182.  Expansion  of  gases.  In  finding  the  thermal  expansibility 
of  a  gas,  care  has  to  be  taken  not  to  permit  elastic  expansion  on 
account  of  diminution  of  pressure.  The  definition,  in  §  177,  with  the 
words  *  at  constant  pressure '  inserted,  applies  to  gases. 

In  a  simple  apparatus  the  gas  partly  fills  a  horizontal  graduated 
capillary  tube,  being  shut  in  between  its  sealed  end  and  an  index- 
tliread  of  sulphuric  acid.  The  volume  of  the  gas  is  proportional  to 
tlie  length  it  occupies  in  the  uniform  tube.  The  tube  is  raised  from 
ice  to  T°,  and  provided  the  whole,  i.e.  the  barometric,  pressure  has 
not  altered,  V^  =  Vq  +  VqcT,  and  e  is  so  large  that  g  of  the  glass 
can  be  ignored.     It  is  a  simple  but  exasperating  experiment. 

In  this  way  Gay-Lussac  and  Charles  found  that  all  gases  expand 
equally,  and  what  is  commonly  known  as  the  *  Law  of  Charles  '  states 
Tliat  AU  gases  expand  1/273  of  their  volume  at  0°  C.  for  each  degree 
rise  of  temperature,  the  pressure  being  constant.  The  gases  must 
not  be  too  near  their  liquefying  temperatures,  and  of  course  no 
tliemical  changes  {e.g.  N2O4  into  2NO2)  are  allowable. 

The  Weight  Dilatometer  method  also  lends  itself  to  the  measure- 
iiient  of  air  expansion.  Dry  a  specific  gravity  bottle  [to  dry  any 
ifottle,  warm  it  gently,  §  277,  and  blow  into  it  through  a  tube  ;  when 
\  isibly  dry,  suck  out  one  breath  instead],  fit  the  stopper  in  with  the 
slightest  trace  of  grease,  and  weigh,  b.  Plunge  under  hot  water  at 
i°,  then,  keeping  it  drowned  all  the  time,  hold  under  the  tap  and 
ultimately  smother  with  plenty  of  broken  ice  ;  remove,  wipe,  and 
weigh  the  bottle ;  write  this  weight  =  b  -^  Wq  —  Wf  Now  fill 
eompletely  with  the  ice-water,  and  weigh,  b  -f-  i^o- 

As  in  §  178,  the  difference  Wt  would  expand  by  Wq  —  Wt  if  heated 
again  to  t°,  and  as  before 

Wt  .t 
There  are  sources  of  inaccuracy,  but  this  is  a  fair  and  not  un- 
common practical  exam  exercise ;    be  careful  to  understand  it,  or 
the  weights  will  trick  you. 


EXAM   QUESTIONS,  CHAPTER  XI 

The  chapter  lends  itself  to  a  good  mttny  calculatory  questions,  in  which 
you  have  to  be  careful  not  to  leave  things  out  :  fortunately  you  meet  them  in 
the  lab.  §  179  is  rather  ancient  history,  §  180  a  mere  tribute  to  the  principle 
italicized  in  Chap.  VI ;  read  §  181,  and  if  asked  for  it  don't  go  on  about  ice; 
§  182,  fellow  to  Boyle's  Law,  is  made  much  use  of  in  the  next  chapter,  where 
Questions  will  be  found. 


134  HEAT 


8.  Calculate  the  increase  in  volume  of  a  litre  flask  between  4°  C.  and  40°  C. . 
and  that  of  a  steel  ship  of  8000  tons  register  (each  =  100  cu.  ft.)  between  30°  Fj 
and  90°  F. 

9.  Define  the  linear  expansibility  of  a  solid.  Find  the  expansion  per  1000 
yd.  of  a  steel  signal  wire,  e  =  0-000012  per  °C.,  between  20°  F.  and  100°  F. 

10.  Describe  a  method  of  measuring  linear  expansibility,  mentioning  any^ 
sources  of  error.  How  is  the  figure  affected  by  (o)  °F.,  (6)  inches  instead  of 
cm.  ?  An  aluminiiun  piston  in  an  iron  cylinder  10  cm.  bore  has  0-03  cm. 
clearance  at  15°  C. ;  at  what  temperature  would  it  seize  up  ?  Expansibilitiei 
0-000011  and  22.     (x  2) 

11.  An  iron  bar  of  cross -sectional  area  10  sq.  cm.  connects  two  unyieldini 
supports  2  m.  apart ;  it  is  heated  to  320°  C. ;  calculate  the  force  it  exerts  wher 
cooled  to  20°  C,  the  expansibility  being  0-000012,  and  Young's  modului 
2  X  10^2  dynes /cm.2 

12.  Why  does  the  rate  of  a  pendulum  clock  depend  on  temperature,  anc 
how  can  it  be  compensated  ?  A  brass  pendulum  keeps  time  at  10°  C. ;  calcu 
late  its  daily  loss  at  25°  C.     (  X  4) 

13.  Give  a  diagram  of  any  modern  temperature-compensated  pendulum 
What  length  of  zinc  tube  is  necessary  to  compensate  the  expansion  of  (itj 
own  length  +  42  in.)  of  steel  ?     (  X  3) 

14.  Calculate  the  proportions  of  a  compensated  pendulum  with  a  pyrex 
glass  rod  and  a  lead  bob. 

15.  How  is  a  barometer  affected  by  temperatm-e  ?  If  one  reads  29-5  in. 
on  a  glass  scale  at  15°,  what  would  it  read  in  a  cold-storage  chamber  at  2°  C.  ? 

16.  A  specific  gravity  bottle  weighing  8-75  gm.  empty  weighs  33-8  gm. 
full  of  liquid  at  0°  and  33-0  full  at  40°.     Find  expansibility  of  liquid. 

17.  What  is  the  effect  on  the  reading  of  a  barometer  of  a  change  of  (a) 
temperature,  (6)  pressiu-e  of  the  atmosphere  ? 

A  barometer  with  a  brass  scale  reads  750  mm.  at  15°  C.  What  will  be  the 
reading  at  0°  C.  ?  The  coefficients  of  cubical  expansion  of  mercury  and  brass 
are  0-000180  and  0-0000567,  respectively.     (  X  2) 

18.  What  are  the  real  and  apparent  coefficients  of  expansion  of  a  liquid  ? 
Establish  a  relation  between  them  and  the  expansion  of  the  material  of 

the  vessel. 

How  would  you  find  the  real  expansion  for  a  liquid,  without  knowing  the 
expansion  of  the  vessel  ?     (  X  3) 

19.  What  do  you  understand  by  the  absolute  expansibility  of  a  liquid  ? 
Show  that  density  at  t°  =  ^^(1  —  et).  A  barometer  which  stood  at  0-75  cm. 
at  0°  stands  at  76-33  cm.  (true)  at  .100°  C.  Adding  on  0-03  cm.  for  vapour- 
pressure  of  mercury  at  100°,  calculate  absolute  expansibility.     (  X  4) 

20.  Explain  how  you  would  determine  the  coefficient  of  thermal  expansion 
of  a  liquid  by  weighing  a  block  of  silica  in  it. 

21.  A  glass  hydrometer  read  s.g.  0-920  in  a  liquid  at  45°  C,  the  liquids 
expansibility  is  0-000525,  and  the  glass,  linear,  0-000008 ;   what  would  be  the 
reading  at  15°  C.  ? 

22.  How  could  you  demonstrate,  and  determine,  a  temperature  of  maxi- 
mum density  for  water  ?     Of  what  importance  is  this  in  the  economy  oCi 
Nature  ? 

23.  Describe  the  change  in  volume  of  water  from  a  '  supercooled '  conditio] 
below  freezing  point  to  nearly  boiling  point;  what  difference  would  it  makeji 
whether  your  observations  were  made  with  ordinary  or  silica-glass  apparatus  ?i 
(X  2) 


PRACTICAL  QUESTIONS. 

Measure  the  coefficient  of  expansion  of  a  metal  rod. 

Measure  the  expansibility  of  a  liquid,  or  of  air,  by  the  s.g.  bottle. 

Compare  the  density  of  water  at  15°  and  30°  C.  by  weighing  glass  in  it. 


I 


CHAPTER   XII 


THERMOMETRY 


§  191.  The  branch  of  our  subject  which  deals  with  the  measure- 
ment of  Heat  from  the  point  of  view  of  hotness  or  temperature  is 
known  as  Thermometry.  It  is  needless  to  define  Temperature,  for 
the  first  physical  necessity  of  active  life  in  any  organism  is  a  certain 
degree  of  warmth,  and  accordingly  a  sense  of  temperature  is  found 
in  all  animals. 

Our  own  temperature  sense  is  located  in  small  '  warm  spots ' 
and  '  cold  spots  '  on  the  body  surface,  sensitive  to  temperatures 
above  and  below  that  of  the  skin  (whatever  it  may  happen  to  be), 
and  together  averaging  about  twelve  per  sq.  cm.  Excited  simultane- 
ously with  the  more  numerous  '  pressure  spots,'  these  tell  us  that  we 
are  touching  a  hot  or  cold  object ;  without  the  pressure  stimulus 
^^c'  feel  hot  or  cold  ourselves.  Their  first  response  is  quick,  but 
rapidly  falls  to  a  much  smaller  value,  and  ceases  to  attract  attention 
if  the  stimulus  is  protracted  :  the  coldness  of  the  water  does  not 
afflict  the  bather  after  the  first  few  seconds.  Accordingly,  the 
temperature  of  the  skin  can  be  altered  considerably  without  our 
l<nowing  much  about  it,  and  a  medium  temperature  which  affects 
t  he  warm  spots  on  a  cold  hand  may  affect  the  cold  spots  on  a  warm 
hand,  as  in  the  familiar  process  of  adjusting  the  temperature  of  the 
bath-water.  And  have  you  never  made  the  quite  sudden  discovery 
that  the  fire  is  out  and  you  are  very  cold  ? 

The  sense  is  curiously  localized,  and  is  cutaneous  only,  as 
appears  from  the  feeling  of  heat  when  perspiring  freely,  while  the 
clinical  thermometer  shows  a  body-temperature  hardly  higher  than 
usual. 

§  192.  Altogether  our  protective  temperature -sense  is  not  to  be 
relied  upon,  and  actions  in  inanimate  matter  have  to  be  employed. 
Those  mostly  made  use  of  are  : 

Solidiiacation,  Melting,  Boiling.  Use  of  these  is  familiar  enough. 
There  is  the  winter  puddle,  telling  us  if  '  it  freezes  ' ;  there  is  the 
problem  of  spreading  the  butter  at  all,  or,  alternatively,  of  rescuing 
it  from  too  long  a  stay  in  the  hearth  ;  there  is  the  sprinkling  of  water 
on  the  hot  flat-iron,  etc.,  etc. 

Colour,  and  other  changes  due  to  chemical  action.  Colour  and 
smell  warn  us  when  things  are  scorching  :  colour  changes  due  to 
thickening  of  the  oxidation  film  are  the  usual  guide  in  tempering 
steel.  Occasionally,  however,  the  sudden  ignition  of  an  oil  is  a 
*^mperature  mark  in  this  and  other  processes, 

135 


136  HEAT  [§  192 

Changes  in  the  colour  and  brightness  of  emitted  light  from  a  hot 
poker,  horseshoe,  fire,  etc.,  are  dealt  with  in  §  974,  on  Radiation 
Pyrometry  ;  these  hold  the  field  above  1300°  C. 

Briefly,  it  may  be  said  here,  that  hot  iron  at  500°  C.  is  just  visibly 
red  in  a  dark  place,  at  700°  it  is  still  dull  red,  at  900°  cherry  red  (for 
steel  quenching),  bright  red  at  1000°,  orange  at  1100°,  yellow  at 
1200°,  '  white  '  at  1300° ;  1400°  is  bright  welding  heat,  and  1500° 
*  dazzUng.' 

For  Change  in  Electrical  Resistance  see  §  778,  and  for  change  in 
power  of  producing  an  electric  current,  §  799 ;  both  these  are  in 
regular  use  from  very  low  temperatures  up  to  high  furnace  heat, 
1300°  C.  This  present  chapter  deals  only  with  the  commonest 
measure  of  temperature  of  all,  Expansion  and  Contraction. 

§  193.  Let  us  now  apply  the  work  of  the  last  chapter  to  the 
Mercury-in- Glass  Thermometer,  which  for  long  gave  the  standard 
measure  of  temperature. 

Filling  a  thermometer.  The  thermometer  consists  of  a  gla^ss 
'  stem  '  of  fine  and  very  uniform  bore,  with  a  suitably  sized  and 
shaped  '  bulb  '  at  one  end,  and  the  problem  is  to  fill  this  narrow- 
necked  bottle.  A  cup  to  contain  some  of  the  liquid  is  formed  at 
the  top  of  the  stem ;  the  bulb  is  warmed,  air  bubbles  out,  and  on 
cooling  some  of  the  liquid  draws  down  to  replace  it.  This  is  re- 
peated, and  then,  as  we  know  perfectly  well  that  air  will  be  sticking 
to  the  glass  (or  dissolved  in  the  liquid),  the  bulb  is  strongly  heated  i 
until  its  contents  have  nearly  boiled  away,  the  vapour  '  washing 
out '  this  air.  As  it  condenses  the  warmed  liquid  descends  and  fills 
the  whole.  It  is  now  heated  a  little  above  the  highest  temperature 
it  is  destined  to  measure,  and  the  top  of  the  stem  below  the  cup  is 
sealed  off  in  the  blowpipe. 

On  cooling,  there  is  only  hquid  and  its  vapour  inside,  but  this 
total  absence  of  permanent  gas  is  not  essential.  Most  common 
thermometers  retain  an  accidental  trace  of  air,  and  high-temperature 
thermometers  have  their  stems  deliberately  filled  with  nitrogen 
before  sealing.  Indeed,  a  mercury  thermometer  with  a  broken  top 
works  until  dirt  gets  in,  or  mercury  spills  out,  but  an  alcohol  one 
would  dry  up. 

Industrial  thermometers,  such  as  those  used  in  large  numbers  in 
food-storage  round  Smithfield  market,  often  have  expensive  '  dis- 
played '  scales,  for  easy  reading.  The  skill  of  the  glass-blowers  in 
blowing  new  tubes  for  these  scales,  in  replacement  of  breakages,  is 
amazing  ;  in  three  tries  they  will  reproduce  a  thermometer  true  to 
1°  F.     Asked  how  :   '  You  start  at  eight  years  old  '  ! 

Annealing  and  ageing.  The  instrument  is  now  baked  for  a  day 
at  its  highest  temperature.  This  annealing  gets  rid  of  strains  ini 
the  glass,  for  glass  gradually  yields  to  stress  even  when  cold — a  long 
tube  resting  at  the  ends  sags  year  by  year — and  this  used  to  show 
itself  in  thermometers  as  an  unsteady  crawl  upward  of  the  readiogi 
for  many  years.  The  high  heat  of  annealing  accelerates  this  to 
hours,  and  leaves  nothing  for  age  to  do. 


§  194]  THERMOMETRY  187 

It  does  not,  however,  prevent  a  temporary  lowering  of  zero  for 
half-an-hour  or  more  after  a  thermometer  has  been  towards  the  top 
of  its  scale,  due  to  a  lag  in  the  complete  contraction  of  the  glass  : 
this  has  to  be  met  by  using  some  better  variety  of  glass.  For 
clinical  or  common  use,  where  thermometers  are  not  taken  quickly 
over  extreme  ranges,  this  need  never  be  feared ;  but  in  testing  a 
thermometer,  try  the  zero  first. 

§  194.  Fixed  points  and  Scales  of  Temperature.  The  earliest 
thermometers  were  air- thermometers,  on  the  principle  of  Fig.  36, 
and  therefore  badly  affected  by  barometric  changes.  Spirit  was 
then  employed,  but  Fahrenheit  at  Amsterdam  in  1720  introduced 
quicksilver  instead.  He  apparently  took  the  greatest  cold  he  ever 
reached  (in  ice  and  salt)  as  zero,  and  the  temperature  under  his 
armpit  as  12°,  subsequently  dividing  these  into  eighths,  like  a  foot- 
rule.  On  this  scale  water  freezes  at  32°  and  boUs  180°  above,  at 
212°  F.,  and  these  prove  to  be  far  more  reliable  standard  fixed 
points,  provided  that  the  boiling  takes  place  at  normal  atmospheric 
pressure.  Anders  Celsius  of  Upsala  devised  a  scale  extending  from 
zero  in  boiling  water  to  100°  in  ice,  but  his  friend  Linnaeus,  the 
botanist,  more  concerned  about  the  growth  of  plants,  induced  him 
to  start  from  the  freezing  point  as  zero  ;  and  this  Centigrade  ther- 
mometer was  described  in  December  1745,  and  daily  observations 
of  temperature  were  published  from  April  1,  1747,  by  which  time  a 
steady  sale  of  thermometers  all  over  the  world  had  been  established 
from  Upsala.  This  most  familiar  of  scientific  instruments  is  less 
than  200  years  old  ! 

It  is  frequently  necessary  to  convert  temperatures  from  one  of 
these  scales  to  the  other. 

To  do  so,  notice  that  the  Fahrenheit  scale  has  a  start  of  32°,  and 
this  must  be  deducted  first  of  all.  Then  its  readings  march  forward 
at  the  rate  of  180  steps  to  the  Centigrade  100,  or  for  every  9  steps  F. 
the  Centigrade  marks  only  5.  Whether,  therefore,  we  divide  the 
temperature  to  be  converted  (after  first  lopping  off  the  start)  into 
9*8  of  F.  degrees,  or  5's  of  C.  degrees,  we  arrive  at  the  same  number  of 
these  intervals,  and  this  is 

F.°-32_C.° 
9  5  ' 

the  Conversion  Formula,  in  which  F.°  and  C.°  stand  for  the  tempera- 
tures being  converted. 

Notice,  however,  that  a  difference  of  9°  F.  is  the  same  thine  as  a 
difference  of  5°  C. ; '  e.g.  expansibilities  F.  are  5/9  of  their  C.  values. 

The  popular  expression  '  ten  degrees  of  frost '  should  mean  ten 
down  from  32°  F. ;  i.e.  22°  F.  actual ;  Centigrade  frost  is  quoted 
in  minus  readings.  The  weather  so  much  cUsUked  in  Canada  is 
*  zero  weather '  F. ;  their  favourite  *  forty  below '  may  be  on  either 
scale,  as  trial  by  the  formula  will  show ;  and  it  is  the  only  common 
point  of  the  two  scales. 


138 


HEAT 


On  the  Absolute  Scale,  °A.,  to  be  introduced  in  §  200,  a  tempera- 
ture  reads  273°  higher  than  Centigrade 

°A.  =  273  +  °C. 

While  less  easy  to  get  used  to  than  the  24-hour  clock,  it  attracts 
meteorologists  because  it  does  away  with  all  risk  of  dropped  minus 
signs  ;  and  we  shall  see  later  that  it  dominates  theory. 

§  195.  Testing  and  graduating  thermometers.  For  the  freezing 
point,  put  the  thermometer  in  ice  broken  small  and  standing  nearly 


RPT 


AVOID 
ERROR  OF 
PARALLAX 


B.P^ 


U 


TESTINGTHE    FIXED    POINTS   OF   THERMOMETERS 


Fig.  64. 

full  of  the  pure  water  of  its  own  melting.  Solid  ice  without  water 
may  be  below  its  melting  point ;  and  water  containing  dissolved) 
salts  lowers  the  melting  point,  §  377.  Barometric  pressure  makes ( 
no  difference. 

For  the  boiling  point  the  whole  thermometer  must  be  in  a  current  1 
of  steam.     For  if  you  let  the  bulb  dip  in  the  water,  you  will  soon 
find  that,  unless  very  special  precautions  are  taken,  the  temperaturei 
of  boiling  water  is  dancing  about,  perhaps  J°,  quite  enough  to  destroy^ 
any  accuracy  of  reading ;  and  besides,  any  dissolved  salts  raise  the* 


§  196]  THERMOMETRY  139 

boiling  point,  §  376,  but  the  rise  does  not  extend  into  the  steam, 
where  the  salts  are  not. 

Commonly  one  sticks  the  thermometer  down  the  long  neck  of  a 
distilling  flask,  a  quarter  full  of  tap-water ;  they  are  easily  pro- 
curable, and  transparent,  but  have  two  objections :  the  ther- 
mometer bulb  is  exposed  to  some  risk  of  cooling  by  radiation,  and 
of  superheating  if  the  flame  gases  chance  to  play  upon  the  steam 
space.  The  metal  Hypsometer  in  Fig.  64  is  preferable  ;  it  is  provided 
with  an  outer  sheltering  jacket,  and  has  a  little  water-gauge  to 
indicate  that  the  steam  pressure  is  not  appreciably  above  the 
atmospheric  (from  too  much  fire  and  a  choked  spout). 

Beside  it  is  the  Barometer.  Allowance  has  to  be  made  that 
(among  ordinary  heights)  the  boiling  point  goes  up  or  down  with  the 
barometer  at  the  rate  of  1°  C.  for  every  2-7  cm.  {roughly  an  inch) 
above  or  below  the  normal  100°  C.  at  76  cm.  of  mercury,  or  1°  F. 
for  0-6  in.  A  thermometer's  boiling  point  cannot  be  tested  ttnthmU 
consulting  the  barometer.     See  Fig.  82. 

*  Hypsometer  '  actually  means  height-meter,  and  little  portable 
hypsometers,  provided  with  sensitive  thermometers  and  spirit- 
lamps,  are  commonly  sold  to  alpinists  in  France  :  reference  to 
Fig.  40  will  show  that  the  boiling  point  falls  1°  C.  for  300  m.  increase 
of  altitude  (and  logarithmically  later),  though  the  modem  aneroid 
is  much  less  trouble.  Nursery  stories  about  the  impossibility  of 
boiling  eggs  on  Mont  Blanc  will  likely  go  on  as  long  as  Humpty 
Dumpty,  but  egg-albumen  coagulates  at  60°,  and  the  boiling  point 
at  15,800  ft.  is  about  82-5°.  Try  it  yourself,  and  at  that  tem- 
perature you  will  cook  your  egg  to  perfection  ;  only  you  must  give  it 
ten  minutes.     And  see  the  complete  Hypsometric  Scale  in  Fig.  40. 

To  take  best  advantage  of  your  tests,  draw  the  thermometer 
scale  on  squared  paper,  and  set  up  or  down  at  each  end  of  it  the 
plus  or  minus  correction  which  has  to  be  added  to  the  false  readings 
to  get  true  temperatures,  e.g.  the  thermometer  shown  reads  —  0-5  in 
ice,  and  the  correction  +  0-5  is  therefore  set  up ;  it  reads  high  in 
steam,  and  —  0-7  is  set  up  {i.e.  0-7  down). 

Rule  the  straight  line  FAB  ;  its  height  above  or  below  the  hori- 
zontal scale  at  any  reading  gives  the  correction  to  be  added  (+)  to 
that  reading. 

I  have  heard  it  urged  that  you  are  no  better  off,  for  perhaps 
the  maker's  scale  has  incidental  errors  bigger  than  your  corrections. 
To  this  one  must  answer  that  it  is  very  probable  that  the  true 
correcting  line  may  resemble  FCB  or  FDB  rather  than  the  straight 
FAB,  but  that  it  is  as  far  out  as  FEB  is  most  improbable.  Let  B 
be  a  beehive  and  F  a  flower  :  no  single  bee  strictly  follows  the  bee- 
line  BAF,  but  they  swarm  along  tracks  like  BCF  and  BDF,  while 
not  one  in  a  hundred  will  go  by  E  half  round  the  garden.  So  in 
a  swarm  of  thermometers  only  very  few  will  be  far  out  in  the  middle 
after  the  ends  are  checked.  And  having  only  two  points,  all  one 
can  do  is  to  draw  a  straight  line  between  them,  and  be  content  that 
the  odds  are  long  that  this  correction  is  better  than  none. 


140  HEAT  [§  196 

§  196.  Stem  error  of  a  thermometer.  A  thermometer  when 
being  tested  is  entirely  immersed,  to  secure  a  uniform  temperature 
all  over,  but  in  common  use  its  long  stem  stands  out  in  a  much 
cooler  place.  The  mercury  in  the  stem  shrinks,  and  the  reading 
is  too  low.  If  the  mean  temperature  of  the  stem  can  be  ascertained, 
a  correction  can  be  calculated  as  in  the  following  : 

Ex.  1.  A  thermometer  sunk  to  its  20°  mark  in  a  bath  reads  90°.  Rest  of 
stem  averages  25°.  Find  true  temperature  of  bath ;  e  of  Hg  in  glass  0-00015. 
How  does  the  error  depend  on  (a)  length  of  degree  divisions,  (6)  expansi- 
bility of  thermometric  liquid,  (c)  increasing  difference  of  temperature  of  stem 
and  bulb  as  that  of  latter  rises  ? 

The  problem  is  to  find  the  length,  at  about  90°,  of  a  thread  of  mercury 
standing  above  the  20°  mark  which,  at  25°,  occupies  (90—20)  degree  spaces. 
The  procedure  for  solid  expansion,  §  173,  is  quite  near  enough. 

Lj,o  =  (90  -  20)  +  (90  -  20)  X  6  X  (90  -  25)°. 
=  70°  +  0-68°  correction. 
.-.  Corrected  temperatm-e  =  20°  +  70*68  =  90-68°. 

Evidently  the  correction  does  not  depend  on  (a)  at  all,  for  we  have  not  had 
to  inquire  their  length  {i.e.  a  long  sensitive  thermometer  does  not  suffer 
excessively) ;  (6)  it  is  proportional  to  the  expansibility  (therefore  large  for 
alcohol) ;  (c)  it  involves  (T  —  low  mark)  x  (T  —  low  temp,  of  stem),  i.e.  is 
about  proportional  to  T*,  becoming  very  serious  in  high-temperature  thermo- 
meters. 

§  197.  Mercury  and  alcohol  in  thermometers.  Mercury  freezes 
at  —  40°  and  boils  at  360°  C,  and  has  therefore  a  long  range,  in- 
cluding the  two  standard  fixed  points,  over  which  its  expansion  is 
'  reasonably  assumed '  to  be  steady.  It  runs  easily,  and  leaves 
nothing  on  the  glass.  It  heats  quickly,  being  a  good  conductor 
and-  having  small  heat  capacity  (§  217).  Its  expansion  is  small, 
permitting  only  a  slender  thread,  but  this  is  perfectly  opaque.  It 
does  not  distil  much  below  300°.  In  thermometers  for  use  above 
this  the  tube  must  be  '  packed  '  with  nitrogen,  which,  compressed 
by  the  expanding  mercury,  practically  prevents  its  vaporization. 
Nitrogen-packed  thermometers  of  hard  glass  are  used  up  to  500°  C, 
when  the  nitrogen  pressure  exceeds  20  atmos.,  and  in  fused  siHca  up 
to  yet  higher  temperatures. 

Alcohol  expands  six  times  as  much  as  mercury,  and  therefore, 
tinted  with  dye,  gives  a  large  bold  column  well  suited  to  domestic 
use,  although  it  is  a  bad  conductor  of  heat,  and  a  large  bulb  of  it 
warms  up  slowly.  It  never  freezes  in  the  most  Arctic  winter 
(f.-pt.  —  150°  C),  but  it  boils  at  78°  C.  (175°  F.),  and  therefore 
cannot  be  carried  to  the  upper  fixed  point,  but  must  be  scaled  against 
a  standard  mercury  thermometer ;  worst  of  all,  it  begins  to  distil 
long  before  this,  so  that  it  is  not  unusual  to  find  a  degree  or  two  of 
it  snugly  hidden  under  the  clip  at  the  top  of  the  stem  (colourless 
perhaps),  and  the  thermometer  reading  too  low  by  that  much. 

Pentane,  lightest  of  '  petrols,'  can  be  used  in  thermometers  right 
down  to  the  temperature  of  boiling  nitrogen,  where  it  is  still  liquid, 
though  very  viscous. 


§  198] 


THERMOMETRY 


141 


§  198.  Forms  of  thermometers.  For  domestic  use  a  wooden 
<(  ale,  firmly  attached  without  possibility  of  slip,  gives  a  bold  reading. 

All-glass  instruments  are  washable  and  non-corrodible.  An 
outer  protecting  tube  enclosing  the  paper  scale  gives  legibility 
and  cheapness ;  for  scorching  heat,  paper  is  superseded  by  a  slip 
of  opal  glass.  But  a  scale  etched  on  the  thick  stem  itself  is  the 
only  sort  sure  not  to  come  adrift.  Avoid  parallax  in  reading  it 
( I'ig.  64),  and  keep  a  twopenny  tube  of  oil-black  for  refilling  the 
marks. 

Registering  Maximum  or  Minimum  thermometers  are  often  useful. 

In  a  pattern  ascribed  to  Rutherford  (Fig.  65,  top)  the  mercury 
pushes  a  little  black  glass  pin  along  the  horizontal  tube,  leaving 
it  with  its  near  end  at  the  highest  point  reached ;   while  in  a  com- 


# 


Maxim  UTTv 


Sixs  mJTz.  8i  mcLX. 


|MM|MM|nM|j,„|,,u|Mn,nn|nM|^n|MMjnn^n,^^ 


ZTtimuTrv 


I II  M  n  1 1 1 II  nil  I  ui  II  HI  II  III  1 1  III  ' 

to    L    Ir.    It.    I».   iL   1.   ,L.r 


Lens-fiont  stem 
HALF  MINUTE-      \    section  ^Z 


■i 

CLINICAL,  (^^^S, 2^) 


Fig.  65. 


panion  alcohol  thermometer  the  spirit  drags  a  submerged  pin  down 
and  leaves  its  head  at  the  lowest  point  reached.  These  are  reset 
daily  by  tilting  them .  But  more  commonly  nowadays  the  maximum 
thermometer  is  constructed  on  the  principle  described  in  Clinical, 
below. 

The  instrument  invented  by  Mr.  Six  in  1782  is  an  alcohol  ther- 
mometer with  a  long  thread  of  mercury  shutting  in  the  spirit. 
Beyond  is  more  alcohol,  and  at  the  end  a  bulb,  containing  an  air 
and  vapour  space.  The  mercury  hardly  alters  in  length,  but  acts 
as  a  flexible  piston,  forced  out  by  the  expanding  alcohol,  or  driven 
back  after  it  by  the  air  pressure  in  the  subsidiary  bulb,  pushing 
either  way  little  black  pins,  and  leaving  them  at  the  highest  and 
lowest  points  reached.  Tiny  wisps  of  steel  wire  keep  them  stuck 
there,  until  dragged  back  daily  by  the  observer's  magnet.  The 
doubled-up  vertical  form  given  to  this  instrument  in  practice  18 


142  HEAT  [§198 

purely  a  question  of  compactness,  and  you  may  find  its  action 
easier  to  understand  if  you  re -draw  the  figure  with  the  tube  straight- 
ened out. 

In  a  Clinical  Thermometer  there  is  a  very  minute  constriction 
between  bulb  and  stem.  The  mercury  is  squeezed  past  this  by 
the  expansion  pressure  in  the  bulb,  but  its  weight  is  not  enough  to 
squeeze  it  back.  The  reading  of  the  patient's  temperature  is  made 
at  leisure,  and  afterwards  the  mercury  is  got  back  far  enough  by 
violent  swinging,  etc.  All  English  clinical  thermometers  are  tested 
at  Kew  :   any  doubtful  one  is  best  tested  in  your  own  mouth. 

The  clinical  is  an  excellent  example  of  a  sensitive  thermometer. 

Its  degree  spaces  are  long.  Each  °  F.  is  only  a  ten-thousandth  the 
volume  of  the  bulb,  but  the  bulb  itself  must  be  small  and  slender 
to  take  up  the  patient's  temperature  quickly.  The  bore  of  the  stem 
must  therefore  be  very  fine  ;  in  the  particular  instrument  drawn  it  is 
elliptical,  1/600  X  1/900  in.,  and  the  glass  of  the  stem  is  shaped  so 
as  to  magnify  its  breadth  when  seen  from  the  front ;   '  lens  front.' 

Small  as  is  the  bulb  of  a  clinical  thermometer,  it  can  '  lie  cold  ' 
in  the  mouth  for  quite  a  time,  for  the  warmth  necessary  has  to  be 
brought  up  to  it  by  the  circulation,  and  by  conduction  through  very 
poorly  conducting  tissues ;  and  the  temperature  is  wanted  fairly 
accurately.  The  safest  way  is  to  use  a  '  1 -minute  '  thermometer, 
keep  it  '  pre-heated  '  in  an  inner  pocket,  and  give  it  two  minutes. 

In  your  Organic  Chemistry  you  will  come  across  the  large-bulb 
Beckmann  thermometer,  graduated  to  0-01°,  and  provided  with  a 
little  overflow  reservoir  at  the  top'  of  the  stem,  by  manipulating 
which  the  thermometer  can  be  adapted  to  read  small  differences  of 
temperature  in  the  neighbourhood  of  any  desired  freezing  or  boiling 
point,  for  molecular  weight  determinations. 

In  a  modern  high-temperature  thermometer,  for  car  radiators, 
steam  boilers,  etc.,  a  steel  bulb,  sometimes  as  big  as  your  thumb, 
is  welded  to  a  stout  but  flexible  steel  tube  6  ft.  long  or  more  ;  the 
bulb,  and  such  narrow  space  as  is  left  between  the  narrow-bore 
tube  and  a  steel  wire  nearly  filling  it,  is  occupied  by  mercury,  and 
the  expansion  of  this  actuates  a  high-pressure  Bourdon  gauge,  like 
Fig.  39,  but  graduated  directly  in  temperatures  ;  up  to  212°  F.  for 
radiator- dashboard  use,  or  to  550°  C.  or  1000°  F.  for  the  engineer 
using  superheated  steam. 

For  '  compound  strip  '  Metallic  Thermometers,  see  §  175. 

§  199.  *  Standard  '  thermometers.  The  degree  Centigrade  was 
long  defined  in  England  as  0-01  of  the  fundamental  interval  on  a 
certain  mercury  thermometer  at  Kew,  which  means  that  the  apparent 
expansion  of  mercury  in  a  certain  glass  vessel  was  made  uniforin  by 
Act  of  Parliament. 

But  a  piece  of  that  same  glass  does  not  expand  quite  uniformly 
with  temperature  as  measured  on  that  thermometer,  which  must 
therefore  give  a  variable  expansion  to  mercury  itself.  The  standard 
scale  becomes  the  difference  of  two  imperfectly  regular  dilations, 


199] 


THERMOMETRY 


143 


a  mixture  of  about  7  parts  mercury  and  1  part  of  some  sort  of  glass. 
And  a  different  sort  of  glass  will  cause  a  discrepancy  of  as  much  as 
0-r  at  40°  C.     That  is  a  poor  state  of  affairs. 

Now,  Gases  expand  very  much,  so  that  change  in  the  containing 
vessel  has  a  much  less  disturbing  effect.  And  they  go  on  expand- 
ing all  very  nearly  alike  at  temperatures  below,  and  far  above,  the 
reach  of  mercury. 

The  constant-volume  gas  thermometer : — 


Fig.  66. 


Fig.  67. 


Let  a  volume  of  gas  which  has  expanded  to  Vq  +  Vq  X  1/273  X  <^ 
by  §  182,  be  now  compressed  into  its  original  volume. 

By  Boyle's  law  PV  is  constant  at  any  fixed  temperature,  so  that 
Po  X  (Vp  +  Vo  X  1/273  X  0  becomes  (Po  -f  Po  X  1/273  X  /)  X  Vo, 
as  is  evident  on  multiplying  out.  That  is,  if  a  gas  is  kept  from 
expanding,  its  pressure  rises  with  the  same  coefficietit,  1/273,  <u  that 
of  increase  of  volume  at  constant  pressure. 

Since  the  volume  remains  constant,  the  whole  of  the  gas  is  now 
kept  in  a  bulb,  the  rise  of  P  in  it  with  temperature  goes  on  uniformly 
throughout,  and  is  easily  read  on  some  form  of  pressure  gauge. 


144  HEAT  [§  199 

This  makes  a  more  manageable  apparatus  than  any  yet  invented  in 
which  the  gas -volume  is  permitted  to  increase. 

An  efficient  laboratory  form  of  constant-volume  air  thermometer 
is  shown  in  Fig.  66.  There  is  a  bulb  and  narrow  connecting 
tube,  the  latter  opening  into  a  wider  vertical  tube,  which  com- 
municates by  a  flexible  pipe  with  a  parallel  open  tube,  both  these 
containing  mercury.  To  maintain  the  constant  volume  the  mercury 
on  the  closed  side  is  kept  to  a  fixed  scale  division  (or  better,  Fig.  67, 
to  touch  a  glass  claw  sealed  inside  the  upper  end  of  the  tube,  where 
the  walls  slope  in  and  the  mercury  surface  is  flat).  This  is  effected 
by  raising  or  lowering  the  open  tube.  Comparing  with  Fig.  51,  you 
will  see  that  this  resembles  a  Boyle's  law  apparatus,  with  the  enclosed 
gas  heated  to  prevent  any  shrinkage. 

As  with  that  apparatus,  the  pressure  on  the  gas  is  H  +  A-  cm.  of 
mercury.  This  alters  steadily  1/273  {=  0-00367)  of  its  value  at  0°  C. 
between  the  most  extreme  temperatures,  e.g.  suppose  H  was  75  and 
h  6-9  at  0°,  making  H  +  )^  =  81-9  cm.,  then  11  -\-  h  rises  or  falls 
0-3  cm.  per  degree,  i.e.  as  long  as  the  barometer  stands  still,  h  alters 
this  much.  To  obviate  the  double  reading  of  H  and  h,  the  barometer 
is  sometimes  incorporated  in  the  apparatus,  as  in  Fig.  67,  a  form 
suggested  as  easy  to  fill  and  to  use,  either  with  or  without  accessory 
barometer,  and  withal  free  from  risk  of  leakage. 

Other  Fixed  Points  in  Modern  Thermometry  are  the  boiUng  points 
(under  760  mm.  pressure)  of  Oxygen  —  183-0°,  COg  —  78-5°, 
Naphthalene  218°,  and  Sulphur  444-55°  C, ;  the  transformation 
point  of  Na2SO4,10H2O  —  Na2S04  32-38°,  and  the  melting  points 
of  Mercury  —  38-9°,  common  salt  801°,  and  Gold  1063°  C. 

§  200.  Absolute  temperature.  The  observation  that  a  gas  alters 
its  volume  or  its  pressure  so  uniformly  by  1/273  of  its  value  at  0°  C, 
per  degree  change,  led  to  the  conception  of  a  temperature  at  which 
the  perfect  gas  would  have  shrunk  273/273rds  of  its  freezing-point 
value — it  would  have  no  volume  at  all  at  —  273°  C.  This  tempera- 
ture has  almost  been  reached,  but  we  have  to  rely  on  gases  for 
cooling,  and,  as  a  matter  of  fact,  all  gases  escape  this  annihilation 
by  liquefying,  and  what  laws  liquid  and  solid  may  follow  then  are 
unknown. 

Later  it  was  found  that  all  Electrical  Effects  in  Metals  bid  fair  to 
become  extinct  at  this  temperature ;  that  Radiation  experiments 
fit  in  with  a  law  which  assumes  the  cessation  of  Radiation  at  this 
same  point,  and  that  Specific  Heats  of  elements  are  converging 
towards  zero  value  there. 

There  seems  something  very  significant  indeed  about  —  273°  C, 
and  it  is  now  called  the  Absolute  Zero  of  Temperature,  and  from  it 
starts  the  Absolute  Scale  in  which  each  temperature  is  the  Centigrade 
plus  273° 

°A.  =  °C.  +  273 

As  closely  as  is  now  known,  the  exact  position  of  Absolute  Zero 
is  -  273-13°  C. 


§  202]  THERMOMETRY  146 

§201.  The  True  Scale  of  Temperature.  While  gas-pressure 
measurement  puts  Thermometry  in  a  much  better  position  than  the 
between-two-stools  one  we  found  in  the  mercury-in-glass  ther- 
mometer, it  still  leaves  the  Question,  '  Why  should  equal  rises  of 
pressure  in  a  gas  be  taken  as  degrees  of  temperature  ?  '  True,  pressure 
is  very  measurable,  but  there  are  other  things  easier ;  platinum 
resistance  thermometers,  §  778,  are  much  less  trouble  ;  what,  then, 
is  there  fundamental  about  gas -pressure  ? 

So  far,  nothing ;  it  is  a  convenient  bit  of  empiricism  open  to 
dispute ;  it  has  not  that  scientific  basis  we  feel  we  have  a  right  to 
demand. 

Read  on.  What  is  Gas  Pressure  ?  We  must  plunge  into  theory  : 
don't  be  alarmed,  glance  at  §  38. 

§  202.  According  to  the  Kinetic  Theory,  all  matter  is  made  up  of 
separate  minute  molecules  in  rapid  motion,  a  Gas  consisting  of  a 
swarm  occupying  a  comparatively  large  space.  In  this  they  fly 
about  at  high  speeds,  and  free  from  one  another's  interference, 
except  for  the  brief  event  of  a  '  collision  '  when  two  of  them  come 
so  close  as  to  change  each  other's  paths.  Actual  collision  between 
hard  particles  is  unlikely,  but  at  any  rate  the  mutual  action  is 
abrupt.  It  is  the  momentum  with  which  the  little  flying  masses 
hit  the  wall  which  constitutes  the  Pressure  of  the  Gas. 

If  the  space  available  for  them  to  fly  back  and  forth  in  is  halved, 
all  else  remaining  the  same,  evidently  they  will  hit  the  end  walls 
twice  as  often,  having  only  half  as  far  to  go ;  i.e.  the  pressure  is 
doubled,  which  is  Boyle's  Law. 

Let  a  cubic  centimetre  of  gas  contain  N  molecules,  each  of  mass  m, 
and  let  their  average  speed  be  v.     Dealing  with  momentum,  we 
may  neglect  the  intermolecular  collisions,  for  at  each  collision  there 
is  a  mere  transference  of  momentum  without  loss  (and  we  cannot 
follow  an  individual  molecule).     We  may  divide  N  into  three  equal 
L-roups,  going  N.  and  S.,  E.  and  W.,  up  and  down,  respectively ; 
rery  molecule  makes  v  journeys  per  second  across  the  1  cm.,  strikmg 
ther  wall  Jv  times  ;  therefore  each  wall  receives  per  second  JN  X 
"  =  JNv  blows.     Each  blow  gives  it  momentum  m  X  2v,  since  the 
iiolecule  is  stopped  and  reversed;    hence  the  forward  momentum 
destroyed  on  1  sq.  cm.  per  sec.  =  pressure  on  wall  in  dynes/cm.* 
=  JNv  X  2mv  =  JNmv^ 
=  I  mass  of  molecules  in  1  c.c.  X  v*, 
or     P  =  i  density  X  (molecular  speed).* 

Or  again 

P  =:  f  [i  (mass  Nm  of  molecules  in  1  c.c.)  X  (their  speed)*] 
=  |N  X  imv^,  the  average  energy  of  motion  of  a  molecule. 
Now  come  in  two  great  generalizations  : — 

The  first  is  Avogadro's  Principle,  with  which  your  Chemistry 
has  made  you  famihar  :  Under  identical  conditions  of  temperature 
and  pressure,  equal  volumes  of  all  gases  contain  the  same  number  of 
molecules. 


146  HEAT  [§  202 

Thus  N  per  c.c.  being  fixed,  P  is  in  a  fixed  proportion  to  (is  |  N 
times)  the  average  flyabout  energy  of  the  molecule  present. 

The  second  is  the  accepted  Statistical  Law,  that  In  a  mixed 
assembly  of  flying  molecules,  the  average  energy  of  all  flying  'particles 
is  the  same,  irrespective  of  their  masses,  §  367. 

Therefore  P  is  in  this  fixed  proportion  to  the  flyabout  energy  of 
any  gas  molecule  whatever. 

Energy  of  motion  is  absolutely  fundamental  in  Natural  Philo- 
sophy. Would  you  know  a  temperature  ?  Raise  a  gas  to  that 
temperature,  watch  a  molecule,  and  study  his  average  energy ; 
if  he  gives  you  the  slip,  no  matter,  watch  the  next  that  comes, 
whatever  it  may  be,  continuing  the  study. 

Now  then.  Make  the  average  flyabout  energy  of  the  gas  molecule, 
the  Temperature. 

From  rest  at  the  absolute  zero,  equal  increments  of  this  energy  mean 
equal  rises  of  temperature. 

That  is  the  true  scientific  Scale  of  Temperature. 

Luckily,  we  have  just  seen  that  the  Pressure  of  a  Gas  which  obeys 
Boyle's  Law — as  we  made  it  do  in  §  199 — ^is  strictly  proportional  to 
this  energy. 

Therefore  the  pressure  in  a  constant -volume  gas  thermometer 
filled  with  a  Perfect  Gas  rises  degree  by  degree  in  a  true  scale  of 
temperature. 

The  snag  is  :  there  is  no  Perfect  Gas ;  all  have  their  little  idio- 
syncrasies, see  §  147.  Choose  one  from  Fig.  52,  then  from  these 
and  other  curves  the  allowances  necessary  to  correct  its  thermometer 
readings  to  the  Perfect  Gas  Scale  are  deducible.  How  big  the 
discrepancies  are  likely  to  be  can  be  inferred  from  this,  that  between 
0°  and  100°  C.  hydrogen  and  nitrogen,  as  they  stand,  differ  nowhere 
more  than  0-02°,  whereas  the  mercury-in-glass  thermometer  is 
0-1°  too  high  at  50°. 

This  '  Kelvin  '  scale  of  temperature  is  adopted  internationally, 
and  absolute  temperatures  °A.,  are  marked  °K.  by  those  who  feel 
sure  of  the  corrections. 

§  203.  The  Law  of  Charles  may  therefore  be  restated  :  The  volume 
of  a  mass  of  gas  atflxed  pressure  is  proportional  to  its  absolute  tempera- 
ture T. 

V  cc  T  when  P  is  constant. 

Now,  Boyle's  Law  states  that  PV  is  constant  at  fixed  temperature, 
so,  having  hitherto  kept  P  constant,  we  stop  at  any  T  we  like,  and, 
altering  P,  V  changes  so  as  to  keep  PV  constant.  We  might,  for 
instance,  force  V  down  to  its  initial  size,  for  which  we  should  have 
to  maintain  a  P  proportional  to  T,  V  being  constant,  as  in  §  199. 

The  two  laws  then  combine  into  one  statement,  the  characteristic 
equation  of  a  perfect  gas,  PV  oc  T,  or 

PV  =  RT 


§204] 


THERMOMETRY 


147 


The  product  of  the  pressure  and  the  volume  of  a  mass  of  gas  is 
equal  to  R  times  its  absolute  temperature,  where  R  is  a  number 
which  depends  on  masses,  units,  etc.,  but  remains  fixed  when  once 
fitted  to  the  particular  case  in  hand. 

Ex.  2.     Find  R  for  1  c.c.  of  air  at  0°  C.  and  76  cm.  mercury. 

PV  =  RT  becomes  76  x  1  =  R  x  273.     .-.  R  =  0-279. 
Ex.  3.     Find  R  for  1  Gram-molecule  (mol.  wt.  in  gm.)  of  any  gas  (occupying 
22,320  c.c.  at  0°  C.  and  1  atmo). 

1  atmo.  =  1,013,000  dynes  per  sq.  cm. 
1,013,000  X  22,320  =  R  x  273,  R  =  82,900,000. 
Now,  PV  =  energy  (§  110)  in  ergs,  .-.  PV/T  =  R  =  ergs  per  degree  =  about 
2  cals.  (§  252)  per  degree,  i.e.  the  '  Capacity  for  heat  of  a  gram-molecule  '   of 
(my  gas  is  about  2. 

A  Corollary  to  this  example  is  this  : 

We  calculated  P  =  |  x  i  mass  molecules  in  1  c.c.  x  v^. 
Multiply  by  22,120,  the  vol.  of  1  gm.-mol. 
.*.  per  gm.-mol.  PV  =  RT  =  f  x  i  (gm.-mol.)  X  v^ 
and  R  =  2  calories,  .*.  3  T  calories  =  J  (gm.-mol.)  v^, 

or  the  average  fly-about  energy  contained  in  the  gram-molecule  of  a 
gas  =  3  calories  X  its  absolute  temperature. 

§  204.  Thermostats.  Thermostats  are  the  automatic  temperature 
regulators  employed  to  control  the  supply  of  cold  or  heat  to  cold 
store,  room,  bath,  incubator,  oven,  or  furnace,  when  it  is  necessary 
to  maintain  these  at  Constant  Temperature. 

Fig.  68  is  a  simple  laboratory  pattern ;  the  long  bulb  contains 
toluol  or  xylol,  which  expands  1-1  parts  per  1000  per  °  C.  rise  of 
temperature,  and  pushes  up  the  mercury  so  as  to  obstruct  the  end 
of  the  pipe  supplying  gas  to  the  heaters.  This  cut-off  can  be  pre- 
arranged to  occur  at  any  desired  temperature  by  raising  or  lowering 
the  pipe,  which  slides  or  screws  in  the  plug. 


Fig.  68. 


Fig.  69. 


Fig.  70. 


In  Fig.  69  a  '  compound  bar,'  §  175,  of  nickel-steel  and  nickel- 
silver,  which  differ  greatly  in  expansibility,  bends  and  unbends  with 
change  of  temperature,  and  makes  or  breaks  an  electric  contact 
which,  through  a  '  relay,'  switches  heaters  in  or  out.  There  is  often 
a  simple  magnetic  accessory  to  prevent  any  '  dithering,'  and  make 
contacts  sharp  and  definite. 


148  HEAT  [§204 

Fig.  70  shows  the  arrangement  usual  in  Incubators  :  a  flat 
capsule,  about  IJ  in.  diameter,  of  thin  german-silver,  is  completely 
full  of  ordinary  ether,  and  lies  inside  the  incubator.  Ether  boils 
normally  at  about  96°,  and  by  the  customary  incubation  tempera- 
ture, 106°  F.,  has  developed  a  considerable  vapour  pressure,  which 
puffs  up  the  capsule,  lifts  the  lever  pivoted  at  X,  and  the  damper 
plate,  and  lets  the  hot  fumes  from  the  lamp  escape  more  freely  up 
the  chimney.  Regulation  of  temperature  by  2°  or  3°  F.  is  effected 
by  sliding  a  weight  on  the  lever,  so  as  to  increase  pressure  on  the 
capsule  and  raise  the  boiling  point. 

In  a  more  elaborate  electrical  contrivance,  rising  temperature 
upsets  the  balance  of  a  Wheatstone  bridge,  §  784,  with  copper  and 
constantan  arms  ;  and  there  are  many  others. 


EXAM   QUESTIONS,    CHAPTER   XII 


A  chapter  you  have  to  study  and  answer  questions  on,  with  some  practical 
experiments.  Skip  the  calculation  of  §  196,  but  pay  much  heed  to  §  200, 
and  read  through,  once,  §§  201,  202,  and  find  how  a  simple  theory  and  a  little 
clear  thinking  lift  you  on  to  an  altogether  higher  plane  of  Science  :  you  will 
want  them  later.  §  203  is  utilized  below ;  §  204  concerns  contrivances  coming 
into  use  everywhere. 

4.  What  are  the  essential  features  of  a  satisfactory  instrument  for  measuring 
temperature  ? 

5.  What  sources  of  error  are  inherent  in  a  mercury  thermometer,  and  how 
do  you  correct  for  them  ? 

6.  Describe  the  construction  and  graduation  of  a  thermometer  to  read 
from  -  5°  to  150°  C. 

If  two  spherical  thermometer  bulbs  have  diameters  as  3  :  2,  and  their  tubes 
bores  as  2  :  3,  compare  the  ratio  of  lengths  of  degree  divisions. 

7.  Discuss  the  relative  merits  of  mercury  and  alcohol  for  filling  thermo- 
meters. A  thermometer  read  —  1°  in  ice  and  102°  in  steam  at  77  cm.  pres- 
sure (2' 7  cm.  raise  b.  pt.  1°);  what  was  the  correct  temperatiu-e  at  reading 
75°? 

8.  Distinguish  true  and  apparent  expansibilities.  Find  the  length  of  1° 
on  a  thermometer  stem  of  0-1  sq.  mm.  cross-section  when  there  is  1  c.c.  of 
mercury  belew  the  zero  mark. 

9.  If  a  Fahrenheit  thermometer  reads  70°  when  a  Centigrade  one  reads, 
correctly,  21°,  what  is  the  correction?     Also  vice  versa? 

10.  Convert  50°  C,  -  40°  C,  and  -  273°  C.  into  Fahrenheit  temperatures, 
and  zero  F.  and  98-5°  F.  into  Centigrade. 

11.  Describe  the  construction  of  a  clinical  thermometer.  How  would 
you  test  its  accuracy  ? 

The  bore  of  the  tube  is  005  mm.  and  from  95°  to  105°  F.  is  5  cm.  Calculate 
the  volume  of  the  bulb. 

What  is  this  range  on  the  Centigrade  scale  and  what  reading  corresponds 
to  98-5°  ?     (  X  5). 

12.  Describe  a  clinical  thermometer,  showing  how  it  is  made  sensitive, 
quick,  and  self -registering.  What  precautions  are  necessary  in  use  ?  If 
the  bulb  contains  1 /300th  cu.  in.  of  mercury,  what  area  of  cross-section  of 
the  stem  would  give  degrees  a  quarter  of  an  inch  long  ?     Exy.  9  x  10'^  1°  F. 


I 


THERMOMETRY  149 

13.  Describe  a  maximum  and  minimum  thermometer. 

14.  What  physical  properties  besides  expansion  can  be  used  to  measiire 
change  of  temperature  ?     Describe  a  thermometer  dependent  on  one.     (  X  2). 

15.  Describe  what  methods  you  would  use  for  the  measurement  of  tem- 
peratures of  about  —  50°  C.  and  of  about  1000°  C.     (  X  4) 

16.  Define  the  degree  Centigrade. 

Describe  any  pattern  of  constant-volume  gas  thermometer,  and  give  reasons 
which  have  led  to  it  superseding  the  merciuy  as  standard.     (  x  3). 

17.  How  would  you  determine  the  freezing  point  and  boiling  point  of 
mercury  ? 

18.  State  the  laws  of  variation  of  pressure,  volume,  and  temperature  for 
the  more  permanent  gases. 

What  deviations  from  the  laws  are  observed  when  great  preesures  are 
employed  ? 

19.  How  is  an  Absolute  Scale  of  Temperature  arrived  at  ? 

A  litre  flask  of  air  at  17°  C.  is  inverted  and  sunk  in  a  lake  to  a  depth  of 
20  ft.  (the  water  barometer  standing  at  34  ft.);  what  is  the  volume  of  the 
air  at  280°  A.  ?     (  X  2) 

20.  Fifteen  litres  of  air  are  cooled  from  45°  to  15°  C,  and  pressure  is  reduced 
from  795  to  760  mm. ;   calculate  the  new  volume. 

In  PV  =  RT  write  the  given  values  (converting  into  °A.) 

Old  conditions  795  x  15  =  R  X  (273  +  45) 

New        „  760  X    V  =  R  X  (273  +  15) 

from  the  first,  R  =  795  X  15/318 

substituting  in  second      V  =  795  X  15/318  X  288/760 
or,  collecting  like  terms    V  =  15  X  old  795/760  mm.  X  new  288/318  °A. 
=  14-22  lit.] 

21.  A  38-c,c.  glass  bulb  with  a  narrow  neck  is  immersed  in  boiling  salt 
solution,  and  its  tip  is  sealed  off;  opened  under  water  at  17°,  9  c.c.  flow 
in;   what  is  the  b.  pt.  of  the  solution  ? 

22.  Define  exactly  what  is  meant  by  N.T.P. 

To  what  temperature  must  carbon  dioxide  be  heated  so  that  its  density 
will  be  the  same  as  that  of  carbon  monoxide,  at  the  same  pressure,  but  at 
0°C.? 

23.  Describe  a  simple  constant -pressure  air  thermometer,  and  show  how 
you  would  determine  the  position  of  Absolute  Zero.  If  it  read  3-4  in  ice  and 
7*3  in  steam,  what  was  the  temperature  of  a  boiling  liquid  8*2  ? 

24.  Describe  a  constant-volume  air  thermometer  and  state  exactly  how 
you  would  use  it  to  determine  the  melting  point  of  vaseline,  or  the  boiling 
point  of  benzene.     (  X  3) 

25.  Justify  the  use  of  the  same  coefficient  for  pressure  increase  as  for  volume 
expansion.  The  closed  limb  of  a  constant-volume  thermometer  standmg 
at  30  cm.,  the  open  limb  reads  32-4  cm.  in  ice,  6M  in  steam,  and  13-7  in  CO, 
enow,  find  the  temperatiu*e  of  the  latter.     (  X  2) 

26.  State  the  Laws  of  Boyle  and  Charles,  and  deduce  the  variation  of 
pressure  with  temperature,  without  change  of  volume.  What  mass  of  tiry 
air  is  contained  in  a  litre  flask  at  21°  C.  and  75  cm.  pressure,  there  bemg 

1-29  gm.  at  0°  and  760  mm. 

27.  The  air  outside  a  chimney  50  m.  high  is  at  0°  C.  and  d  =  0-00129. 
Calculate  the  reduction  of  pressure  in  the  furnace  if  the  chimney  ga»e«  are 
at  273°  C.  and  contain  an  additional  5%  weight  of  carbon. 

28.  Find  the  change  in  the  weight  of  air  in  a  room  when  the  barometric 
height  falls  from  75  to  72  cm.,  and  the  temperature  at  the  same  time  tails 
from  13°  to  0°  C.  «o       J  ,^ 

The  room  is  10  x  8  x  3  m.,  and  the  density  of  air  at  0  and  76  cm.  is 
1-293  gm.  per  litre. 


160  HEAT 


29.  It  is  desired  to  keep  a  test-tube  containing  a  culture  at  30°  C.  within 
narrow  limits.     Explain  carefully  how  to  do  this,  discussing  precautions. 

30.  Describe  a  thermostat  suitable  for  an  incubator.     Whether  it  is  easier    f,i 
to  maintain  a  steady  temperature  of  36°  C.  or  —  36°  C.  ?  I| 

PRACTICAL  QUESTIONS 

Test  the  fixed  points  on  a  mercury  thermometer. 

Find  the  temperature  corresponding  to  a  mark  made  on  an  ungraduated 
thermometer. 

Find  the  temperatiu"e  of  the  room,  or  the  boiling  point  of  salt  water,  by 
ditto. 

[Recollect  how  boiling  point  involves  reading  the  Barometer  :  the  corrections 
for  water  and  salt-water  may  be  taken  as  equal.] 

Measure  the  coefficient  of  increase  of  pressiu-e  with  temperature  of  a  gas. 

Find  the  temperature  of  the  room  by  the  air  thermometer;  or  the  b.  pt. 
of  a  liquid ;  or  the  barometric  pressure. 


CHAPTER   XIII 


CALORIMETRY 


§  211.  A  body  is  observed  to  get  hotter  or  colder  without  either 
burning  away,  or  melting,  or  freezing,  and  without  being  hammered 
or  filed  or  worked  on  in  any  way.  It  is  natural  to  suppose  that 
some  entity  passed  into  it  and  raised  its  temperature  ;  or  left  it,  as 
it  cooled.  That  entity,  up  to  the  middle  of  the  nineteenth  century, 
was  called  Caloric,  but  the  word  became  too  closely  identified  with 
a  theory  which  had  to  be  abandoned,  and  it  fell  out  of  use,  and 
now  we  call  it  Heat. 

We  use  the  word  in  that  sense  when  we  open  the  window  '  to  let 
the  heat  out,'  until  presently  someone  wants  it  shut  because  it  is 
letting  '  the  cold  '  in.  In  autumn  we  '  start  a  little  heat '  in  the 
greenhouse  '  to  keep  out  the  frost.'  The  refrigerating  engineer 
blows  cold  air  round  the  hold  and  reckons  out  how  much  heat  he 
has  extracted.  Opposite  sides  of  the  same  thing,  Caloric,  Heat  : 
in  physics.  Cold  is  reckoned  as  merely  deficiency  of  Heat. 

Temperature  is  measured  in  Thermometry  by  its  degree,  like 
height  of  water-level ;  Heat,  in  Calorimetry,  by  quantity,  like  water 
by  the  gallon. 

Heat  is  always  contained  in  Matter,  and  gives  it  a  temperature. 
Empty  space  cannot  contain  heat  and  cannot  have  temperature ; 
although  it  may  be  full  of  Radiation. 

Heat  travels  from  one  portion  of  matter  to  another  without 
change  in  total  quantity,  and  a  body  cooling  from  one  temperature 
to  a  lower  gives  out  the  same  amount  of  heat  as  would  raise  it  from 
the  lower  to  the  higher. 

A  large  mass  of  substance  can  contain  more  heat  than  a  small, 
indeed,  Quantities  of  Heat  are  proportional  to  the  masses  of  a  standard 
substance  which  they  can  warm  from  one  to  another  fixed  temperature. 

The  Capacity  for  heat  of  a  whole  body  is  the  number  of  units  of 
heat  that  must  be  poured  into  it  to  raise  its  temperature  1°. 

The  standard  substance  is  Water. 

The  unit  quantity  of  heat,  called  the  calorie,  warms  1  gramme 
of  water  1°,  viz.  from  14-5°  to  15-5°  C. 

The  great  or  kilogram  Calorie  =  1000  of  these  gram -calories. 

The  engineer's  '  British  Thermal  Unit  '  warms  1  lb.  of  water 
r  Fahrenheit.     It  is  453-6  gm.  x  5/9  =  252  calories. 

The  Therm,  the  unit  in  which  gas-bills  are  reckoned,  contains 
100,000  British  thermal  units. 

The  heat  specific  to  a  substance,  its  capacity  for  heat  \^v  gramme, 
called  the  Specific  Heat  (sp.  ht.)  of  a  substance,  is  the  fraction  of  a 
oalorie  that  warms  1  gramme  of  substance  V, 

J5l 


162  HEAT  [§  211 

The  specific  heat  of  water  (1  at  15°  by  above  definition)  exceeds 
that  of  all  other  substances  except  hydrogen  and  helium.  Specific 
heats  vary  a  little  with  temperature,  but  not  excessively,  and  fori 
most  things  we  can  speak  of  a  '  mean  specific  heat  between  0°  and  | 
100°,'  and  reckon  it  constant.  For  water  itself  this  gives  a  '  mean  j 
thermal  unit '  which  fortunately  talHes  with  the  15°  calorie  ■: 
within  0-0005.  J 

The  Quantity  of  Heat  required  to  warm  a  body  is  therefore  the 
product  of  its  capacity  for  heat  and  its  rise  of  temperature,  i.e.  the 
product  of  its  Mass,  Specific  Heat,  and  Rise  of  Temperature, 

Quantity  of  heat  is  'primarily  measured  by  catching  it  in  water ^ 
when  it  =  mass  of  water  X  1  X  rise  of  temperature  produced. 

§  212.  Measurement  of  quantities  of  heat.  Method  of  constant 
heat  supply.  Some  of  the  earUest  calorimetric  experiments  were 
made  by  Black  at  Glasgow  {ca.  1760)  by  using  a  clear  charcoal  fire 
which  supphed  heat  at  a  presumably  constant  rate.  The  reckoning 
of  the  power  of  a  stove  by  the  shortness  of  time  in  which  it  brings 
a  certain  kettle  to  the  boil  is  familiar  enough  ;  it  is  just  one  step 
further  in  scientific  exactness  to  express  it  in  calories  per  minute 
=  grammes  of  water  in  the  kettle  x  rise  of  temperature  per  minute. 

Black  showed  that  quicksilver  heated  much  faster  than  the 
same  weight  of  water,  so  that  its  specific  heat  is  small  (as  mixing 
hot  quicksilver  and  cold  water  also  proved) — so  small  that  even  an 
equal  bulk  of  the  heavy  metal  did  not  take  as  much  heat  per  degree 
as  water.  Indeed,  it  took  not  much  longer  to  melt  a  pot  full  of  lead 
than  to  bring  the  same  pot  full  of  water  to  the  boil,  at  a  far  lower 
temperature. 

You  will  find  in  §  225  one  of  his  experiments  on  latent  heats  is 
given  as  performed  in  the  laboratory  with  a  gas-burner,  and  also 
the  modern  development  of  this  method  in  which  the  heat  is  both 
supplied  and  exactly  measured  by  electrical  means,  a  development 
which  has  changed  a  rough-and-ready  method  into  an  accurate  one. 

§  213.  The  method  of  mixtures  is  frequently  employed  in  finding 
specific  heats,  etc.  A  weighed  mass  M  of  substance  is  heated  in 
some  sort  of  steam  or  vapour  jacket  swaddled  in  wool  (or  for  high 
temperatures  an  electric  oven  or  furnace)  until  it  reaches  a  steady 
temperature  T,  and  is  then  dropped  quickly  into  a  '  calorimeter  * 
held  for  the  moment  close  under  the  heater  (much  closer  than  in 
Fig.  71).  This  calorimeter  is  a  little  pot  of  thin  poHshed  copper  or 
aluminium,  about  two-thirds  filled  with  W  gm,  of  water,  and 
furnished  with  a  stirrer  and  a  deUcate  thermometer.  It  is  sheltered 
from  draughts  and  from  stray  warmths  (hand,  flames,  etc.),  and 
from  conduction,  by  standing  on  pointed  corks  inside  a  larger 
jacket.     See  Fig.  71. 

Just  before  dropping  the  hot  body  in,  the  water  is  observed  to 


§214] 


CALORIMETRY 


163 


be  at  <i°  (near  the  room  temperature) ;  and  soon  after,  it  rises  to 
a  maximum  ^2°  (kept  stirred).  Therefore  the  hot  body  has  lost 
Ms{T  —  t^),  the  product  of  mass,  specific  heat, 
and  fall  of  temperature,  and  the  water  has  gained 
W(<2  —  ^i)  calories,  and  to  a  first  approximation 
these  are  equal.     Hence  s. 

Ex.  1.  Find  sp.  ht.  of  metal  of  which  300  gm.  at  100° 
raise  500  gm.  water  from  14°  to  20°  C. 

Calories  lost  by  metal  300  x  s  X  (100  —  20)  fall  =  24,000*. 
„     gained  by  water  500  X  1  X  (20  —  14)  rise  =  3000. 
These  are  equal,  .*.  s  =  0-125. 

§  214.  Allowances  for  vessel  and  for  cooling.    But 

the  water  has  not  captured  and  held  all  the  heat. 

I.  Some  was  lost  into  the  cooler  air  as  the  hot 
body  passed  from  heater  to  calorimeter ; 

II.  Some  went  to  heat  thermometer,  stirrer,  and 
metal  calorimeter ; 

III.  Some  has  already  been  lost  from  its  walls, 
for  as  soon  as  it  rose  in  the  least  degree  above  its 
surroundings  it  began  to  send  them  heat. 

I.  Of  these,  the  first  must  be  minimized  by  a 
short  quick  transfer,  without  splash. 

II.  The  second  is  allowed  for  by  adding  in  the 
capacity  for  heat,  or  Water  Equivalent,  as  it  is 
called,  of  the  pot,  etc.  The  good  conducting  metal 
speedily  rises  to  the  same  temperature  throughout 
as  the  water,  therefore  multiply  its  weight  c  by 
its  specific  heat  (0-1  Cu,  0-25  Al)  and  add  the  pro- 
duct on  to  W.  And  add  0-5  gm.  for  each  c.c.  of 
thermometer  submerged. 

III.  Cooling  correction.     With  material  chopped 

small,  so  that  it  readily  parts  with  its  heat  when  stirred  up  with 
the  water,  and  enables  the  final  temperature  to  be  read  in  a  few 
seconds,  no  cooling  correction  need  be  made. 

But  with  larger  lumps  of  poor  conducting  material,  go  on  watching, 
after  reading  ^g,  for  half  the  time  it  took  to  rise  from  t^,  and  any 
small  (fraction  of  a  degree  of)  cooling  observed,  add  on  to  increase, 
and  correct  t^.  (See  Cooling,  §  232,  from  which  it  will  also  appear 
that  it  is  an  advantage  to  work  on  a  rather  large  scale,  and  to  }>e 
content  with  a  small  rise  of  temperature,  delicately  measured.) 
Hence  s  from 

M5(T  —  ^2)  =  (W  +  w.  eq.  of  cal.,  etc.)  X  (^  corrected  -  /,). 

Ex.  2.     In  Ex.  1  the  copper  calorimeter  (sp.  ht.  0-1)  and  stirrer  weighed 
160  gm.;    submerged  part  of  thermometer  =  2  c.c.  bulk;    and  it  cooled  to 
19-8°  in  half  the  time  of  experiment  afterwards  (.-.  add  20  —  19-8 
on  to  the  highest,  20°). 


W^ 


Fio.  71. 


0-2' 


164  HEAT  [§214 


Calories  lost  by  metal         300  X  5  X  (100  -  20)  =  24,000s 
,,     gained  by  water    500  X  1     "1 

„      cal.andst.  160  X    -U  x  (20-2  -  14)  =  3200 
„  „      thermom.      2  X     '5) 

These  are  equal,  .*.  s  =  -133. 


§  215.  It  might  be  thought  that  glass  or  crockery  calorimeters, 
being  bad  conductors  of  heat,  would  be  better  than  metal.  But 
their  bad  conductivity  is  itself  the  objection  :  they  get  only  partly 
warmed,  and  one  does  not  know  how  much  to  allow.  The  Water 
Equivalent  of  any  calorimeter  can,  however,  be  found  by  pouring 
hot  water  into  it,  and  observing  the  cooling,  just  as  you  run  water 
hotter  into  the  bath  than  you  want  it,  or  people  make  tea  in  cold 
tea-pots.  A  vacuum-flask,  of  water-equivalent  measured  in  this 
way,  makes  a  calorimeter  which  is  valuable  on  account  of  its 
comparative  freedom  from  cooling  correction,  see  Fig,  76. 

Ex.  3.  Into  an  empty  calorimeter  at  15-8°,  100  gm.  of  water  at  54-2" 
were  poured,  and  the  final  temperature  was  52-2°.  Calculate  water  equivalent 
of  calorimeter. 

Calories  gained  by  metal  =  w.  equiv.  x  (52-2  —  15-8) 
„      lost  by  water       =  100  x  (54-2  —  52-2) 

These  are  equal,  .'.  water -equivalent  =  5-5  gm. 

Simple  experimental  variations  in  this  method  of  mixtures  occur, 
and  often  form  the  basis  of  exam  questions.  Liquids  may  be 
heated  in  a  beaker  and  poured  into  the  water ;  but  if  they  are 
soluble  or  chemically  active,  paraffin  oil  can  be  used  instead  of 
water,  and  its  sp.  ht.  found  afterwards.  It  may  be  preferable  to 
heat  the  water  and  pour  it  into  inflammable  substances,  and 
unstable  substances  can  be  cooled  instead  of  heated. 

Ex.  4.  50  gm.  water  at  90°  stirred  into  200  gm.  paraffin  oil  at  15°  bring 
the  mixture  to  40-7°.     Omitting  corrections,  find  s  of  oil. 

Water  loses  50  X  1  X  (90  —  40-7)  =  2470  cals. 
Oil  gains  200  X  s  X  (40-7  —  15)     =  5140s  cals. 
These  are  equal,  .*.    s  =  0-48. 

Ex.  5.     40  gm.  sodiiun  removed  from  oil  at  90°  are  dropped  into  200  gm. 
of  this  paraffin  oil  and  raise  it  from  15°  to  23°.     Find  specific  heat  of  sodium. 
Sodium  loses  40  X  s  x  (90  -  23)  =  2680s  cals. 
Oil  gains  200  X  0-48  x  (23  -  15)  =  768  cals. 
s  =  768  -^  2680=  0-286. 

§  216.  Specific  heat  of  liquids  by  cooling.  Any  particular  hot 
body,  unaltered  as  to  its  external  surface  and  cooling  always  under 
exactly  the  same  environing  conditions,  will  pass  out  heat  always 
at  the  same  rate  at  the  same  temperature.  If  this  results  in  the 
temperature  falling  off  faster  one  time  than  the  other,  this  must 
mean  that  its  available  internal  supply  is  less. 

A  small  closed  calorimeter  contains  the  Uquid  of  sp.  ht.  s  ;  and  in  a 
second  experiment,  the  same  volume  of  water.  It  hangs  in  a  cold 
enclosure,  and  in  each  case  is  timed  as  it  cools  from  60°  to  50°. 


§  218]  CALORIMETRY  155 

The  rate  of  losing  heat  depending  solely  on  the  outside,  if  it  takes 
lialf  as  long  with  liquid  s  inside,  then  evidently,  to  give  up  calories 
at  the  same  rate,  the  liquid  has  to  cool  twice  as  fast  as  the  water, 
i.e.  it  contains  per  cubic  centimetre  only  half  as  much  heat  as  water. 
Its  heat  capacity  per  c.c.  is  therefore  0-5,  but  the  c.c.  weighs 
(density)  gm.  .-.  s  (which  refers  to  grams)  =  0-5  -^  density.  To 
}_a^neralize,  for  half  read  I /a,  and  for  twice  read  a  times.  Densities 
(an  be  compared  on  the  spot  by  weighing  the  equal  volumes  of 
liquids  employed. 

§  217.  Dulong  and  Petit  discovered  in  1819  that  the 

Specific  Heat  of  a  Solid  Element  x  its  Atomic  Weight  =  6-4. 

This  is  only  approximately  true,  since  specific  heats  decrease  some- 
what with  temperature  fall.  The  product  6-4  is  called  the  Atomic 
Heat,  it  is  evidently  the  Capacity  for  heat  in  calories  of  a  Gram-atom 
of  any  solid  element.  [The  chemistry  books  discuss  also  the  relation 
that  the  Molecular  Heat  of  a  compound  is  the  sum  of  the  atomic 
heats  of  its  constituent  atoms.]  The  serious  exceptions  are 
Beryllium  3-5,  Boron  2-7,  Carbon  1-8,  and  Silicon  3-8,  but  all  show 
a  rise  of  specific  heat  with  temperature  which  lifts  them  about  1 
per  100°,  up  to  5-5  ;  their  abnormality  has  had  interesting  theoretical 
results.  In  the  other  direction,  all  atomic  heats  converge  towards 
zero  as  the  temperature  approaches  absolute  zero. 

Referring  to  Ex.  3,  corollary,  §  203,  the  gas  molecules  bom- 
barding a  metal  wall  will  be  in  temperature  equilibrium  with  it 
when  the  kinetic  energy  of  the  moving  particles — which  are  atoms 
of  metal — ^that  they  hit,  is  equal  to  their  own  (the  Statistical  Law 
of  that  §),  3  T  calories  per  gm.-mol. 

Now,  a  pendulum  has  its  energy  alternately  kinetic  and  potential, 
and  so  has  any  vibrating  mass,  so  that  the  energy  of  a  million  such 
would  at  any  moment  be  half  kinetic  and  half  potential.  The 
atoms  of  a  sohd,  instead  of  flying  free,  are  vibrating  under  those 
mutual  attractive  forces  which  keep  it  a  solid  ;  and  6-4  T  being 
roughly  double  3  T,  you  see  that  their  average  speed  of  movement 
is  the  same  as  a  gas  speed,  while  they  have,  in  addition,  an  equal 
quantity  of  potential  energy  stored  in  their  electrical  pulls  on  one 
another. 

§  218.  Some  Specific  Heats  at  ordinary  temperatures  are : 
Aluminium  0-22;  iron,  nickel  0-11;  eureka  010;  copper,  brass, 
zinc  0-093  ;  silver,  tin  0055  ;  bismuth,  mercury,  platinum,  tungsten, 
lead  0032  ;   other  metals,  divide  6-4  by  their  atomic  weight. 

Ice  0-5,  average  wood  042,  ebonite  and  bakelite  0-33,  porcclam. 
brick,  earth,  marble  and  most  stone  0-22— 0-19,  glass  0-19— 0-16 
(ordinary). 

Sea  water  0-94,  brine  (s.g.  1-2)  0-7,  alcohols  and  glycerine  0-60, 
paraffin  and  many  oils  0-5,  benzene,  xylol,  turpentine,  etc.  0-4. 

Hydrogen  3-4,  helium  1-26,  steam  0-48,  air  0-24,  oxygen  0-22, 
CO2  0-20 ;  but  at  constant  volume  (§  252),  hydrogen  2-4,  heUum 
0-76,  air  0-17,  oxygen  0-155. 


166  HEAT 


EXAM   QUESTIONS,  CHAPTER   XHI 

A  Laboratory  Chapter. 

Read  the  end  of  §  217,  and  see  how  the  theory  you  met  in  §  202  accounts  for 
a  new  relation  altogether. 

6.  Define  specific  heat,  thermal  capacity,  and  calorie.  Fifty  grams  of 
water  at  60°  were  poured  into  50  gm.  at  10°  in  a  calorimeter,  and  the  final 
temperature  was  32°.  What  was  the  thermal  capacity  or  '  water  equivalent ' 
of  the  calorimeter  ? 

7.  A  mass  of  100  gm.  copper  at  100°  is  put  into  a  copper  calorimeter  of 
weight  105  gm.  containing  300  gm.  water  at  20°  C,  and  the  temperature  rises 
to  22-4° ;  how  do  you  account  for  this  ?     Take  sp.  ht.  copper  0-1. 

8.  A  pint  of  boiling  water  is  poured  into  a  20-oz.  earthenware  teapot  at 
15°,  sp.  ht.  0-2 ;  supposing  this  badly  conducting  vessel  is  4/5ths  heated  through 
during  the  critical  period  of  making  tea,  calculate  how  far  off  the  alleged 
necessary  '  boil '  the  water  was,  and  compare  with  a  15-oz.-avdp.  silver 
teapot. 

9.  Iron  shot  weighing  82-8  gm.  at  100°  were  poured  into  71-2  gm.  of  water 
in  a  28-gm.  aluminium  calorimeter  at  12-6°  C.  The  final  temperature  was 
21-4°  C,  calculate  sp.  ht.  of  iron. 

10.  How  would  you  measure  the  specific  heat  of  a  soluble  solid  ?  200  gm. 
lead  at  100°  were  put  into  132  gm.  oil  at  15°  and  the  temperature  rose  to  21°. 
Find  sp.  ht.  oil.     (  X  2) 

11.  A  platinum  ball  weighing  80  gm.  is  raised  to  incandescence  in  a  furnace 
and  then  rapidly  transferred  to  a  calorimeter  containing  300  gm.  of  water 
at  15°  C.  If  the  temperature  of  the  water  rises  to  21°  C,  what  was  that  of 
the  furnace  ?     Discuss  the  errors  likely  to  arise.     (  X  2) 

12.  How  much  coal  per  24  hr.  will  raise  to  15°  C.  the  air  of  a  building  50  m. 
X  20  m.  X  25  m.,  the  whole  air  being  replaced  twice  an  hour  by  fresh  air 

entering  at  5°  C,  and  the  coal  yielding  6000  cals./gm.     Air  sp.  gr.  0-0013, 
sp.  ht.  0-24. 


PRACTICAL    QUESTIONS 

Measure  the  water  equivalent  of  a  calorimeter ;   or  of  a  vacuum  flask. 
Measure  the  heat  evolved  in  mixing  water  and  alcohol. 
Measure  the  specific  heat  of  a  solid. 

Find,  calorimetrically,  the  temperatiu'e  of  a  limip  of  metal  which  has  cooled 
in  the  air  for  a  minute  after  being  boiled. 

Find  the  specific  heat  of  a  liquid  knowing  that  of  a  metal. 


CHAPTER  XIV 
LATENT-HEAT  CALORIMETRY 

THE  CALORIMETRY  OF  CHANGES  OF  PHYSICAL  STATE 

§221.  The  chapter  on  Change  of  State  must  be  anticipated 
thus  far  : — 

As  a  solid  is  supplied  with  heat,  its  temperature  presently  ceases 
to  rise,  and  remains  at  a  steady  melting  point  until  all  is  melted, 
the  heat  meanwhile  '  going  into  hiding,'  so  to  speak.  And  again 
\\  hen  the  liquid's  temperature  reaches  a  boiling  point  it  stops  rising 
and  the  liquid  gradually  disappears  into  vapour. 

The  calories  that  have  hidden,  per  gramme  of  substance,  con- 
stitute the  Latent  Heats  of  these  changes  of  physical  state — Fusion 
and  Vaporization. 

If  now  heat  be  steadily  abstracted  from  the  vapour,  its  latent 
heat  reappears  at  the  same  temperature  as  it  disappeared,  entirely 
holding  up  the  natural  fall  of  temperature  until  all  is  returned  ; 
then  during  the  fall  through  the  liquid  state  its  specific  heat  is  in 
play ;  then  another  stoppage  as  the  latent  heat  of  liquefaction 
reappears,  without  further  fall  of  temperature,  while  the  liquid 
freezes  solid. 

Notice  the  complete  discontinuity ;  you  must  cease  climbing 
stairs  and  pay  an  entrance  fee  to  the  next  theatre  of  events 
before  you  can  go  up  even  one  step  higher  :  only,  there  is  this 
difference,  you  get  your  money  back  in  full  when  you  come  out 
again. 

The  measurement  of  these  Latent  Heats  is  of  importance.  And, 
once  measured,  they  open  up  new  and  convenient  calorimetric 
methods. 

§  222.  The  Latent  Heat  of  Melting  of  Ice  is  easily  found  by 
dropping  dry  lumps  of  it  into  the  calorimeter  :  use  at  least  a  pint 
of  water,  and  ice  half  as  big  as  your  fist.  To  avoid  a  big  correction 
for  heat  derived  from  the  air,  including  condensation  of  dew  on  the 
outside,  and  to  expedite  the  experiment  very  much,  follow  Hum- 
ford's  plan — ^warm  the  water  nearly  twice  as  far  above  the  air  at 
first  as  you  will  cool  it  below  the  air  at  last.  Then  the  loss  of  heat 
while  hotter  will  be  about  balanced  by  the  slower  gain  during  the 
much  longer  time  it  is  colder  than  the  air.  Don't  drag  on  ;  rather 
fish  out  the  reluctant  bit  of  ice  with  forceps. 

Ex.  1.  Into  a  calorimeter,  w.  oq.  12,  containing  600  gm.  wat«r  mi  26*» 
86  gm.  dry  ice  at  0"  were  stirred,  and  temperatiune  fell  to  10*. 

167 


168 


HEAT 


[§222 


Lost  by  water  and  vessel  =  512  X  15°  =  7680  cals 
Gained  by  ice  in  melting  86  X  L 

„      after  melted,  in  rising  to  10°,  86  X  10 

.*.  L  H-  10  =  7680  -^  86.     .*.  L  =  79-5  cals.  per  gm. 


}  = 


86  X  (L  +  10)  cals. 


Notice  how  the  86  gm.  multiplies  both  the  L  and  the  10°,  but  the  L  and 
the  10°  remain  quite  separate  from  each  other. 

Or  the  procedure  may  be  reversed.  A  hollow  is  scooped  in  a  large 
block  of  ice  (Fig.  72).  Into  the  dried  concavity,  water  at  the 
ordinary  temperature  is  run,  covered  with  an  ice  lid,  and  left  to 
cool  to  0°.  Then  it  is  pipetted  out,  the  last  traces  are  removed  with 
a  dry  ice-cold  sponge,  and  the  whole  is  weighed,  when  the  extra 
weight  is  that  of  the  ice  melted  by  the  heat  brought  in  by  the  water. 


Ex.  2.     20  gm.  of  water  at  16°  are  run  in  and  24  gm.  at  0°  removed. 
Lost  by  water  20  X  16  =  320  cals. 
.  Gained  by  ice     4  x  L  =        „  .*.  L  =  80. 


i 


§  223.  Ice  Calorimeters. 

Black's,  Fig.  72.  The  specific  heat  of  a  body  may  be  found  by 
dropping  a  known  mass  of  it  at  a  known  temperature  (say  the 
room  temperature)  into  the  dry  cavity  in  the  block  of  ice  of  §  222, 
and  after  leaving  covered  with  the  ice  lid  for  several  minutes, 
removing  and  weighing  the  water  produced. 

Weight     of    body   X   sp.  ht.   X 

temp.  °C.  =  ice  melted  X  80. 

Bunsen's,  Fig.  73.  In  this  the 
well-known  contraction  of  ice  on 
melting  is  made  to  indicate  the 
weight  melted.  An  inner  tube  is 
surrounded  by  a  sealed- on  jacket 
containing  air-free  water,  mercury 
fills  the  bend  and  extends  along  a 
graduated  capillary  tube. 

A  freezing  mixture  is  circulated 
through  the  inner  tube  until 
a  cap  of  clear  ice  forms  round  it,  then  the  instrument  is  packed  round 
with  melting  ice  and  left  for  a  day  or  two  to  settle  down  to  0°. 
Then : 


I  M  '  '  '  I  I 


Fig.  72. 


■r> 


Ex.  3.  5  c.c.  of  water  at  15°  are  run  into  the  inner  tube,  the  mercury 
thread  retreats  and  comes  to  rest  150  mm.  nearer  the  bulb.  1  gm.  of  platinum 
at  100°  is  now  dropped  in  and  the  thread  retreats  6-5  mm.  further.  Find 
sp.  ht.  platinum. 

Water  emits  5  X  15  =  75  cals.  as  it  cools  to  0°. 


Index  moves  150  mm., 
.*.  1  gm.  X  s 


'.  1  mm.  corresponds  to  75  -~  150  =  0-50  cal. 
X  100°  =  6-5  X  0-50  cal.;   s  =  0-0325. 


The   mercury  thread  may  be  driven  out  along  the  scale, 
further  experiments,  by  squeezing  in  the  cork  C. 


for 


X 

0-62 

X 

(16-6 

-5r 

=      7190  cals. 

X 

46-4 

=  46,400      „ 

X 

0-50 

X 

(25- 

16-6)° 

=      4200       „ 

Total  57,790  cal. 

§  226]  LATENT-HEAT   CALORIMETRY  159 

§  224.  Owing  to  the  prevalent  use  of  ice  as  the  typical  solid, 
one  is  apt  to  overlook  the  fact  that  not  all  solids  are  on  the  point 
of  spontaneously  melting,  but  require  warming  up,  as  solids,  before 
any  question  of  latent  heat  comes  in.  Then  the  following  example 
shows  that  the  Melting  Point  and  the  specific  heats,  both  as  solid 
and  as  liquid,  must  have  been  found  already  : 

Ex.  4.  How  much  heat  warms  1  kg.  of  glacial  acetic  acid  from  5°  to  25°  T 
It  melts  at  16-6°  with  latent  heat  46-4;  sp.  ht.  solid  0-62,  liquid  0-50. 

Solid  absorbs  before  melting   1000 
Melting  requires  1000 

Liquid  absorbs  after  melting  1000 


§  225.  The  Latent  Heat  of  Vaporization  of  a  substance  can  be 
measured  by  (Black's)  Method  of  constant-heat  supply : 

Ex.  5.  (Very  rough.)  A  little  calorimeter  contained  dry  ice,  broken 
small.  It  was  placed  over  a  steady  bunsen  and  sheltered  from  draught. 
In  2  min.  the  ice  had  just  disappeared,  at  4-5  min.  water  boiled,  at  19  min. 
all  boiled  away.     Find  latent  heats  of  melting  and  boiling. 

Water  rose  100°  in  4-5  —  2  =  2-5  min.  /.  bunsen  supplies  each  gramme 
of  it  with  40  cals.  per  min. 

.*.  it  takes  40  X  2  =  80  cals.  to  melt  1  gm.  ice. 

and   40  X  (19  —  4-5)  =  580   cals.  to   boil  away    1    gm.    water    at    100"*,    a 
high  result,  due  to  the  neglect  of  '  cooling  '  loss. 

Ex.  6.  An  electric  lamp  using  0-50  ampere  at  94  volts  was  immersed  in 
a  can  of  liquid  which  it  kept  steadily  boiling  at  34°.  The  can  stood  on  a 
balance  pan,  and  the  time  at  which  it  tilted  above  the  counterpoise  was 
noted.  5  gm.  were  removed  from  the  counterpoise,  and  the  next  time  of 
passing  of  the  pointer  over  zero  was  noted,  and  so  on.  The  intervals  were  60 
sec,  50,  50,  50,  52,  50,  48,  48. 

On  the  average,  then,  1  gm.  was  boiled  off  every  10  sec.  by  an  energy  supply 
(94  X  0-50  X  10)  -r  4-2  cals.  (§  812), 

or  L  =  112  cals.  per  gm. 

§  226.  More  usually,  the  Latent  Heat  of  Vaporization  is  measured 
when  it  is  all  being  given  out  again  during  liquefaction.  In  the 
common  noisy  laboratory  experiment,  a  jet  of  steam  from  a  boiler 
is  plunged  into  the  cold  calorimeter  water  and  allowed  to  warm  it 
about  20° ;  then  the  increase  in  weight  is  the  steam  condensed. 
Corrections  as  in  §  214  are  required,  and  also  it  is  very  necessary 
to  avoid  condensation  of  steam  in  the  supply  pipe  :  this  should  be 
very  short,  and  wrapped  in  wadding,  or  else  a  simple  '  steam-trap  * 
{i.e.  water-trap)  may  be  inserted.  Work,  for  preference,  with  a 
pint  of  water,  a  bunsen  full  on,  and  a  steam  nozzle  2  or  3  mm. 
bore.  The  steam  is  reckoned  as  giving  up  its  latent  heat  as  it 
condenses  to  water  at  100°,  and  then  this  hot  water  mixes  in,  giving 
up,  per  gram,  (100— final  temp.)  calories. 


160  HEAT  [§  22( 

Ex.  7.  A  copper  calorimeter  weight  159-8  gm.  weighed  571-0  gm.  when! 
containing  water  at  the  room  temperature  16°  C.  Steam  at  100°  blown  in] 
raised  it  to  27-3°  in  2  min.,  and  it  afterwards  cooled  at  the  rate  of  0-15°  perl 
min.     (§  214  III.)     Final  weight  579-0  gm. 

679-0  —  571-0  =  8  gm.  of  steam  gave  up  8  X  [L  +  (100°  -  27-3°)]  cals. 
Calorimeter  received  [159-8  X  0-1  +  (571  -  159-8)]  x  [(27-3  +  0-15)  -  16]* 

=  4900  cals. 
Equating  these 

8L  =  4900  —  582.     .-.  L  =  540  cals.  per  gm. 

§  227.  In  Joly's  Steam  Calorimeter  the  cold  body  the  specific 
heat  of  which  is  to  be  measured  lies  on  a  light  balance  pan,  hun£ 
inside  a  box,  to  which  steam  at  100°  is  admitted  by  a  large  pipe.] 
The  steam  condenses  on  the  body,  warming  it  to  100°. 

Ex.  8.  180  gm.  of  metal  originally  at  21°  increase  to  a  weight  of  183  gmj 
Of  this  0-15  gm.  is  known  to  be  due  to  condensation  on  the  pan  itself.  Fine 
specific  heat  of  metal. 

2-85  gm.  steam  condensing  to  water  at  100°  emit  2-85  x  540  —  1539  cals. 
.*.  180  X  5  X  (100  -  21)°  =  1539.        .-.  s  =  0-108. 

[Joly  measured  the  Specific  Heats  of  Oases  at  Constant  Volume. 
From  both  sides  of  the  balance  hung  3-in.  copper  spheres,  with 
catch-pans,  in  the  steam  enclosure.  Into  one  a  few  grammes 
of  gas  were  compressed,  and  the  increased  weight  (about  0-1  gm.) 
condensed  on  this  side  was  due  to  the  heat  absorbed  by  this  gas. 
He  found  for  air  0-172,  oxygen  0-155,  hydrogen  2-40.] 

§  228.  Another  calorimetric  operation  is  the  measurement  of 
the  heat  absorbed  or  produced  during  Solution,  Combustion,  or 
Chemical  Action  of  any  sort. 

Quantities  of  the  powdered  salts  to  be  dissolved  are  dropped  into 
water  in  a  large  calorimeter,  stirred  around,  and  the  changes  of 
temperature  noted.  Allowance  is  made  in  calculation  if  the  specific 
heat  of  the  solution  is  sensibly  different  from  1.  This  would  have 
been  measured  as  in  §  215. 

Or  the  reagents  in  two  separate  tubes  immersed  in  the  calori- 
meter (so  as  to  start  at  its  temperature)  are  gradually  mixed  in  one. 

Or  the  substance  to  be  burned  (a  food-stuff  e.g.)  is  enclosed 
in  a  submerged  steel  '  Bomb  '  with  compressed  oxygen,  and  fired 
electrically.     This  method  is  largely  used. 

Fuel- Gas  is  passed  through  a  delicate  meter  and  burnt  in  a 
miniature  '  geyser.'  Coal  gas  for  public  supply  is  kept  constantly 
under  test  by  the  Gas  Referees,  and  now,  in  1934,  their  chief, 
Sir  C.  V.  Boys,  one  of  the  most  brilliant  experimental  instrument  con- 
structors ever  born,  seems  to  have  succeeded  in  building  a  perfectly 
automatic  gas  calorimeter  capable  of  defying  corrosion  and  coping 
with  all  other  difficulties,  year  in  and  year  out,  working  to  an 
accuracy  of  1  in  1000. 

Of  course,  all  weights  and  temperatures  have  to  be  observed, 
usually  to  0-01°,  with  '  water  equivalents '  and  cooling  corrections. 


§229]  LATENT-HEAT   CALORIMETKY  161 

Some  Heats  of  Combustion  are,  in  calories  per  gramme  :  hydrogen 
34,000,  fuel  oil,  paraffin  and  petrol  11,000,  anthracite  8400,  common 
coal  7000—8000,  coke  7000,  coal  gas  7000  (1  cu.  ft.  is  about  20  gm.), 
fat  and  butter  9000,  lean  meat  and  proteins  5000,  sugar  4000, 
(methylated)  alcohol  6500,  sulphur  2300,  iron  1600,  zinc  1300, 
dynamite  1300,  black  gunpowder  700. 

§  229.  Animal  calorimetry.  A  small  animal  may  be  put  in  a 
perforated  biscuit-box  inside  a  ventilated  ice-safe,  and  the  increased 
rate  of  draining-away  of  water  from  the  safe  measured,  grams  x  80 
=  calories.  The  heat  removed  in  the  regulated  ventilating  current 
must  not  be  lost  sight  of.  The  apparatus  is  standardized  by 
burning  inside  it  a  known  weight  of  alcohol. 

Elaborate  experiments  with  the  human  subject,  in  large  calori- 
metric  apparatus  of  this  character,  have  shown  that  the  body 
converts  the  net  potential  energy  of  food  (as  measured  by  heats  of 
combustion)  into  thermal  and  mechanical  energy,  as  quantitatively 
as  does  any  inorganic  engine.  The  output  of  energy  as  hard 
mechanical  work  may  be  about  one-eighth  of  the  energy  given  off 
as  heat.     See  also  §  234. 

EXAM  QUESTIONS,  CHAPTER  XIV 

For  some  Latent  Heats  see  Table  §  270. 

You  are  not  likely  to  meet  with  §§  223,  225,  227  in  practice.  This  Chapter 
was  split  off  from  XIII  on  account  of  the  mass  of  questions  :    Verb.  aap. 

9.  Define  Specific  Heat  and  Latent  Heat. 

How  can  the  heat  of  fusion  of  ice  be  determined  with  accuracy  ? 

10.  A  copper  calorimeter  of  41-5  gm.  weighed  118-0  gm.  when  containing 
water  at  18-0°  C.  Ice  was  put  in  until  the  temperature  was  10-4°  and  the 
weight  124-82  gm.     Find  latent  heat  of  ice. 

11.  Define  Latent  Heat  of  Fusion,  and  of  Vaporization. 

A  lump  of  ice  weighing  80  gm.  and  at  —  10°  C.  is  dropped  into  water  at 
0*  C,  6  gm.  of  water  freeze  on  to  the  liunp ;  calculate  the  specific  heat  of  ice. 

12.  How  much  sea-water  at  6°  is  required  to  melt  at  —  2-6°  C.  100,000 
tons  of  ice  ? 

13.  4  lb.  ice  (sp.  ht.  0-5)  at  —  20°  C.  were  mixed  with  3  lb.  paraffin  oil 
(sp.  ht.  0-67)  at  17°  C.     Find  temperature. 

14.  If  specific  gravity  of  ice  is  0-918,  at  what  rate  per  square  metre  is  boat 
escaping  from  a  lake  when  a  layer  of  ice  2  mm.  thick  is  formed  in  an  hour  «m 
its  surface  ? 

15.  280  gm.  of  copper  at  100°  were  dropped  into  120  gm.  of  ice  ami  water 
in  a  copper  calorimeter  of  40  gm.  The  final  temperature  was  8°,  how  much 
ice  was  there  ? 

16.  A  portable  polar  stove,  loaded  with  2  kg.  ice  at  —  40°,  boils  it  in  10 
min.  with  the  expenditiu^  of  150  gm.  of  tdcohol,  of  calorific  value  6000  cals./gm. 
Sp.  ht.  ice  0-5;   calculate  efficiency. 

17.  The  capillary  of  a  bunsen's  ice  calorimeter  is  0076  cm.  diam.  6  grn. 
of  copper  at  100°  C.  is  placed  in  the  calorimeter.     The  mercury  moves  1 1-8  cm. 

Calculate  the  density  of  ice. 

18.  Describe  some  form  of  ice  calorimeter,  and  its  use.  Wliat  mass  of 
ether  must  be  evaporated,  with  latent  heat  96,  to  freere  1  kgm.  of  ice  ? 

O 


162  HEAT 


I 


19.  The  air  of  a  1000-c.m.  room  is  changed  every  10  min.,  being  cooled  from 
25"  to  18°  over  ice;  how  much  is  melted  per  hour?  Air  at  18"  weighs  1'21 
kg. /cm.,  sp.  ht.  0-24. 

20.  If  12  gm.  of  an  alloy  at  15°  C.  are  stirred  into  216  gm.  of  the  liquid 
alloy  at  100°  C,  the  temperature  of  the  mixture,  which  is  all  liquid,  is  85°  C. 
If  sp.  ht.  =  0-04  both  solid  and  liquid,  calculate  latent  heat. 

21.  What  do  you  understand  by  latent  heat  of  solution  ?  10  gm.  of  a 
salt,  sp.  ht.  0'3,  are  dissolved  in  100  gm.  water  in  a  calorimeter  of  water 
equivalent  10-5  gm.  The  dry  salt  was  at  20-1°,  and  it  cooled  the  water  from 
15-2°  to  11-1°;   what  was  its  latent  heat  of  solution? 

22.  Assuming  that  a  northerly  wind  500  km.  wide  and  2  km.  deep  flowed 
over  this  country  for  8  weeks,  at  10  km.  per  hour,  how  much  ice  would  be  melted 
to  keep  this  4°  cooler  than  usual  ? 

23.  How  would  you  determine  the  latent  heat  of  steam?  How  does  it 
vary  with  change  in  boiling  point  ? 

24.  Explain  how  you  would  determine  the  specific  heat  and  latent  heat  , 
of  alcohol.     Point  out  sources  of  error,  and  precautions.     (  X  2) 

25.  A  calorimeter,  water  equivalent  6  gm.,  contained  101*2  gm.  water  at 
14-5°.     3-38  gm.  of  steam  at  100°  were  condensed  and  raised  temperature  ■. 
to  32-3°.     The  heating  took  3  min.,  IJ  min.  later  temperatiu-e  had  fallen 
0-3°.     Find  latent  heat  of  steam.     (  X  2) 

26.  Find  the  approximate  weight  of  steam  that  would  warm  from  0"  to 
20°  C.  a  room  15  X  12  X  10  ft.,  air  weighing  0-08  lb.  per  cu.  ft.,  and  total 
latent  heat  of  steam  at  20°  being  590. 

27.  Compare  the  quantities  of  water,  originally  at  15°  C,  necessary  to 
condense  100  tons  of  steam  at  39°  C.  when  its  total  latent  heat  is  580,  and  at 
26°  C.  when  its  total  latent  heat  is  588. 

28.  Steam  at  100°  is  blown  into  100  gm.  of  water  at  20° ;  how  much  will 
condense,  in  all,  and  how  much  ice  would  bring  back  the  temperature  to  20°  ? 

29.  Gas  being  tenpence  per  therm,  what  would  it  cost  to  boil,  and  to 
evaporate,  100  gallons  of  water  at  60°  F.,  if  90%  of  the  heat  is  utilised  ?    (  X  2) 

30.  Steam  condenses  on  a  kilogram  iron  weight  initially  at  15°,  to  a  weight 
of  19  gm.;   calculate  sp.  ht.  iron. 

31.  Heat  is  steadily  supplied  by  an  electrical  heater  to  100  gm.  of  a  liquid, 
and  raises  it  in  12  min.  from  15*2°  to  boil  at  76-7°.  In  17^  min.  more  it  is 
all  evaporated  ;  calculate  sp.  ht.  if  lat.  ht.  =  46.     (  X  2) 

32.  Superheated  steam  (sp.  ht.  0-3)  at  150°  C.  is  blown  into  a  ton  of  grease 
at  0°  which  melts  at  45°  with  latent  heat  25,  and  has  sp.  ht.  0-5  both  solid 
and  liquid ;  how  much  is  required  to  raise  the  mass  to  80°  C.  ? 

33.  How  much  steam  at  100°  C.  will  just  melt  8  lb.  of  ice  at  —  10°  C.  ?  (  X  2) 

34.  A  calorimeter,  water  equivalent  15  gm.,  contains  400  gm.  of  water  at 
20°  C,  and  200  gm.  of  poimded  ice  at  0°  C.  are  poured  in.  Steam  is  passed 
in  imtil  the  contents  are  again  at  20°  C.  If  the  increase  in  weight  is  32-4  gm., 
determine  the  latent  heat  of  vaporization  of  water.     (  X  3) 

35.  Ten  grams  of  steam  at  100°  are  blown  into  150  gm.  of  partly  melted 
snow,  and  the  result  is  water  at  10°,  find  the  weight  originally  unmelted. 
(X2) 

36.  A  nickel  half-gram  weight,  at  14°,  is  dropped  into  a  vacumn  flask  of 
liquid  nitrogen  at  its  boiling  point,  —  196°  C,  and  the  gaseous  nitrogen 
evolved  measm-es  162  c.c.  at  14°  C.  and  740  mm.  pressure;  find  its  latent 
heat  of  vaporization. 

Sp.  ht.  nickel  0-11 ;  density  of  nitrogen  at  0°  and  760  mm.  0-00125. 

PRACTICAL  QUESTIONS. 

Find  the  specific  heat  of  a  liquid  by  dropping  ice  in ;  or  by  blowing  steam  in. 
Find  the  latent  heat  of  fusion  of  ice ;  or,  of  steam. 


CHAPTER  XV 


I       THE  MOVEMENT   OF  HEAT  FROM  PLACE  TO  PLACE 


COOLING 


There  are  various  processes  by  which  heat  travels  from  place 
to  place ;  their  joint  effect  in  promoting  the  cooling  of  a  hot  body 
will  first  be  considered.     Afterwards  they  will  be  taken  individually. 

§  231.  Sir  Isaac  Newton  was  led  by  his  experiments  to  a  state- 
ment, now  known  as  Newton's  Law  of  Cooling,  that  The  rate  of 


Fio.  74. 

cooling  of  a  hot  body  is  proportional  to  the  excess  of  its  temperature 
above  that  of  its  surraiiJidings. 

A  calorimeter  of  hot  water  is  stood  inside  a  larger  vessel  through 
the  double  walls  of  which  cold  water  is  circulated,  so  as  to  get 
surroundings  at  a  definite  temperature.  The  hot  water  is  stirred 
and  its  temperature  is  read  every  minute,  and  a  curve  like  Fiff.  74 
plotted,  a  curve  which  always  has  the  same  general  shape,  though  it« 

163 


164 


HEAT 


[§  2311 


actual  gradients  depend  on  the  particular  apparatus  employed, 
becoming  flatter  for  a  larger  vessel,  for,  of  course,  this  has  less  cooling 
surface  per  unit  cubic  content  than  has  a  smaller  one.  Per  contra^ 
reflect  how  quick  you  have  to  turn  out  of  bed  if  they  stint  you  with 
only  a  milk- jug  of  hot  water. 

According  to  the  smooth  curve  plotted,  the  temperature  fell 
during  the  first  2  min.  from  82-5°  to  75°,  i.e.  the  rate  of  cooling 
was  7-5°  per  2  min.  The  average  temperature  meanwhile  was 
78-5°,  which  was  67-5°  in  excess  of  the  surrounding  11°. 


7-5^ 


rate  of  cooling     _        height  of  2-min.  step 
temperature  excess  ~  total  height  to  be  gone  down  ~~  67-5° 


111 


This  ratio  has  been  worked  out   on  the  diagram  for  several 
2-min.  intervals.     According  to  the  law,  it  should   be  constant 


n 

50° 

5-o/ 

^ 

^Q 

HE  ATI 

NG 

^y 

^^ 

\^ 

LING 

V 

< 

:oc 

30 

3-3/ 

V\ 

^ 

r 

20 

I'l/^ 

-   1 

1 

\6 

/  \ 

1 
1 
1 

1 
1 

/ 

Z     1 

._J 

3     N 

^^J 

4    U 

5    T 

-- 

E 

1 
7    S 

_J 

8 

0 

1    M 

6 

Fig.  75. 


it  is  only  roughly  so,  being  greater  at  the  higher  temperatures. 
For  this  discrepancy  there  are  two  reasons  :  first,  that  the  unchecked  I 
evaporation  from  the  water  surface  obeys  a  law  of  its  own,  being 
much  greater  at  high  temperatures,  see  Fig.  82  ;  second,  that  the 
hotter  vessel  raises  a  more  rapid  draught  in  the  air,  and  so  cools 
itself  better.  The  experiment  is  put  forward  as  an  actual  average 
case  of  cooling  ;  had  we  stood  a  closed  hot  vessel  in  a  strong  draught, 
as  Newton  did,  we  should  have  found  a  closer  adherence  to  his  Law. 
Limiting  ourselves  to  small  differences  of  temperature,  and  recol- 
lecting the  variety  of  processes  involved  in  ordinary  cooling, 
Newton's  Law  is  good  enough  for  common  uses. 

§  232.  Cooling  corrections  in  heat  experiments.  The  instruction 
given  in  §  214  was  to  add  to  the  top  temperature  the  fall  that 
afterwards  took  place  in  half  the  time  of  heating.  For,  heating 
steadily  from  the  room  temperature,  the  average  excess  has  been 


^  233]  TRANSFERENCE   OF   HEAT  165 

half  the  final  excess  above  the  surroundings,  and  by  the  Law, 
(ooling  has  averaged  half  as  fast  as  it  is  now  going  on  at  the  finish. 
The  fall  from  the  top  temperature  for  half  the  time  is  therefore 
( (|iial  to  the  fall  from  the  average  temperature  for  the  whole  time. 
I'liis  is  near  enough  for  the  purposes  of  this  book,  in  the  figure  it 
gives  4/5ths  of  the  correction  worked  out  minute  by  minute  on  the 
way  up,  a  2%  error  on  the  total.  This  figure  is  a  far  more  extreme 
case  than  you  are  likely  to  meet  with  in  practice. 

§  233.  The  processes  referred  to  as  promoting  Cooling  are  those 
by  which  Heat  travels  from  place  to  place  : 

Evaporation,  Convection,  Conduction,  and  Radiation. 

Evaporation  from  wet  surfaces  has  been  already  met  with.  On 
the  small  scale  it  helps  cool  your  tea,  on  a  large  scale  it  assists  in 
the  great  cooling-towers  of  electric  power-stations ;  and  on  the 
largest,  it  makes  all  Weather. 

A  little  evaporation  takes  away  a  lot  of  latent  heat :  conversely, 
condensation  of  de'w  on  a  cold  body  warms  it  effectually. 

At  the  boiling  point  it  becomes  all- important,  suddenly  taking 
complete  charge. 

Convection  currents  in  quiet  air  account  for  seven-eighths  of  the 
cooling  of  a  closed  hot-water  vessel,  etc.,  and  in  a  good  draught 
for  a  much  greater  proportion. 

The  effect  of  Conduction  varies  very  greatly  according  to  the 
things  in  contact  with  the  hot  body.  Except  with  metal  it  is  slow. 
An  instance  is  the  comforting  application  of  cold  metal  or  stone 
to  a  bruised  or  inflamed  surface.  Another,  where  it  is  quite  abnor- 
mally effective,  is  this  :  a  short  copper  fuse-wire  takes  two  or  three 
times  the  calculated  current  to  melt  it,  on  account  of  conduction 
into  the  thick  metal  clamps  at  its  ends.  And  everyone  knows 
the  part  it  played  in  the  Davy  lamp. 

Radiation  is  distinguishable  by  going  on  equally  in  all  directions, 
quite  independent  of  gravity,  etc. ;  it  passes  through  a  vacuum, 
and  is  merely  hindered  more  or  less  by  the  presence  of  matter. 
Radiant  '  heat '  is  therefore  quite  unlike  the  heat  that  we  have 
measured  in  a  calorimeter,  and  the  whole  subject  is  left  to  the  last 
I  chapter  of  the  book. 

I  That  radiation  plays  but  a  small  part  in  cooling  at  temperatures 
below  100°  is  seen  from  the  fact  that,  contrary  to  the  usual  state- 
'ments,  a  blackened  metal  vessel  cools  only  about  one-eighth  faster 
than  a  polished  one,  which  radiates  much  less ;  and  is  emphasized 
I  by  the  success  of  the  popular  Vacuum  Flasks,  which  cool  only  by 
the  radiation  from  a  silvered  surface  traversing  a  vacuum  jacket. 

But  Radiation  becomes  enormously  effective  above  a  red  heat. 
While  the  common  hot-water  '  radiator  '  merely  warms  the  air 
rising  in  convection  currents  past  it,  and  sends  but  little  *  in  rays  * 
j  to  cold  hands  held  in  front  of  it,  an  open  fire  warms  by  radiation 


166 


HEAT  [§  233 

So  also  do  electric  bowl-fires. 


only,  unless  '  the  chimney  smokes.' 
etc. 

The  figure  shows  the  results  of  a  lecture-table  experiment  in 
which  a  small  calorimeter  of  hot  water  cooled  from  75°  by  thei 
amounts  shown  in  15  minutes. 


A    *CU  P     OF     TEA* 
IN    1/4  H  R.  C  0  0  L  ED     FROM    fS 


C 


mt 


yL 


%v 


B 


D  E 

Fig.  76. 


A  exposed  to  all  losses,  cooled  32°. 

B    ditto,   but   Radiation,   presumably    doubled   by    blackenii 

surface,  caused  only  1°  more  cooling ;    you  see  how  smal 

a  part  Radiation  plays  below  100°. 
C  ditto,  except  Conduction  to  table  checked  by  soft  mat,  saved 

one -eighth. 
D    Convection    only ;     Conduction    and    Evaporation   checked, 

saved  2/5ths. 
E  '  cosied  '  in  a  cloth,  reducing  Convection  also  ;   saved  3/5ths. 
F  '  cosied,'  but  Conduction  encouraged  into  cold  iron,  compare 

E^F  with  A^C  into  teak. 
G   small  Vacuum- jacketed   flask,    left   open,    evaporating,    11* 

(compare  C^D). 
H  small  Vacuum- jacketed  flask,  corked ;    Radiation  only,  1-5° 

(compare  A^B). 

§  234.  The  Heat  Loss  of  the  human  body.  But  the  best  possible 
observations  of  these  four  processes  are  obtainable  under  experi- 
mental conditions  of  unique  simpKcity.  You  require  no  apparatus 
whatever ;    on  the  contrary,  you  divest  yourself  of  everything. 

You  are  aware  of  an  immediate  disinclination  to  step  on  metal  i 
or  stone,  on  smooth  oilcloth,  or  into  a  splash  of  water.  You  arei 
avoiding  good  conductors,  and  also  that  closeness  of  contact  thati 
enables  even  an  indifferent  conductor  to  snatch  away  a  little  heat,  i 

You  do  not  stand  in  the  cold  wind.  Cold  air  merely  rising  pasti 
you  in  streams  caused  by  your  own  warmth  is  tolerable,  but  a  more 
active  ventilation  conveys  away  more  heat  than  you  care  to  lose. 

When  the  water  happens  to  be  on  the  cold  side,  you  may  be  gladi 
of  the  protection  which  even  a  thin  costume  affords,  as  the  badly 
conducting  film  entangled  in  the  meshes  of  its  fabric  fends  off  the 
rush  of  water.  And  a  woolly  swim-suit  comforts  you  in  water  or  out. 

The  speed  with  which  you  towel  down,  in  the  friendly  lee  oi 
anything  that  breaks  the  wind  a  little,  is  an  admission  of  the  great 
additional  effectiveness  of  evaporation  as  a  cooling  agent. 


§  235]  TRANSFERENCE   OF   HEAT  167 

Finally,  if  you  are  lucky  and  the  sun  shines  out,  a  long  scorch 
in  its  radiant  warmth  compensates  all  your  hardships.  Shivering, 
an  involuntary  (reflex)  exercise  of  the  muscles,  counteracting  the 
restriction  of  their  blooc^supply,  ceases ;  the  constricted  arterioles 
rlilate  as  the  need  for  protection  against  chill  is  mitigated,  the 
whole  surface  warms  and  convection  increases ;  and  perspiration 
ultimately  brings  evaporation  to  the  rescue  of  a  system  the  (internal 
heat  production  +  radiation  received)  of  which  cannot  be  otherwise 
disposed  of. 

[England,  that  was  written  in  :  on  a  West  Indian  beach  in 
August,  reverse  everything.] 

Experiments,  both  direct  calorimetric  and  also  via  the  heat  of 
oxidation  of  food-stuffs,  indicate  that  a  10-stone  man  under  the 
conditions  of  ordinary  life  loses  from  2 J  to  3  million  calories  per 
24  hr.  Of  this  the  warming  of  expired  air  accounts  for  3-5  %, 
evaporation  from  lungs  7-5  %,  very  variable  evaporation  from  skin 
14-5%,  and  convective  losses  (including  trifling  pure  radiation) 
73%.  Hard  muscular  work  means  an  all-round  increase,  as  the 
A\  hole  surface  becomes  flushed  and  moist,  and  the  less  permeable 
jirticles  of  clothing  are  thrown  off. 

To  get  an  idea  of  what  this  output  of  heat  means,  reflect  that  in 
the  hottest  and  stickiest  shade  you  ever  suffered,  smothered  in  the 
most  unsuitable  garb,  it  was  your  own  heat  production — possibly 
•^^led  by  the  internal  calorific  value  of  ice-creams — that  was 
iibling  you ;  not  the  heat  of  the  weather,  unless  that  touched 
ifi*'  F. 

Birds,  having  a  relatively  larger  cooling  surface,  give  off  heat 
faster — a  sparrow  a  dozen  times  as  fast  per  gramme.  Their 
physiological  activities  must  be  intense — they  are  at  106°  F. — and 
their  appetites  notoriously  correspond. 


CONVECTION  OF  HEAT 

§  235.  Convection.  By  this  is  meant  the  conveying  of  heat  from 
place  to  place  in  a  fluid  by  the  bodily  movement  of  heated  portions  of  it. 

The  motion  may  be  mechanically  forced,  like  the  forced  circula- 
tion of  water  in  a  motor-boat  engine,  or  forced  draught  in  a  flue,  or, 
simply,  a  wind;  but  often  the  word  suggests  only  the  natural 
rising  of  expanded  heated  portions  in  a  fluid,  under  the  control  of 
gravity,  as  in  the  so-called  '  thermo-syphon  '  water  circulation  of  an 
ordinary  car  engine. 

When  a  fluid  is  heated  locally,  neighbouring  portions  usually 
expand,  and  therefore  becoming  lighter,  are  lifted  by  the  sinking 
of  the  colder  denser  portions  around.  The  rising  stream  conveys 
its  heat  with  it,  and  constitutes  the  convection  current.    Meal, 


168 


HEAT 


[§235 


thrown  in,  shows  these  currents  in  a  saucepan  of  water  over  a 
burner — up  over  the  hot  places  and  down  all  round ;  flame  and 
light  ashes,  smoke,  or  the  well-known  rippling  appearance  of  '  hot 
air  rising,'  mark  their  track  in  air.  In  any  case,  the  warmth  of  the 
rising  stream  can  be  felt  by  the  hand. 

The  Convective  Circulation  of  Heat  evidently  depends  upon  : 

(a)  How  much  heat  the  fluid  takes  up  per  gram  (its  sp.  ht.). 
(6)  How  much  it  expands,  i.e.  what  Ufting  force  acts  on  it. 
(c)  Its  viscosity  ;   the  less  viscous  the  quicker  it  moves. 
{d)  The  size  and  length  of  the  pipes  and  channels  through  which 
the  stream  flows. 

Water  stands  high  in  respect  of  (a),  but  in  (6)  at  4°  it  fails  alto- 
gether, and  generally  in  (5)  and  (c)  is  far  excelled  by  air.  Yet 
'  water-cooling  '  is  quickest,  for  1  c.c.  of  water  will  remove  as  much 
heat  as  2500  c.c.  of  air,  and  it  is  difficult  to  get  this  great  bulk  past 
a  small  hot  surface.  Hence  the  risk  of  melting  out  the  seams  of  a 
kettle  full  of  air  only,  and  hence  the  need  for  extensive  air-cooling 
surfaces,  seen  in  the  gilled  cylinder  of  a  bicycle  motor,  or  thOj 
honeycombed  miscalled  '  radiator  '  of  a  car.  i 

Fig.  77  shows  the  water- circulating  system  of  a  small  motor. 
Hot  water  rises  from  the  jacket  sur- 
rounding the  hot  cylinder  and  then  de- 
scends through  gilled  pipes,  whence  its 
heat  is  carried  off  by  the  wind. 

Domestic    hot-water   heating   systems 
are  merely  magnifications  of  essentially 
the  same  arrangement.     The  cold-water 
supply  is  admitted  to  the  tank  at  the  top  > 
of  the  house,  whence  it  sinks  down,  with  . 
the  general  circulation,    to    the    boiler. 
Taps  and  heaters  should  be  on  the  up 
pipe,   so  that  a  whole  boilerful  of  hot ! 
water  is  available  at  once  for  a  bath. 

In  hard-water  districts,  boiler  pipes  ■ 
get  furred  up,  sometimes  to  the  size  of 
a  quill,  and  circulation  is  choked. 
StoMng-up  then  raises  steam,  which 
blows  through  and  condenses  in  the 
colder  water  above,  and  the  noise  of 
this  at  last  scares  the  householder  into  having  the  pipes  cleaned 
out. 

The  languid  and  dilatory  hot-water  supply  even  yet  too  common' 
in  hotels  is  accounted  for  by  the  pipe-system  being  too  extensive 
to  be  kept  active  by  the  feeble  changes  in  density  of  the  half -warmed 
water. 


Fig.  77. 


§  236.  In  Fig.  78  is  sketched  a  present-day  hot-water  system  for 
a  hospital,  which  copes  with  both  these  difficulties,  and  leaves  littfen 
to  chance.  ., 


li 


236] 


TRANSFERENCE   OF  HEAT 


169 


The  electric  blower  B  supplies  oil-fuel,  and  the  necessary  20 — 25 
1  imes  its  mass  of  air,  to  the  closed  furnace  of  the  Boiler  (one  of  four), 
and  ultimately  blows  the  thin  smoke  up  the  chimney.  Boiler 
steam  at  50  lb.  pressure  passes  to  half-a-dozen  Calorifiers  ;  and  also 
drives  all  necessary  pumps.  Passing  through  closed  pipes  in  the 
calorifier,  it  heats  surrounding  water,  and  is  itself  condensed  and 
returned  as  warm  water  to  the  boiler  by  the  feed-pump  FP.  No 
fresh  water  enters  the  boiler,  and  it  therefore  remains  free  from 
calcareous  '  scale.' 

From  the  top  of  the  Calorifier  the  hot  water  is  forced  into  the 
supply  pipes  by  a  silent  centrifugal  pump  CP,  driven  by  a  little 
steam-turbine  T,  the  exhaust  steam  from  which  also  goes  to  warm 
the  calorifier.     CP  maintains  a  rapid  circulation  throughout  the 


IFIERS  Jlf     BOILERS 

< 

WASH  IMG 


O^FP 


Fia.  78. 


building,  the  cooled  water  returning  by  the  pipe  on  the  left  to 
re-enter  the  cooler  part  of  the  calorifier.  The  one  shown  is  supplying 
baths  and  washing  water ;  and  a  cold  water  make-up  feeds  in  as 
required,  from  an  attendant  pump,  which  always  maintains  a 
steady  pressure. 

Live  steam  admission  to  the  calorifier  is  governed  by  an  auto- 
matic thermostatic  valve  TV  (cf .  §  204)  set  to  a  temperature  below 
that  at  which  CaCOg  would  be  thrown  out  of  solution,  so  that 
water  and  pipes  remain  clear. 

No.  2  Calorifier  similarly  suppUes  the  heaters  throughout  the 
building,  a  closed  circulation  which  remains  clear  because  very 
little  make-up  water  need  ever  be  admitted.  All  heaters  (the 
miscalled  '  radiators  ')  are  comiected  at  both  ends  to  the  horizontal 
supply  pipe,  '  in  parallel '  with  it,  and  have  hand  valves  for 
regulation. 

No.  3  sends  hot  water  through  the  heating  pipes  of  a  swimmmg- 
bath,  maintaining  its  ultra-violet-sterilized  water  at  74*  F. 

Nos.  4  and  5  supply  another  building,  and  No.  6  is  a  stand-by. 


170  HEAT  [§ 


CONDUCTION  OF  HEAT 

§  237.  Conduction  of  heat.  For  this  process  the  presence  of 
matter  in  the  path  is  a  necessity,  as  for  convection ;  but,  unhke 
convection,  there  is  no  perceptible  motion  in  that  matter ;  and  the 
process  is  at  its  worst  in  common  fluids,  and  reaches  its  best  in  those 
dense  solids,  the  metals.  It  is  a  process  perfectly  independent  oi 
changes  of  density,  and  therefore  of  gravity.  Difference  of  tern 
perature  is  its  sole  and  direct  cause. 

Substances  dififer  very  greatly  as  regards  the  faciUty  with  which 
heat  travels  through  them,  i.e.  in  conducting  power.  Specia 
apparatus  to  show  this  is  not  worth  while  :  just  take  2-in.  pieces 
of  stout  copper  wire,  of  iron  nail,  of  solder,  lead,  brass,  and  electrical 
resistance  wire,  of  chalk,  and  glass  tubing,  stick  the  end  into  a 
flame  from  which  the  fingers  are  shielded  by  a  card,  count  seconds^ 
until  you  have  to  drop  them,  and  you  will  have  some  notion  of 
relative  conductivities.  Matches,  paper,  sealing-wax,  and  string 
you  need  not  drop  until  flame  itself  reaches  your  fingers ;  they  are 
bad  conductors,     try  this. 

Many  substances  conduct  so  much  worse  than  the  metals  that  in 
common  parlance  they  are  '  non-conductors.'   Chief  among  them 

is   STILL  AIR. 

Wool,  fur,  and  feather  owe  most  of  their  value  as  clothing  to  the 
Air  they  entangle  and  prevent  from  drifting  off  in  convection 
currents.  Hard-woven  calico  is  a  chilly  integument  compared 
with  '  cellular  '  cloth  of  the  same  weight  (but  one's  outer  clothing 
had  need  be  more  wind-proof,  lest  the  air  retained  in  loose-woven 
stuff  be  violently  blown  out).  Under  the  microscope,  Down  is  a 
most  formidable  entanglement  of  tiny  barbs ;  and  look  also  at  the 
air-cell  structure  of  a  thin  section  of  Cork. 

Asbestos,  slag-wool,  and  light  magnesia  owe  their  value  as  steam- 
pipe  laggings  to  their  air-retaining  porosity.  To  cool  iron  slowly 
the  smith  buries  it  in  loose  sand.  Iron,  or  even  copper,  fih'ngs 
conduct  very  badly,  the  good  contact  essential  for  good  conduction 
is  lacking.  Aluminium  is  one  of  the  best  of  conductors,  but  sheet 
aluminium,  paper-thick,  crumpled  up  by  hand,  and  packed  2  or 
3  in.  thick,  keeps  steam  pipes,  etc.,  warm  as  well  as  any  other  lagging, 
besides  being  proof  against  heat,  damp,  mould  and  vermin,  and  is 
now  much  in  use. 

The  hay  and  sacking  wrapped  round  pipes  and  plants  in  winter 
act  as  air-retainers.  These  wrapped-up  things,  maintaining  no 
vital  heat  of  their  own,  must  ultimately  freeze  in  a  long  frost ; 
but,  like  a  mantle  of  snow,  the  wrapping  will  make  the  temperature 
changes  more  gradual,  probably  preventing  local  choking  in  pipes, 
and  the  sudden  thaw  in  the  morning  sun  so  disastrous  to  an  otherwise 
hardy  plant. 


§238]  TRANSFERENCE   OF  HEAT  171 

In  this  question  of  Clothing,  Conduction  and  Convection  meet. 
Indeed,  Convection  can  never  he  complete  without  Conduction,  through 
thin  adherent  surface-layers  and  thin  strata  of  fluid ;  precisely  as 
mechanical  mixing  has  to  be  completed  by  Diffusion. 

Let  us  dispose  at  once  of  the  hoary  fable  about  Light  and  Dark 
Clothing.  Fig.  76  has  shown  you  how  trifling  is  Radiation  from 
temperatures  even  a  good  deal  hotter  than  your  own  :  the  difference 
between  the  radiation  you  lose  outwards  through  light  and  dark 
clothes  is  perfectly  insensible. 

But  it  is  otherwise  with  incoming  radiation  from  the  Sun,  which 
is  of  a  higher  *  quaUty  '  altogether,  Chapter  LVI ;  and  merely 
throwing  a  white  handkerchief  over  your  head  in  sunshine  shows 
in  a  few  seconds  that  light  clothes  reflect  away  most  of  its  heat, 
whereas  dark  surfaces  absorb  it  freely,  often  to  your  discomfort. 
But  unless  you  are  deliberately  going  to  stand  still  and  warm 
yourself  in  pale  winter  sunshine,  dark  clothes  have  nothing  thermally 
to  recommend  them  whatever,  at  any  time,  place  or  season — ^not 
even  for  a  *  black  brother.* 

§  238.  Thermal  conductivity.  Let  heat  be  travelling  straight 
through  a  plate  of  area  A  (Fig.  79,  left)  from  a  hot  face  at  /'°  to  a 
cold  one  at  t°.  So  long  as  the  conditions  remain  everjrwhere  the 
same,  you  will  admit  that  the  same  quantity  of  heat  enters  each 
square  centimetre  of  plate,  and  the  total  is  A  times  that  of  1  sq.  cm. 

And  you  will  admit  that  the  total  quantity  transmitted  is  propor- 
tional to  the  time  T  seconds  of  observation. 

It  is  found  by  experiment  that  the  flow  is  proportional  to  the 
difference  of  temperature  t'  —  t  between  the  faces. 

And  it  follows  that  it  is  inversely  as  the  distance  D  it  has  to 
travel.  For  the  plate  can  be  supposed  split  into  D  successive 
plates  1  cm.  thick,  each  with  I/D  of  the  total  temperature  difference 
between  its  faces. 

CAT(^' tV* 

/.  quantity  transmitted,  H  calories  =  ^ 


where  C  is  the  constant  depending  on  the  material,  its  Conductivity. 
In  this  relation,  puttmg  all  else  =  1,  H  =  C,  or  C  =  H,  which 
means,  you  see,  that  the  Thermal  Conductivity  of  a  material  is  the 
fraction  of  a  calorie  conducted  from  one  face  to  the  opposite  face^ 
r  cooler,  of  a  \-cm.  cube,  in  1  sec.  Fig.  79  right. 

Here  is  a  formula  which  can  be  read  straight  off,  which  gathers 
together  a  number  of  things  easy  to  overlook,  which  contains  no 
mathematical  complexities  such  as  squares,  etc.,  but  carries  you 
through  all  conduction  calculations.  Exceptionally,  it  is  worth 
memorizing,  and  if  it  is  not  mnemonic  enough  in  itself,  think  of  the 
CAT,  with  pair  of  tickly  whiskers,  V  —  t,  reclining  upon  D  the  dog, 
in  front  of  H  the  heat  of  the  fire. 


172 


HEAT 


[§239 


§  239.  The  Conductivity  of  poorly  conducting  substances,  Fig.  80. 
A  simple  method  of  carrying  the  foregoing  into  practice  for  a  poor 
conductor  appears  in  the  following  example  : 

Ex.  1,  Fig.  80,  left.  The  under  side  of  a  tile  0"7  cm.  thick  is  kept  hot 
by  a  steam  jet.  On  the  tile  stands  a  calorimeter  with  a  flat  bottom 
20  sq.  cm.  area  kept  in  good  thermal  contact  with  the  tile  by  a  smear  of  water, 
and  wrapped  in  wadding  to  hinder  accidental  access  of  heat.  In  300  sec.  it 
and  its  contents,  110  gm.  of  water,  rise  from  14°  to  23*5°.     Find  C.  of  tile. 

Here  H  =  110  x  (23-5  -  14)°  =  1045  cals. 

AT{t'~t)II)  =  20sq.cm.  x  300  sec.  x  [100  -  ^(23-5  +  14)]° -^  0-7=700,000. 
.-.  C  =  1045  -^  700,000  =  0-0015. 

[    The  I  (23-5  +  14)  is  the  average  temperature  of  upper  side  of  tile. 


^ 

\ 

1 

1 

\ 

>  c 

•per  sec. 


t*1  t 


Figs.  79  and  80. 


In  the  little  apparatus  shown,  on  a  larger  scale,  on  the  right,  the 
same  experiment  is  being  carried  out :  a  copper  block  is  heated 
electrically,  and  the  heat  travels  down  through  the  test  plate  and 
warms  a  lower  block ;  temperatures  are  read  by  thermo-junctions, 
§  799,  stuck  in  fine  holes  drilled  in  the  blocks.  With  the  aid  of  a 
little  retaining  ring,  Liquids  can  be  used  instead  of  the  solid  plate  : 
being  heated  from  the  top,  there  are  no  convection  currents  to 
interfere.  They  are  poor  conductors  ;  try  holding  a  test-tube  full 
of  water  by  its  bottom  end  while  you  boil  the  top. 

Gases  are  measured  in  another  way.  A  large  thermometer  bulb 
hangs  in  an  enclosure  and  cools  by  conduction,  convection,  and 


§  241]  TRANSFERENCE   OF   HEAT  173 

radiation.  Convection,  usually  the  most  effective,  nearly  vanishes 
below  10  cm.  gas  pressure,  while  conductivity  remains  constant 
(according  to  kinetic  theory).  Radiation  is  found  by  repeating  in 
a  high  vacuum,  and  subtracted,  leaving  conduction  alone. 

§  240.  Conductivity  of  good  conductors.  For  good  conductors 
the  plate  method  gives  bad  results.  Heat  cannot  be  got  into  or 
out  of  the  plates  fast  enough  to  keep  them  near  the  outside  tem- 
l)eratures  :  the  *  Emissivity  '  is  much  less  than  the  conductivity. 
A  boiler  plate  is  far  below  the  flame  temperature,  and  calculating 
from  the  observed  rate  at  which  it  passes  heat  through  to  evaporate 
the  water,  you  see  in  Ex.  23  below  how  little  its  opposite  faces 
differ  in  temperature.     See  also  Exx.  21,  22,  27. 

You  can  see  for  yourself  that  a  bunsen  flame  does  not  actually 
touch  any  cold  thing  held  in  it :  a  film  of  gas  conducts  worse  than 
an  inch  of  copper. 

It  is  like  a  Channel  crossing  :  if  you  swim,  or  sail  in  a  small  boat, 
the  passport  and  customs  formahties  either  side  do  not  add  appre- 
ciably to  the  total  difficulty,  but  if  you  fly  they  make  almost  the 
whole  of  it. 

The  method  has  been  adapted  to  good  conductors  by  using  a 
thicker  plate  and  drilling  the  holes  for  the  thermo- junctions  actually 
just  inside  the  plate  itself.  Other  methods  for  good  conductors 
are  quite  beyond  this  book. 

§  241.  It  may  be  noted  that  the  rate  at  which  heat  first  spretids 
or  diffuses  depends  not  only  on  Conductivity,  but  also  on  Specific 
Heat.  For  if  the  latter  is  small,  so  that  only  a  small  fraction  of  a 
calorie  need  be  left  in  each  cubic  centimetre,  a  little  heat  can  soon 
travel  a  long  way.  The  first  flush  passes  rapidly  through  lead 
(C.  =  0-08,  sp.  ht.  0-034 ;   compare  iron  C.  0-17,  sp.  ht.  0-11). 

Other  things  being  equal,  it  can  be  shown  that  the  distance  to 
which  heat  spreads  is  proportional  to  the  square  root  of  the  time 
occupied.  Or,  putting  it  the  other  way  about,  the  time  it  takes  is 
proportional  to  the  square  of  the  distance  travelled. 

Hence,  even  a  poor  conductor  can  momentarily  snatch  a  little 
heat  from  a  body  suddenly  coming  into  good  contact  with  it — 
your  bare  foot  on  oilcloth.  On  the  other  hand,  you  know  how  long 
a  large  cup  of  coffee  remains  too  hot  to  drink,  and  how  a  lavish 
helping  of  pie  holds  the  heat. 

The  daily  wave  of  warmth  may  penetrate  perceptibly  half  a  yard 
into  the  ground  (work  your  hand  down  into  the  shingle)  :  English 
water-pipes  are  laid  from  2  to  3  ft.  deep,  and  in  Canada  are  safe  all 
winter  through  at  5  ft. ;  the  frozen  muskegs  of  the  north  never 
thaw  to  this  depth  :  in  no  soil  is  the  annual  wave  noticeable  below 
50  ft. 

And  consequently,  large  masses  which  must  necessarily  scatter 
their  heat  far  afield,  take  a  long  time  to  cool.  Drawn  threads  of 
molten  glass  sohdify  almost  instantly;    the  1000- ton  anvil,  cast 


174 


HEAT 


[§241 


in  situ,  of  a  large  hammer,  was  unapproachable  for  6  months 
and  the  earth,  with  its  temperature  gradient  of  1°  C.  in  about 
100  ft.  of  depth,  emits  only  an  eighth  of  a  calorie  of  its  internal] 
heat  per  square  centimetre  per  day. 


§242. 


Table  of  Conductivities  for  Heat. 


Silver  .      . 

10 

Copper 

0-9 

Aluminium 

0-5 

Iron      .      . 

0-14 

Lead    .      . 

0-08 

Brass,  zinc 

0-26 

Manganin,    ^^.^g 
Constantan  J 


Rocks   .      .  0-003— 0-008 

Brick,  tile,  slate    .      -j    n.nn^ 

Most  glasses  .  .  .  0-0025 
Sand,  dry  soil,  snow,^.^^^^ 

asbestos       .  .    /"""" 

Leather,  paper,  wood,  ^  0-0003 — 

vulcanised  rubber  .  J   0-0005 


Mercury    .  .   0-020 

Ice       .      .  .   0-004 

Water       .  .   0-0014 
Many  organic/  0-0003 — | 

liquids  \  0-0005 

liir  X  r  0-00003— 1 

Most  gases  |   ^^^^^^^   ' 

Hg,  He      .  .   0-00035 


Lagging  materials  for  steam  pipes,  such  as  asbestos-  or  slag- wool,' 
light  magnesia,  crumpled  Al  foil,  with,  for  Cold  Storage,  slab  orj 
granulated  cork,  balsa  wood,  expanded  ebonite,  wool,  etc.,  about; 
000013. 


EXAM   QUESTIONS,   CHAPTER   XV 


The  questions  explore  the  chapter.  Q.  10  shows  how  the  English  open 
fire  is  a  success  in  your  home,  but  fails  on  a  larger  scale  :  you  see  how  much 
can  be  made  out  by  very  simple  calculations.     Fig.  78  you  can  redraw  simpler. 

1.  Explain  the  various  processes  by  which  a  hot  body  loses  heat,  taking 
as  instances  a  jug  of  hot  water  and  a  gas  flame,  or  red-hot  coke.     (  X  2) 

2.  State  and  explain  Newton's  Law  of  Cooling. 

Show  how  to  ascertain  the  final  temperature  which  a  calorimeter  would 
have  reached,  in  the  absence  of  heat  losses  to  the  suroundings,  when  heated 
from  the  room  temperature  for  a  few  minutes  by  a  steady  source  of  heat. 

3.  How  does  the  rate  of  cooling  of  a  hot  body  depend  on  its  mass,  surface, 
specific  heat,  and  temperature,  and  on  the  temperature  of  its  sm-roundings  ? 

4.  A  vessel  of  hot  water  is  suspended  in  an  enclosure  :  describe  its  various 
ways  of  losing  heat,  and  discuss  the  conditions  affecting  loss  by  each.  How 
would  you  minimize  the  losses  ?     (  X  2) 

5.  Explain  Newton's  Law  of  Cooling.  Compare  the  times  taken  by  equal 
volmnes  of  alcohol  and  water,  in  succession,  to  cool  from  60°  to  50°  in  the  same 
closed  calorimeter,  alcohol  having  a  sp.  ht.  0-55  and  sp,  gr.  0-8. 

6.  A  calorimeter  (w.  eq.  20)  containing  200  gm.  of  turpentine  cooled  from 
65°  to  60°  in  196  sec,  the  equal  volmne  of  water  weighed  230  gm.  and  took 
435  sec.  for  the  same  interval  of  temperature.     Find  sp.  ht.  turpentine. 

7.  Describe  in  detail  the  principles  of  construction  of  the  vacuum  flask. 
Explain  how  it  is  suitable  for  keeping  hot  liquids  hot  or  cold  liquids  cold. 
(X  2) 

8.  What  do  you  understand  by  '  free  convection,'  and  by  *  forced  con- 
vection '  ? 

Describe  (o)  the  hot-water  system  of  a  house,  (6)  the  cooling  system  of  a 
car,  (c)  the  ventilation  of  a  hall.     (  X  3) 

9.  A  small  volume  of  a  fluid,  of  expansibility  a,  is  at  T ;  rest  of  fluid  at  /. 
If  density  at  0°  =  d  find  resultant  force  per  c.c.  on  the  warmer  portion. 
What  forces  diminish  resulting  motion  ? 


TRANSFERENCE   OF  HEAT  176 

10.  [On  Convection,  etc.]  A  chimney  10  m.  high  contains  gases  at  60°  C. ; 
the  outer  air  is  at  6°  C.  (Densities  at  0°,  gases  00013,  air  000129.)  Calculate 
(o)  the  reduction  in  pressure  inside  the  room  with  doors,  etc.,  shut ;  (6)  the 
volume  of  gases  passing  through  a  9-in.  (23-cm.)  chimney-pot,  door  open; 
(c)  the  coal  consumed  hoiu-ly  in  heating  these  gases  (of  sp.  ht.  0-33),  if  1  gm. 
gives  6700  cal. ;  (d)  total  work  done  per  hour  and  (e)  the  fraction  this  represents 
of  total  heat  produced. 

(o)  Weight  of  1  cm.  square  column  in  chimney  =  1000  X  0*00 13    X  273/333. 

outside        =1000x000129x273/278. 
Difference  =  driving  pressiu-e  (or  reduction  in  room) 

=  1000(0-001260  -  0001065)  =  0-195  gm./cm.» 
=  2  mm.  water  =192  dynes /cm. * 

(b)  Momentum   imparted   to    chimney   gases   per   sec.    per   cm.*  =  driving 

pressure. 
.'.  mass  entering  per  sec.  x  velocity  =  volume  x  density  x  velocity 
=  density  x  (velocity)"  =  0-001065  X  r«  =  192. 
.'.  velocity  =  425  cm, /sec.  at  most. 
.'.  volume  =  velocity  x  area  =  425  X  ir  X  11-5*  =  176,000  c.c.  per  sec. 
=  6  cu.  ft. 

(c)  Coal  =  176,000  x  0-001065  x  0-33  X  (60°  -  5°)  x  3600  sec.  -r  6700 

=  1840  gm. 

{d)  Work  done  =  lifting  176,000  X  0001065  gm.  1000  cm.  high  per  sec. 

=  6800  kg.-m.  per  hr. 

(e)  =  fraction  6-8  X  10^  x  981/1840  X  6700  X  (4-2  x  10') 

=  0-00129  total  heat  energy  produced  :    a  chimney  is  a  wasteful  way  of 
creating  a  draught. 

This  is  our  open  coal  fire,  warming  and  ventilating  a  room  for  half-a-dos«i 
people  admirably  in  our  comparatively  mild  climate,  but  quite  imsmtable 
for  larger  use. 

11.  Three  thermometers  (a)  next  skin,  (6)  between  vest  and  shirt,  (c) 
between  shirt  and  coat  read  30-1°,  24-8°,  and  21-4°  C.  Vest  and  shirt  being 
equally  thick,  calculate  their  relative  conductivities. 

12.  Explain  fully  (o)  why  a  wet  finger  freezes  on  to  cold  metal  but  not 
to  wood,  (6)  why  a  tall  chimney  gives  a  better  draught,  (c)  why  snow  melts 
slowly  on  the  mountain  tops,  even  in  the  sun. 

13.  Calculate  the  daily  loss  of  internal  heat  per  square  metre  of  the  earth** 
surface,  if  the  conductivity  of  the  crust  is  0-004  and  the  temperature  increaaea 
1*  C.  for  every  32  m.  of  depth. 

14.  Define  thermal  conductivity  and  describe  a  method  of  determining  it. 
Calculate  the  amount  of  heat  extracted  daily  from  a  chamber  6  m.  cube, 

having  walls  15  cm.  thick  of  thermal  conductivity  000016  c.g.s.  units,  in 
order  to  maintain  it  at  —  5°  C,  the  average  temperature  outside  being  25''  C. 

15.  A  30-cm.  cube  box  contains  ice;  it  is  wrapped  in  sacking  2-5  cm.  thick, 
conductivity  0  0002 ;  how  much  ice  will  melt  per  hour  if  the  outer  air  is  at 
20°  C.  ?     (  X  2) 

16.  A  plate  4  m.  square  and  1  cm.  thick  transmits  a  million  calories  per  sec. 
when  its  opposite  faces  are  at  100°  and  85°.     Calculate  conductivity. 

17.  A  stream  10  c.c.  of  water  per  sec.  is  passed  at  16°  C.  into  a  glass  tube 
60  cm.  long,  1  cm.  in  diameter  and  0-5  mm.  thick. 

The  outside  is  at  100°  C. :  what  will  be  the  temperature  of  the  outflowing 
water,  C.  of  glass  being  0001  ? 


176  HEAT 

18.  Ice,  of  sp.  gr.  0-918,  has  formed  4  cm,  thick  on  a  pond;  its  conductivity 
is  0-005  and  the  air  temperature  is  —  5°  C. ;  how  fast  is  it  getting  thicker  ? 
(X  3) 

19.  The  walls  of  a  thatched  cottage  are  24  cm.  thick,  of  total  area  200  sq.  m., 
and  of  conductivity  0003 ;  how  much  wood,  producing  3000  cals.  per  gm., 
must  be  burnt  daily  to  keep  the  interior  5°  C.  warmer  than  the  outer  air, 
supposing  that  half  the  heat  is  immediately  lost  up  the  chimney  ? 

20.  On  what  does  the  warmth  of  clothing  depend  ?  Supposing  the  body 
at  37°  C,  clad  in  woollen  3  mm.  thick,  and  of  conductivity  0-0001,  the  outer 
air  16°  C,  and  the  outer  surface  1-5  sq.  m. ;  how  many  calories  are  lost  per 
16  hr.? 

21.  Why,  though  glass  is  a  bad  conductor,  does  most  of  the  heat  escape 
from  a  warm  room  through  the  window-panes  ?  Calculate  the  hourly  heat 
loss  through  3-mm.  thick  glass,  the  room  being  10°  C.  warmer  than  the  outer 
air. 

22.  What  is  the  fallacy  in  the  foregoing  question,  and  why  is  it  possible 
to  keep  a  greenhouse  adequately  warm  at  night,  without  any  excessive 
expenditure  of  fuel,  in  spite  of  the  thinness  of  the  glass  ? 

23.  Describe  experiments  illustrating  the  difference  between  good  and  bad 
conductors. 

The  metal  of  a  boiler  is  1-5  cm.  thick;    find  the  difference  of  temperature, 
between  its  faces  if  32  kg.  of  water  is  evaporated  per  hour  per  sq.   metre, 
the  conductivity  being  0-16.     (  X  2) 

[32,000  X  540  =  0-16  x  10,000  sq.  cm.  X  3600  sec.  x  {f—  t)  -^  1-5,  whence 
(t'  —  t)  —  4-5°.  This  is  average  steam  boiler  practice,  and  you  see  how 
small  is  the  temperatiu-e  gradient  in  the  iron.  The  flue  gases  are  something 
like  600°  hotter  than  the  water.] 

24.  The  end  of  a  cold  metal  rod  is  stuck  into  boiling  water ;  describe  the 
distribution  of  temperature  (a)  as  time  goes  on,  (6)  after  several  minutes,  at 
various  distances  along  the  rod. 

25.  An  iron  rod  is  held  in  a  flame  at  the  lower  end;  why  do  parts  remain 
cooler  than  the  flame  ?  Would  it  heat  at  a  different  rate  if  the  flame  were 
applied  at  the  top ;  and  if  so,  why  ? 

26.  A  copper  bar  is  20  cm.  long  and  6  sq.  cm.  cross-section;  one  end  is 
kept  at  100°  and  the  other  at  40°  by  cooling  water  which  flows  continuously 
away,  having  been  warmed  5° ;   how  fast  is  the  flow  ? 

27.  How  much  heat  would  escape  in  a  minute  through  a  copper  steam 
pipe,  surface  area  of  1  sq.m.,  thickness  4  vara..,  if  the  steam  is  at  160°  C.  and 
the  external  temperature  is  157°  C.  ? 

If  8  kg.  of  condensed  steam  flow  away,  what  was  the  latent  heat  of  steam  ? 
(X  2) 

28.  A  steam-pipe  at  170°  C.  is  lagged  with  light  magnesia  5  cm.  thick. 
The  air  is  at  30° ;  if  the  conductivity  is  0-00025,  calculate  the  hoiu-ly  con- 
densation of  steam,  at  latent  heat  500,  per  metre  of  pipe  50  cm.  mean  circum- 
ference. 


8 


I 


CHAPTER   XVI 
THE  MECHANICAL  EQUIVALENT   OF  HEAT 


§  25L  The  theory  in  favour  even  up  to  the  middle  of  the  nine- 
teenth century  was — as  expounded  by  Black  in  the  middle  of  the 
eighteenth — that  Heat  was  an  igneous  fluid  or  caloric,  permeating 
the  pores  of  all  substances.  It  was  admitted  that  caloric  was 
weightless,  for  a  balance  bearing  a  bottle  of  water  counterpoised  by 
brass  weights  continued  in  equiUbrium  after  a  stay  overnight  in  a 
cold  room  had  frozen  the  water,  and  thus  caused  it  to  give  up  latent 
caloric  amounting  to  more  than  a  hundred  times  that  lost  by  the 
brass  weights. 

Benjamin  Thompson,  6.  1753,  supported  himself  as  schoolmaster 
at  Rumford  (now  Concord),  Mass.,  while  studying  for  medicine, 
but  was  driven  from  the  country  in  1776  for  alleged  pro-English 
sjrmpathies.  After  seven  years'  research  in  gunnery,  he  reorganized 
mihtary  service  in  Bavaria,  found  work  in  Munich  for  its  thousands 
of  beggars,  and  transformed  the  State  from  disorder  and  shiftlessness 
to  prosperity  and  content.  Created  Count  Rumford,  he  continued 
his  interest  in  the  study  of  Heat,  and  invented  the  first  economical 
cast-iron  open  domestic  stoves,  still  common  in  English  cottages. 
In  1796  he  was  called  back  virtually  to  rule  Bavaria,  retired  in  1799, 
took  the  leading  part  in  founding  the  Royal  Institution  of  London, 
married  Lavoisier's  widow  in  1804,  but  left  her,  and  resided  at 
Auteuil  until  1814. 

Struck  by  the  heat  developed  in  boring  cannon  in  the  arsenal — 
doubtless,  Uke  most  of  us,  he  had  picked  up  borings  fresh  from  the 
tool — he  made  experiments  to  find  out  whether  the  current  explana- 
tion, that  caloric  had  been  squeezed  out  of  the  soUd  metal,  was 
probable.  By  the  now  familiar  specific-heat  experiment  he  could 
find  no  difference  in  the  capacity  for  heat  of  solid  metal  and  of 
borings,  and  in  1798  he  set  a  horse  to  work  a  blunt  boring  tool  on  a 
oannon  '  casting-head  '  immersed  in  water  and  exultantly  records 
his  friends'  astonishment,  when,  in  2  J  hrs.,  2  gallons  of  water  boiled^ 
^ile  only  a  pound  of  chips  had  been  produced. 

Sir  Humphry  Davy  in  the  following  year  rubbed  together  two 
pieces  of  ice  in  a  frosty  atmosphere  (and  even  in  vacuo)  and  showed 
that,  with  no  possible  access  of  heat  from  without,  the  friction 
continuously  melted  the  ice,  actually  producing  a  liquid  which,  it 
was  agreed,  contained  not  less  but  more  caloric  than  the  ice. 

§  252.  But  it  was  not  until  1840  that  Joule  of  Manchester,  and 
others,  began  to  make  accurate  experiments  on  the  relation  of  work 
and  heat,  and  to  find  that  in  whatever  way  they  effected  the  con- 

177 


178  HEAT  [§  262 

version — by  compressing  air,  by  churning  water,  by  grinding  metal 
plates  together,  by  hammering  lead,  by  way  of  electro-magnetic 
induced  currents,  etc. — a  perfectly  definite  quantity  of  mechanical 
work  completely  converts  into  one  unit  of  heat.  This  quantity  is  termed 
the  Mechanical  Equivalent  of  Heat  {dynamical  equivalent,  Joule's 
equivalent,  J). 

Heat  is  thus  a  '  mode  of  motion  ' — a  form  of  energy. 

Joule's  favourite  apparatus  in  his  earlier  experiments  was  one 
in  which  falhng  weights  drove  a  paddle  and  churned  water,  a  grand- 
father clock  power- supply  which  made  the  experiment  exquisitely 
tedious.  He  found  that  772  ft.-lb.  produce  one  British  thermal 
unit  (pound  °F.).  Subsequent  allowance  for  discrepancy  between 
his  sensitive  mercury  thermometers  and  the  hydrogen  scale,  and  for 
gravity  at  Manchester,  raised  this  to  777  ft.-lb. 

Him  went  to  work  the  opposite  way ;  he  found  that  there  was 
a  greater  difference  between  the  heat  contained  in  the  live  and  the 
eidiaust  steam  from  an  engine  when  it  was  working  hard  than 
when  running  light.  He  measured  this  and  found  1391  ft.-lb. 
=  1  lb.  °C. 

Mayer,  a  physiologist,  in  1842,  made  an  estimate  as  follows  : 
The  specific  heat  of  air  allowed  to  expand  at  atmospheric  pressure 
as  it  is  heated  is  0-239.  According  to  theory,  this  is  1-4  times  its 
specific  heat  when  expansion  is  prevented  (the  elasticities  ratio  of 
§  415).  Now,  1  gm.  of  air  at  0°  and  1  atmo.  occupies  1/0-001293  = 
773  c.c.  and  expands  1/273  of  this  =  2-84  c.c.  when  heated  1**. 
It  therefore  does  work  in  lifting  the  atmosp?iere 

=  pressure  x  expansion  =  1,013,000  dynes  X  2-84  =  2-88  milHon 

ergs. 

Assuming  that  this  work  represents  the  additional  heat  energy 
absorbed  by  the  expanding  gas,  0-239  cal  x  0-4/1-4  =  2-88  million 
ergs. 

.*.  1  calorie  =  42  milHon  ergs.  , 

§  253.  In  1900,  experiments  were  carried  out  in  Manchester  with 
a  '  hydrauhc  '  brake,  a  sort  of  reversed  turbine.  Two  large  co-axial 
saucer-like  wheels  closely  face  each  other,  each  is  partitioned  up 
inside,  into  a  ring  of  radial  pockets  slanting  to  meet  those  on  the 
other  wheel.  Water  run  in  gets  caught  up  and  flung  violently 
from  wheel  to  wheel,  whereby  one  tends  to  drag  the  other  round. 
(It  is  the  contrivance  which  reappears  30  years  later,  with  fixed 
quantity  of  fluid  and  variable  speed,  as  the  '  fluid  flywheel.') 

The  one  was  rotated  by  a  100-h.p.  engine,  and  the  other  pre- 
vented from  following  it  by  a  load  on  a  radial  steelyard.  Here  the 
*  circumference  '  in  the  calculation  of  §  67  is  that  of  the  circle  on 
which  the  load  hangs,  and  in  which  it  would  have  been  hoisted. 
To  avoid  thermometer  vagaries  the  water  ran  in  from  an  ice  tank 
and  came  out  boiling  into  a  weighing  tank.     All  the  inevitable 


§264]  MECHANICAL   EQUIVALENT  179 

heat  leakages  were  most  carefully  gauged  and  allowed  for,  and  the 
result  is 

J  =  4186  X  10'  ergs  per  calorie  (15°). 
=  4-186  joules  per  calorie 
or  1400  ft.-lb.  per  lb.  °C. 

or  777-7  ft.-lb.  per  lb.  °F.  =  1  British  thermal  unit. 
[Units  of  heat  x  J  =  units  of  energy.] 

§  254.  But  you  will  make  measurements  of  the  Mechanical 
Equivalent  in  the  laboratory  ;  and  it  is  a  far  better  thing  to  describe 
to  an  examiner  what  you  have  done  yourself,  than  to  seek  to  beguile 
him  with  historical  essays. 

In  a  crude  experiment,  take  the  temperature  of  some  lead  shot, 
in  its  bottle,  and  then  pour  half-a-pound  or  more  of  it  into  a  card- 
board tube  2  or  3  ft.  long,  corked  at  the  ends.  Invert  the  tube, 
say,  60  times,  so  that  the  shot  crashes  from  end  to  end,  falling  a  total 
vertical  height  60  X  inside  length  of  tube,  whereby  each  gram  of 
lead  has  done  on  it  that  many  gram-centimetres  of  work,  or  g  times 
as  many  Ergs.  Pour  out  the  shot  round  the  thermometer-bulb,  in 
a  httle  cup,  and  observe  the  small  ultimate  rise  of  temperature; 
this,  multiplied  by  the  sp.  ht.  0-03,  gives  the  calories  produced 
per  gram  of  lead ;  divide  this  into  the  ergs,  and  the  quotient  is  J. 
See  Question  9,  below. 

A  better  experiment  utilizes  the  apparatus  of  Fig.  7,  §  67.  Into 
the  double-cone  friction-clutch  20  c.c.  of  cold  water  are  put,  and  the 
steady  temperature  of  the  whole  is  taken.  The  mill  is  run  for,  say, 
700  revolutions,  then  the  water  well  stirred  and  the  highest  tempera- 
ture recorded,  subsequently  waiting  for  half  the  time  of  running 
and  adding  the  small  cooling  which  ensues  on  to  the  top  of  that 
(§232).    Then 

W(yrk  MgrZN  ergs  =  J  x  {wt.  of  dutch  X  sp.  ht.  -h  20  gma.  Aq.)  X 

{corrected  rise  of  temperature). 
See  Question  14  below. 

Experiments  like  these,  or  youthful  attempts  to  imitate  the  fire- 
sticks  of  the  South  Seas,  serve  at  any  rate  to  impress  on  the  user 
how  small  a  heat  a  great  labour  kindleth. 

For  an  electrical  method  see  Chapter  L. 

Note  particularly  the  form  of  statement  employed  above, 
•.  .  .  vx)rk  completely  converts  into  .  .  .  heat.*  This  is  not  reversible, 
see  §  294,  and  this  form  should  be  adhered  to. 


180  HEAT 


EXAM  QUESTIONS,  CHAPTER  XVI 

1.  What  is  meant  by  the  equivalence  of  heat  and  mechanical  work  and  by 
what  experiments  is  it  suggested  ?  How  may  the  mechanical  equivalent 
of  heat  be  determined  in  the  laboratory  ?     (  X  6) 

2.  Why  is  the  specific  heat  of  a  gas  greater  when  it  is  allowed  to  expand 
with  heating  ?     What  theoretical  use  has  been  made  of  this  ? 

3.  Define  the  term  Mechanical  Equivalent  of  Heat. 

If  1  gm.  of  water  when  vaporized  at  100°  C.  becomes  1700  c.c.  of  steam, 
atmospheric  pressure  being  76  cm.  of  merciiry,  calculate  approximately  how 
much  of  the  heat  supplied  is  used  in  producing  this  increase  of  volume. 

4.  When,  how,  and  where  is  heat  developed  as  the  equivalent  of  work 
done  when  a  man  (a)  jumps  down  on  soft  ground,  (6)  slides  slowly  down  a 
rope,  (c)  walks  downstairs  ? 

5.  Waterfall  is  78  ft.  high,  water  at  top  is  at  40°  F.  What  would  be  tem- 
perature of  water  (a)  half-way  down,  (6)  at  bottom  ? 

6.  Calculate  the  rise  of  temperature  at  the  base  of  a  waterfall  100  m.  high. 

7.  State  the  principle  of  the  Conservation  of  Energy. 
Give  two  examples  of  the  transformation  of  energy. 

How  much  heat  is  produced  when  a  mass  of  500  kg.  falls  5  m.  on  to  the 
head  of  a  pile  ? 

8.  Describe  how  you  have  measured  the  Mechanical  Equivalent  of  Heat. 
A  5-kg.  hammer  falls  on  0-5  kg.  of  iron  from  a  height  of  1  m.,  under  an 

acceleration  double  that  of  gravity,  30  strokes  per  minute.     How  fast  will 
the  iron  be  heated  if  half  the  energy  goes  in  warming  it  ? 

9.  Some  shot,  density  11-4,  sp.  ht.  0-03,  is  contained  in  a  cardboard  tube 
5  ft.  long,  which  is  so  manipulated  that  the  shot  falls  from  end  to  end  40 
times.  It  is  then  poured  round  a  thermometer  and  found  4-7°  warmer  than 
before.     Calculate  J. 

Work  spent  among  shot  =  40  x  5         =  200  ft. -lb.  (per  lb.  shot). 

Heat  obtained     =  0-03  X  4-7  =  0-141  lb.  °C.    „ 
Neglecting  losses,  these  are  equal.     /.  1420  ft. -lb.  =  1  lb.  °C. 

10.  How  much  hotter  will  a  quantity  of  lead  become  in  consequence  of  a 
fall  of  300  m.,  supposing  it  retains  only  half  the  heat  generated  ?  How  and 
when  is  this  heat  most  likely  to  be  generated  ?     (  X  4) 

11.  Merciu-y  weighing  275  gm.  is  contained  in  a  tube,  of  water  equivalent 
3-3  gm.,  and  15-8  cm.  long;  it  is  inverted  100  times  and  the  mercury  rises 
0-85°  C.     Calculate  J. 

12.  A  block  of  ice  falls  from  the  end  of  a  glacier  which  is  just  melting,  and 
0-5%  of  the  ice  is  thereby  melted.  From  what  height  must  the  ice  have 
fallen?     (x  2) 

13.  A  snowball  hits  a  wall  at  15  m.  per  sec,  and  sticks;  what  fraction  of 
it  is  melted  by  the  impact  ? 

14.  200  gm.  hangs  from  the  rim  of  a  brake  wheel  80  cm.  circumference, 
and  is  just  kept  suspended  by  the  friction  between  the  cones  of  a  shpping 
friction  '  clutch,'  which  forms  part  of  a  calorimeter  of  total  water  equivalent 
40  gm.  The  calorimeter  warms  9-0°  during  1000  revs,  and  subsequently 
cools  0-3°  in  half  the  time.     Calculate  J. 

Work  spent  in  friction  =  200  X  80  X  981  X  1000  ergs  =  1570  joules. 
Heat  obtained  =  40  X  (9-0  +  0-3)°  =  372  cals. 

15.  Describe  a  good  method  for  J.  A  leaden  bullet  at  50°  hits  a  target, 
and  is  melted.  Calculate  its  minimum  speed,  given  m.  pt.  335°,  lat.  ht.  6-4, 
etc. 


MECHANICAL   EQUIVALENT  181 

16.  State  precisely  in  well -recognized  terms  what  is  implied  by  the  expree* 
sion  '  the  mechanical  equivalent  of  heat.' 

An  engine  of  10  H.P.  is  employed  to  grind  150  kg.  of  com  per  hour.  Find 
the  rise  in  temperature  of  the  meal  produced,  given  that  1  H.P.  =  746  joules 
per  sec,  4-2  joules  =  1  calorie,  specific  heat  of  meal  =  0-4,  and  half  the  power 
of  the  engine  is  wasted. 

17.  A  meteorite  initially  at  0°  C.  meets  the  earth's  atmosphere  and  is 
\  aporized  by  frictional  heating.  If  its  mean  sp.  ht.  were  0-2,  its  b.  pt.  3000°  C, 
and  latent  heat  of  vapour  50,  and  0-9  of  the  heat  were  simultaneously  loat, 
find  minimum  speed  at  first  contact. 

18.  Water  at  15°  C.  and  1000  atmos.  pressure  escapes  through  a  porous 
plug  into  the  atmosphere.     Find  its  temperature. 

Work  per  c.c.  =  1000  x  1,016,000  X  1  ergs  =  418  X  10'  X  1  X  (<  -  15). 

19.  Water  under  a  head  of  21  m.  is  drawn  through  a  half-open  tap  into  a 
pail.     Calculate  its  rise  of  temperature. 

20.  It  is  said  that  heat  and  mechanical  energy  are  mutually  convertible; 
put  this  more  precisely  and  accurately. 

Milk,  of  sp.  gr.  1-03  and  sp.  ht.  0-97,  is  being  forced,  by  a  pressure  of  200  kg. 
per  sq.  cm.,  through  fine  jets  into  an  open  vessel.  Calculate  its  rise  of  tem- 
perature. 

21.  A  basin  of  water  at  40°  F.  is  warmed  for  washing  the  hands  by  pouring 
in  a  quart  of  boiling  water.  What  addition  of  energy  in  foot-tons  does  this 
represent  ? 

22.  A  steam  pile-driver  burnt  3/4  ton  of  coal  (7000  B.Th.  units  per  lb.) 
while  delivering  2000  blows  with  a  2-ton  monkey  falling  3  ft.  Calculate 
its  efficiency.  The  concrete  pile  was  driven  15  ft.,  it  was  specified  to  carry 
30  tons.     Estimate  the  theoretical  efficiency  of  the  whole  process. 

23.  The  wall  of  a  cold-storage  chamber  is  of  area  1200  sq.  m.  and  is  lagged 
with  15  cm.  of  slag-wool,  of  thermal  conductivity  000013;  the  chamber  is 
to  be  kept  at  —  2°  C.  against  an  outside  temperatm-e  of  20°  C.  If  the  refrigerat- 
ing machinery  is  1  /3  efficient,  calculate  the  driving  power  required. 


PRACTICAL  QUESTION. 
The  faUing  shot^tube ;  or  some  form  of  friction  mill. 


CHAPTER   XVII 
CHANGE   OF  STATE— MELTING  OR  FUSION 


§  261.  If  a  thermometer  is  put  into  a  vessel  among  fragments 
of  a  solid,  such  as  naphthalene  or  wax,  the  whole  steadily  suppH(  ' 
with  heat,  and  the  thermometer  watched,  its  steady  rise  presently 
ceases.  On  inspection,  what  has  happened  is  that  the  substance 
has  begun  to  melt — to  change  its  physical  state  from  Solid  to  Liquid. 
And  provided  that  it  is  kept  well  stirred,  so  as  to  expedite  th( 
sluggish  spreading  of  heat  through  the  mixture,  and  prevent  local 
overheating,  the  thermometer  moves  hardly  at  all  until  all  th( 
soUd  has  melted,  then  resumes  its  steady  rise.  Repeating  the 
experiment  as  often  as  you  like  with  the  same  material,  the  ther-j 
mometer  will  always  stick  at  this  same  Melting  Point  of  temperature. 
Further,  if  the  liquid  is  allowed  to  cool  and  congeal,  the  falling^ 
thermometer  will  stand  steady  for  some  time  at  this  same  tempera- 
ture, now  a  Freezing  Point. 

Clearly  the  transition  of  any  particular  substance  from  its  solid 
to  its  liquid  condition  : 

(a)  takes  place  reversibly  at  a  definite  temperature  ; 

(6)  involves  the  absorption  and  disappearance  of  a  characteristic' 
quantity  of  heat,  and  conversely  its  reappearance  during  sohdification. 
For  on  the  way  up  heat  is  poured  into  the  substance,  without  affect- 
ing its  temperature,  for  a  time  proportional  to  the  amount  to  be 
melted ;  and  on  the  way  down  the  body  goes  on  giving  out  heat  toi 
its  surroundings  at  the  usual  rate  for  some  time  without  any  diminu- 
tion of  temperature. 

When  the  solid  has  been  brought  up  to  the  melting  point  already,^ 
the  number  of  calories  then  required  to  melt  1  gramme  of  it  is  calledt 
its  Latent  Heat  of  Liquefaction,  or  the  Latent  Heat  of  the  substance  ini 
its  liquid  state. 

The  Melting  Point,  or  more  conveniently  the  Solidifying  Point,i 
of  any  substance,  is  determined  in  precisely  the  way  suggested 
above,  by  finding  where  the  heating  or  Cooling  Curve  (cf.  Fig.  74)i 
of  a  potful  of  it  shows  a  horizontal  '  flat.'  This  is  a  weU-lmowni 
laboratory  experiment. 

The  measurement  of  this  Latent  Heat  of  Fusion — or  the  equa; 
development  of  heat  on  solidification — ^has  been  described  iii 
Chapter  XIV. 

§  262.  The  process  of  Fusion  is  not  always  so  simple  as  outhnec 
above.  Often  the  substance  begins  to  soften  long  before  it  melts 
from  plastic  solid  it  passes  by  slow  stages  into  very  viscous  liquid 

182 


§  263]  FUSION  183 

having  all  the  time  an  increased  specific  heat,  and  the  temperature 
at  which  it  finally  takes  up  the  small  remainder  of  its  latent  heat, 
and  satisfactorily  liquefies,  may  or  may  not  be  sharply  marked. 
Of  crystalline  substances,  Platinum  and  Iron  are  plastic  and  weld- 
able  500°  before  melting,  but  melt  sharply  at  last ;  Silica  (quartz) 
softens  at  1500°,  can  presently  be  worked  in  the  oxy-gas  flame  like 
glass,  can  later  be  shot  or  blown  into  threads,  and  has  no  well- 
defined  melting  point.  The  '  colloid,'  Glass,  is  at  best  a  treacly 
liquid  slowly  hardening,  through  working  and  annealing  stages, 
to  its  usual  condition,  from  which  long-continued  heating  enables  it 
to  pass  on  by  progressive  devitrification  to,  ultimately,  a  crystalline 
stony  mass. 

Substances  of  mixed  composition  often  give  two  or  more  flats 
on  a  slow  Cooling  Curve — solidifying  points  of  Fractions  of  definite 
composition  crystallizing  out  of  the  fluid.  This,  of  course,  means  a 
period  of  plasticity.  The  fusible  alloys  used  as  solders  show  this 
very  well,  the  plumber's  joints  are  '  wiped  '  when  in  a  clay-like 
condition  of  solid  grains  and  fluid  metal.  The  solidification  of 
paraffin  wax  may  take  place  in  three  closely  succeeding  steps  ;  and, 
near  the  other  end  of  the  paraffin  series,  the  lightest  petrol  is  such  a 
mixture  that  it  has  only  reached  the  viscous  liquid  stage  at  —  190°  C. 

Melted  Sulphur  falls  below  its  melting  point  before  beginning  to 
freeze.  It  is  known  to  occur  in  two  crystalline  forms,  and  it  solidifies 
into  a  mixture  of  these  two,  different  from  the  stable  one  originally 
melted.  The  warm  soUd  now  cools  more  slowly  than  fits  the  normal 
cooling  curve,  revealing  the  fact  that  an  unstable  crystalline  form  is 
rapidly  changing  into  the  stable  one,  setting  free  a  Latent  Heat  in 
the  process.  In  this  way.  Cooling  Curves  have  been  of  great  value 
in  the  study  of  steels,  and  other  alloys. 

§  263.  Frequently  a  liquid  cools  below  its  freezing  point  without 
any  signs  of  freezing,  but  this  under-cooled  condition  is,  of  course, 
unstable.  Sooner  or  later  rapid  solidification  begins,  and  setting 
free  latent  heat,  raises  the  whole  mass  up  to  its  true  freezing  point, 
and  continuing  more  slowly,  keeps  it  there  until  all  is  sohd.  This 
under-cooled  condition  is  most  easily  induced  if  the  liquid  is  dis- 
persed in  drops  through  another  fluid,  sulphur  in  zinc  chloride 
solution  has  been  cooled  even  to  0°  C.  without  solidifying,  and  water 
in  oil  to  —  20°  C.  Much  of  the  water  in  ordinary  Clouds  exists  as 
these  under-cooled  drops,  persisting  until  this  temperature. 

Under-cooling  is  often  a  convenience,  rather  than  not,  in  finding 
freezing  points,  for  the  sudden  rise  of  the  thermometer,  with  sub- 
sequent steadiness,  makes  the  determination  very  definite. 

The  condition  is  that  of  the  '  Supersaturated  Solutions  '  of  the 
chemist,  made  by  dissolving  silver  nitrate,  say,  in  a  minimum  of  hot 
water,  or  by  melting  sodium  sulphate,  thiosulphate,  etc.,  in  their 
own  '  water  of  crystallization  '  plus  a  very  little  more.  These 
solutions  habitually  refuse  to  crystallize  spontaneously,  but  ^ojj 
when  violently  shaken  up  or  when  a  crystal  of  the  solid  is  dropped 


184  HEAT  [§  263 

in,  and  thereupon  get  warm  from  the  liberated  Heat  of  Solution.  For 
their  crystals  are  hardly  more  than  nasty-flavoured  ice,  and  to  liquefy 
them  in  any  way  calls  for  its  latent  heat :  recollect  how  intensely 
cold  the  bottle  becomes  when  dissolving  up  '  hypo  ' ;   see  §  267. 

§  264.  It  is  well  known  that  a  distinct  Change  of  Bulk  accompanies 
fusion  :  ice  floats,  most  solids  sink  in  their  melted  liquid.  The 
change  is  most  easily  ascertained  by  measuring  the  densities  s  and  I 
of  solid  and  liquid  near  the  melting  point,  by  any  s.g.  method. 
Then  Specific  Volume  {i.e.  volume  of  1  gm.),  I/5  melts  to  volume 
l/l,  and  this  change  is  the  fraction  (I/5  —  l/Z)/(l/5)  of  the  original. 

The  change  of  bulk  lays  the  process  of  Fusion  open  to  the  influence 
of  mechanical  Pressure.  For  evidently  if  an  obstacle  is  put  in  the 
way  of  the  sudden  free  expansion  of  a  body,  by  imposing  a  heavy 
pressure  which  it  must  force  back,  it  must  be  given  the  power  to 
do  this  external  work  by  increasing  its  molecular  activity,  i.e.  by 
heating  it  hotter.  Hence  substances  which  expand  on  liquefying 
will  have  their  melting  points  raised  by  pressure  ;  while  ice  and  other 
substances  which  contract  on  liquefying  have  their  melting  points 
lowered  by  the  pressure  which  is  helping  them  shrink. 

The  equilibrium  melting-freezing  point  under  heavy  pressure  is 

found  by  putting  the  liquid  under  pressure  in  a  steel  '  bomb,'  and 
finding  the  '  flat '  on  its  cooling  curve.  An  extra  turn  of  the  screw, 
and  the  determination  is  repeated  at  higher  pressure  still. 

Theoretical  calculation  applied  to  the  question  gives,  approxi- 
mately, this  very  reasonable  result : 

R-ise  of  M.  Pt.  _  expansion  per  gm.  on  melting  X  pressure  [dynes] 
M.  Pt.  ^Absolute  ~  Latent  heat  expressed  in  ergs 

which  says  that  the  necessary  increase  of  molecular  activity  (measured 
by  temperature)  is  in  the  same  proportion  to  the  total  molecular 
activity  as  the  extra  work  to  be  done,  in  lifting  the  outer  pressure, 
is  to  the  total  work  spent  in  freeing  the  molecules  from  solid  bondage. 
Putting  1  atmo.  =  1,013,000  dynes/cm.2  and  42,000,000  ergs. 
=  1  cal.,  the  formula  becomes 

Rise  of  M.  Pt.  per  atmos.  _  expansion  per  gm.  on  melting 
M.  Pt.  ° Absolute         ~  41  x  Latent  heat  in  calories 

and  by  this  the  bracketed  figures  in  the  table  were  calculated. 

Seeing  how  small  is  the  expansion,  and  therefore  how  small 
an  amount  of  external  work  is  done  even  against  heavy  pressure, 
it  is  evident  that  the  effect  of  pressure  can  he  only  very  small.  Thus 
the  figures  in  the  table  at  the  end  of  this  chapter  show  that  for 
naphthalene,  30  atmos.  would  raise  the  melting  point  1°  C,  while 
for  water,  with  its  great  latent  heat,  it  takes  1-^0-0072  =  139  atmos. 
to  lower  the  freezing  point  merely  one  degree. 

[Plainly,  you  need  not  worry  about  pressure  when  testing  ther- 
mometers in  ice.]  i 


265]  FUSION  185 


§  265.  Ice.  The  rather  exceptional  properties  of  Ice  have  so 
profound  an  influence  in  Nature  that  they  demand  special  notice. 

The  latent  heat  of  water  being  so  great  makes  its  freezing  a 
slow  process,  and  even  small  quantities  take  a  considerable  time 
to  freeze  solid.  Conversely,  the  melting  of  ice  in  mass  takes  a  very 
long  time,  icebergs  drift  far  into  warmer  seas  and  stronger  sunshine 
before  their  dissolution,  and  we  are  all  famiHar  with  the  long-drawn- 
out  chill  of  a  slow  Thaw. 

Consisting,  as  it  probably  does,  of  water- substance  in  the  tri- 
hydrol,  (H20)3,  condition,  Ice  assumes  a  typically  hexagonal  crystal- 
line form  which -you  have  doubtless  s6en  in  numerous  pictures  of 
snow  crystals — direct  low- power  micrographs  of  minute  snowflakes, 
taken  on  the  spot — and  of  ice  flowers,  cavities  melted  in  clear  ice 
by  lantern  heat.  Also,  though  not  with  the  same  geometrical 
regularity,  in  hoar-frost,  and  the  breath  of  Jack  Frost  on  the  window- 
pane. 

If  water  be  blown  from  a  fine  spray  into  an  atmosphere  below 
—  20°  C,  however,  and  the  frozen  spherules  collected  under  a 
polarizing  microscope,  no  trace  of  crystalline  chargicter  appears  in 
them  ;  there  wasn't  time  to  pack  properly.  The  addition  of  colloids 
has  the  same  effect,  a  little  gelatine  gives  ice-cream  its  velvety 
smoothness  by  preventing  the  formation  of  gritty  crystals  of  ice. 
Just  in  the  same  way  sulphur  is  left  in  microscopic  yellow  beads  by 
the  rapid  evaporation  of  its  solution  in  carbon  disulphide  thickened 
with  Canada  balsam. 

In  freezing  to  clear  ice,  without  the  slightest  contamination  of 
micro-organism,  muddy  particle,  or  dissolved  solid,  liquid,  or  gas — 
the  way  par  excellence  of  obtaining  Aqua  pura — ^water  expands 
exactly  one-eleventh  in  bulk,  going  down  to  a  density  of  0-9167, 
according  to  the  best  modem  determinations.  Consequently  Ice 
floats  ;  and  as  more  usually  it  is  whitened  by  the  myriad  air-bubbles 
thrown  out  of  solution  during  freezing,  or  in  glaciers  never  eliminated 
during  the  imperfect  consolidation  of  the  original  neve,  and  forming 
even  up  to  a  seventh  its  bulk,  their  calved-off  icebergs  may  be 
floating  with  two-ninths  their  volume  out  of  sea- water,  s.g.  1*025, 
instead  of  the  single  ninth,  or  less,  that  they  tell  about  who  forget 
that  ice  is  bubbly  and  that  sea  is  salt.  The  extreme  of  this  occurs, 
of  course,  in  fresh-fallen  snow,  a  foot  thickness  being  commonly 
equivalent  to  only  an  inch  of  rain ;  while  an  artificial  instance  is 
modern  ice-cream,  whipped  up  during  manufacture  to  a  frothy 
*  swell '  of  60—80%- 

The  very  converse  is  found  on  Kanchenjunga,  where  the  daily 
'  heat-treatment '  by  sun  and  frost,  going  on  year  after  year,  pro- 
duces ice  of  such  toughness  that  the  labour  and  delay  of  cutting 
steps  in  it  have  contributed  largely  to  the  defeat  of  recent  expeditions. 


186  HEAT  [§  266 

Forming  a  firm  floating  layer,  of  which  an  inch-and-a-half  will 
carry  a  man,  and  a  foot-and-a-half  a  railway  train,  ice  shields  the 
water  from  the  wind  which  was  rippling  and  stirring  up  the  surface. 
Hence,  and  also  as  it  is  a  poor  conductor  of  heat  (0-004),  the  rate  of 
loss  of  heat  from  a  pond  once  well  frozen  over  is  much  less  than  it 
was  before  freezing  began,  and  the  total  formation  of  ice  in  a  frost 
is  a  mere  fraction  of  what  it  would  be  if  ice  sank,  while,  as  in  §  181, 
the  depths  of  the  water  beneath  remain  well  above  the  freezing 
point,  at  4°  C.  or  39°  F. 

In  some  swift-running  clear  rivers,  however,  such  as  the  chalk 
stream  of  the  Avon  at  Christchurch,  or  the  St.  Lawrence,  which 
has  left  its  silt  in  the  Great  Lakes,  the  whole  of  the  tumbling  water 
may  get  cooled  to  zero,  or  even,  in  this  paucity  of  '  nuclei  of  crystalU- 
zation  '  (cf .  §§  263,  312),  '  super-cooled '  perceptibly  beyond,  and 
then  one  or  other  of  two  new  effects  may  ensue  : 

During  long  clear  nights  parts  of  the  bed  of  the  stream,  especially 
dark  rocks  or  weed-beds,  radiate  through  the  pellucid  water  to  the 
cold  vault  of  sky,  and  to  these  super-chilled  surfaces  adheres  a  quickly- 
growing  deposit  from  the  super-cooled  water,  of  ground  ice,  or 
anchor  ice,  to  a  thickness  of  even  5  ft.  in  a  night,  with  long  tentacles 
straggling  up  like  a  growth  of  weeds.  The  morning  sun,  striking 
through  the  clear  water,  loosens  this,  and  it  floats  up,  to  the  annoy- 
ance of  the  early  fisherman,  often  carrying  a  collection  of  its  anchor- 
age with  it :  thousands  of  tons  drift  down  the  St.  Lawrence  every 
morning.  Why  a  little  sunshine  is  so  effective  is  by  no  means 
evident,  seeing  that  practically  all  its  heat  is  absorbed  by  the  upper 
fathom  of  water ;  Barnes  maintains  that  some  particular  green  or 
blue  wave-length  is  specifically  destructive  of  trihydrol. 

The  other  possibility  is  that  a  mist  of  colloidal  icy  particles  forms 
throughout  the  supercooled  water,  exactly  as  a  mist  of  water  drops 
forms  in  supercooled  air,  §  284,  and  these  grow  to  crystallization, 
just  as  minute  '  liquid  crystals  '  can  be  watched  clotting  into  angular 
solid  ones  under  the  polarizing  microscope,  forming  '  lolly  '  or 
frazil  ice.  This  thickens  the  water  (as  a  strong  hot  saline  solution, 
kept  stirred,  thickens  with  granular  crystals  as  it  cools),  grows  in 
fringes  on  the  canoe  paddle,  and,  as  the  Harbour  Master  of  Montreal 
writes,  '  has  a  pecuHar  effect,  something  like  a  spider's  web,  though 
much  heavier,  becoming  a  form  of  glue  on  ships'  sides  and 
bottoms ;  vessels  anchored  here  in  cold  weather  have  a  great 
deal  of  difficulty  in  moving  once  this  has  got  hold  of  them.'  The 
Lachine  rapids  get  the  credit  for  most  of  this  (1  sq.  ft.  of  the  river 
has  been  reckoned  capable  of  producing  from  10  to  15  cu.  ft.  of  ice). 

Coming  down  and  conglomerating  with  the  growlers  of  anchor 
ice  underneath  the  ice  bridge  which  forms  downstream  of  the 
harbour,  it  causes  so  much  obstruction  that  the  otherwise 
floodless  St.  Lawrence  has  often  risen  15  ft.  in  the  effort  to  drive 
through. 


§  265]  FUSION  187 

The  expansion  in  freezing  has  an  effect  on  domestic  water-pipes 
only  too  unpleasantly  apparent  when  the  subsequent  thaw  releases 
their  contents.  Against  this  the  thick  lead  service  pipes  (J-in.  bore 
6  lb. /yd.,  f -in.  9  lb. /yd.),  now  insisted  on,  are  a  real  protection ; 
they  are  uniformly  strong,  so  that  one  particular  spot  does  not 
readily  bulge  and  weaken ;  water,  shut  in  between  earlier- frozen 
parts  in  these  pipes,  may  rise  to  such  a  pressure  as  it  freezes,  as  to 
])artly  melt  the  ice  plugs,  and  escape  back  into  the  mains.  But  the 
stoutest  pipes  exposed  to  quick  hard  frost  soon  split. 

The  investigation  of  the  lowering  of  the  freezing  point  with 
pressure  was  carried  up  to  700  atmos.,  with  a  drop  of  5°  as  expected, 
but  at  the  enormous  pressure  of  2000  atmos.  the  plunger  of  the 
hydrauUc  cylinder  makes  a  sudden  move  inwards,  and  our  familiar 
dilated  ice  becomes  another  crystalline  solid,  a  new  ice  3%  denser 
than  water.  With  increasing  pressures  there  are  further  jolts, 
to  an  ice  6%  denser  than  water,  and  finally  one  9%  denser. 

Nobody  has  ever  seen  these  ices,  nor  presumably  ever  will ;  they 
can  exist  only  under  as  many  tons  to  the  square  inch  as  we  do 
pounds ;  let  us  return  to  the  natural  brand. 

The  action  of  freezing  again  as  soon  as  the  pressure  is  relieved 
is  called  Regelation.  In  a  well-known  Experiment,  a  Block  of  Ice 
is  bridged  between  two  stools,  and  a  heavy  weight  is  hung  in  a 
loop  of  thin  steel  wire  round  the  middle  of  the  block.  The  wire 
slowly  cuts  through  the  ice,  but  leaves  it  as  solid  as  ever,  with  only 
a  slight  filmy  appearance  marking  its  track.  The  pressure  under 
the  wire  lowers  the  melting  point,  the  ice  melts,  the  water  escapes 
past  the  wire  and  re-freezes  above  it,  its  latent  heat  being  con- 
ducted down  through  the  wire  (all  below  0°)  to  the  cutting  side, 
which  is  a  fraction  of  a  degree  colder.  Catgut,  a  bad  conductor, 
fails,  by  not  returning  this  heat  fast  enough ;  mere  pressure,  of 
course,  cannot  go  on  liquefying  indefinite  quantities  of  ice,  and 
the  energy  of  fall  of  the  weight  is  also  quite  inadequate  :  no  regelation, 
no  cutting.    See  this  done. 

The  weight  on  a  skate-blade,  or  that  of  a  curling-stone,  liquefies 
a  surface  film  at  the  areas  of  contact,  and  the  skater  or  the  stone 
glides  on  a  thin  lubricant  produced  exactly  when  and  where  it  is 
wanted,  and  the  more  freely  the  harder  the  pressure — an  ideal 
system  of  lubrication  occasionally  imitated  by  orange-peel  on  the 
Ijavement. 

Regelation  confers  on  Snow  its  binding  power.  Very  cold  snow 
is  typically  fine,  and  will  not  bind ;  in  a  less  frigid  atmosphere  the 
flakes  are  larger — already  clung  together — and  bind  into  admirable 
snowballs  and  miniature  roof -glaciers.  The  pressure  of  crystal  on 
crystal  melts  the  points  of  contact,  and  squeezes  out  water. 
which  immediately  re-freezes  all  round  them  and  seals  the  grains 
together. 


188  HEAT  [§  265 

In  this  way  the  soft  snow  of  the  snowfields  gradually  compresses 
and  combines  into  the  clear  ice  of  the  Glacier.  The  weight  of  the 
glacier  on  its  sloping  bed  bears  hard  on  projecting  bosses  of  rock, 
crushes  and  partly  liquefies  the  ice  there,  and  squeezes  it  round  them 
to  re-freeze  again  on  the  lee  side.  From  this  action,  together  with 
the  existence  of  '  gliding  planes  '  in  the  ice  crystals,  §  144,  the  whole 
glacier  of  hard  elastic  ioe  streams  on  like  a  river  of  very  viscous 
liquid,  at  a  speed  averaging  perhaps  18  in.  a  day  in  the  Alps,  but 
reaching  as  much  as  80  ft.  in  the  ice  sheet  of  Greenland,  the  great 
iceberg  factory  of  the  Atlantic.  Embedded  in  its  under  surface, 
by  the  same  action,  are  the  hard  fragments  of  rock,  which  so  slowly 
grind  its  bed  to  the  polish  that  may  endure  for  scores  of  thousands 
of  years  after  the  glacier  has  disappeared. 

Probably  the  warmth  carried  down  the  crevasses  by  falls  of  sun- 
melted  water  has  a  deal  to  do  with  keeping  the  lower  surface  of 
the  icy  blanket  near  enough  to  0°  C.  for  pressure -melting  to  be 
practicable. 

Below  —  9°  C.  Ice  is  a  thoroughly  hard  and  stony  substance, 
good  enough  to  mend  roads  with  (as  exceptionally  in  winter  here, 
and  cf.  §  385),  but  above  that  temperature  it  rapidly  loses  both 
strength  and  hardness,  to  become  the  ultra-fragile  soUd  that  every- 
one attacks  with  the  feeblest  of  weapons,  another  plain  instance  of 
that  essential  physical  continuity  insisted  on  in  §  145,  strikingly 
abrupt  though  the  final  transition  may  be. 

Per  contra,  one  finds  some  refrigerator-made  ice  in  the  rotten 
condition  that  drives  every  skater  to  the  bank  ;  and,  on  trial,  having 
a  latent  heat  of  liquefaction  short  by  a  dozen  calories  ;  evidently  a 
honeycomb  mass  of  columnar  crystals  soaking  with  water. 

Ice  is  a  very  volatile  solid,  giving  off  even  at  —  10°  C.  as  much 
as  2-4  gm.  of  vapour  into  a  cubic  metre  of  air,  and  at  0°  twice  this 
amount,  see  Fig.  82,  a  far  greater  volatility  than  that  of  camphor, 
naphthalene,  etc.  Recollect  how  ice  and  snow  disappear  from  the 
paths  during  a  few  days'  windy  frost,  and  how  sheets  from  the  wash, 
which  went  stiff  as  boards  when  first  hung  out  to  dry,  become  soft 
again  in  a  few  hours. 

For  '  Heavy  Ice  '  see  §  925.  .^ 

§  266.  After  Ice,  Iron.     The  molten  metal,  density  6-9,  solidifies 
to  a  density  6-5,  a  6%  expansion,  which  enables  it  to  press  into  all  I 
corners  of  the  mould  at  the  last  moment,  and  produce  sharp  castings  ; 
as  do  type-metal  and  similar  alloys,  for  the  same  reason. 

Accordingly,  Iron  exhibits  a  Regelation.  Iron  bars  at  a  bright 
welding  heat,  1400°  C,  had  their  ends  jammed  together  suddenly 
at  \  ton  per  sq.  in.,  the  temperature  fell  57°,  and  the  bars  welded 
together  as  the  pressure  was  released.  It  is  the  same  process,  then, 
that  unites  white-hot  iron  under  the  hammer  of  the  smith,  and  cakes 
snow  into  lumps  under  the  feet  of  the  wayfarer.  ^ 

I 


{ 


§  268]  FUSION  189 

§  267.  Freezing  mixtures.  We  have  already  noticed  that  the 
liquefaction  of  a  substance  by  solution  in  water  usually  demands 
a  supply  of  heat ;  e.g.  per  gramme  common  salt  20-7  cals. ;  sodium 
thiosulphate  44 ;  sodium  sulphate  cryst.  57  ;  ammonium  sulpho- 
cyanide  75  ;   ammonium  nitrate  79  cals. 

Hence  a  soluble  salt  rapidly  dissolved  in  cold  water,  and  absorbing 
this  '  latent  heat  of  solution,'  will  bring  the  temperature  down  very 
low  for  a  time. 

Half  a  pound  of  powdered  ammonium  nitrate  stirred  into  half 
a  pint  of  cold  water  may  reduce  it  to  —  15°  C,  and  equal  parts  of 
powdered  sulphate  of  soda  and  diluted  sulphuric  or  hydrochloric 
acid  will  have  about  the  same  effect.  A  more  domestic  one  consists 
of  about  equal  parts  of  washing  soda  and  sal  ammoniac.  These  are 
the  only  ice- less  freezing  mixtures  practical  enough  to  be  worth 
mention  [unless  one  includes  solid  carbon  dioxide,  *  dry  ice,'  dis- 
solving in  ether,  at  —  79°]. 

Doubly  effective  are  mixtures  of  ice  and  a  solid  salt,  where  both 
Uquefy.  1  part  of  coarse  common  salt  and  3  parts  of  broken  ice 
will  reach  —  22°  C,  and  3  parts  of  crystallized  calcium  chloride 
and  2  of  ice  reach  —  55°,  easily  freezing  mercury.  The  action 
is  that  the  salt  continuously  dissolves  to  a  saturated  solution  in 
the  liquefying  ice,  and  the  temperature  reached  is  the  melting 
point  of  ice  in  equilibrium  with  saturated  solution  of  the  salt. 
For  how  it  comes  about  that  this  is  much  lower  than  its  melting 
point  in  equilibrium  with  pure  water,  consult  §  377. 

§  268.  Arctic  ice,  frozen  hastily,  entangles  residual  brine,  but 
this  gradually  leaches  out  when  it  is  piled  in  hummocks  ;  the  crj'stals 
themselves  are  pure.  Sea- water  and  ice  form  evidently  a  dilute 
Freezing  Mixture  ;  the  equilibrium  freezing  point  is  about  —  2**  C, 
and  this  persists  throughout,  the  temperature  of  maximum  density 
of  salt  water  being  —  3°  C. 

In  an  extant  fragment  of  my  father's  diary,  for  1860,  the  bleakest 
winter  on  record,  it  is  noted  that  '  as  the  light  improved  we  were 
surprised  to  see  that  the  salt  water  was  frozen,  but  we  were  saved 
the  trouble  of  deciding  whether  to  bathe  or  not  by  slipping  off  the 
springboard,'  and  thereafter  he  and  his  young  cousins  made  a 
morning  practice  of  warming  up  after  their  dip  bv  sliding  on  the 
shallow  end.  The  elder  told  me,  half  a  century  later,  how  these 
unforgettable  experiences,  mellowing  with  years,  led  him  to  intro- 
duce '  Daylight  Saving  '  to  the  House  of  Commons ;  so  that,  in  a 
manner  of  speaking,  England  owes  her  '  Summer  Time  '  to  the  bitter 
cold  of  icy  salt  water. 


19tf 


HEAT 


[§27 


a  -p 

'©  o 
^  ft 


fl  o  g 

<S     ©     rj 


m 


o 


<N  (M 
+    I 


CO  o  CO  53 

00    I     I  o  p 

66  I    I 

I    I 


60 
+  + 


O  (N  CO  I— I  O        rH  i-l  O  »0 
o     iX>  CO  (N  Tt<  (N  i-H 


"^  9  T"  .^9  'f  '^  '9 

t-»bs^obo666>oi2 


00        CO  ^ 

lOO(Mi-H«OOi  U32 

•000000  I    ""I  *? 

!•  666666  66 

""          III  + 


o »«      »o  »o 

ec  ooooousi-i^rH 

'<*'^0-<*QpOOa005x 
CSl6666i-HrHpH66 


•^         O 

CO   »0  V  Tt< 

»i3  I— I  »0  <M  »0  00 

u:)tHooc<iioo50wjOc» 
c<ii-H6t^66i-H'^pH6 

I— i  «o 

05 


CO  "«!»< 

00  CO  I    CO 

<NO  I    O 

66  6 


O  CCl 

»o  (N  -'^i  r- 
■-•666  6 


<M  <N  CO 

<NeOCCOi.-l05(Ml>(N 
(NOOOi-HTjHCOi-ICOCO 

6666666666 


31 


.1^  5  g 

^        ft 


o 


^^^  CO  o 

©543 
o 


bo 


^ajo 


m 


ft-^  5" 


So  U5  l>  O  <N  00    I      I 


;-■;  »o  o  »o  r*(  ,-H  »o   , 

<->  CO  CO  Oi  t*  (M  t^ 
o     (M  (M  pH  CO  CO(N     ' 


COQOOOeCO<N<M 
OiOOJrt^OOOCOCO 
(M  »0        .-H        CO 


.0»OOOOOC<>rJ< 

oco»Oi-toeoo»oio 

•COC^-^t^CO-^rHrH 


(N      u5  OS  eo 

lOOS-^lOOiOCOlO 
fHCO<NOCO<NCOCO 

00000000 
00000000 

66666666 


»o  CO  00 

tJH  r-l  0>  10  t^  (M  »0 

l>'^COOI>(NI>QO 

6i-l666rH<Ni-t 


10 

.I-H        10        00        Gs?  »0 

Ocibr-i'^o6coi>-^ 

t-COCOO-^'^lO'^ 
r-(  1-H  CO  t1< 


CO 
I— I  CO  10  Tf<  CO  CO 

COCNI»CiOtJH(MO<N 

666>-h6666 


FUSION  191 


EXAM   QXJESTIONS,  CHAPTER  XVII 

One  way  or  another,  Ice  always  seems  a  naturally  interesting  substance, 
once  away  from  latent  heat  calculations. 

1.  Give  a  brief  account  of  change  of  state. 

2.  What  is  the  efifect  of  pressure  on  (a)  the  boiling  point,  (6)  the  meltini;^ 
point,  of  a  substance  ? 

Describe  experiments  to  illustrate  the  effect,  pointing  out  any  differences 
in  the  behaviour  of  different  substances.     (  X  2) 

3.  Describe  a  method  of  determining  specific  heats  by  means  of  cooling 
curves.  If  a  substance  solidifies  during  the  experiment,  show  how  its  cooling 
curve  enables  you  to  find  both  its  freezing  point  and  its  latent  heat.     (  X  3) 

4.  How  would  you  make  an  attempt  to  ascertain  what  two  different  kinds 
of  wax  were  present  in  a  mixture  ?  How  can  you  very  simply  find  out  whether 
a  substance  expands  or  contracts  on  melting  ? 


PRACTICAL  QUESTIONS 

Graph  a  cooling  curve  and  deduce  an  '  emissivity,'  i.e.  the  loss  per  sq.  cm. 
of  surface  per  1°  warmer  than  the  surroimdings  :  determine  a  melting  point, 
or  a  temperature  of  coagulation. 

Find  the  specific  heat  of  a  liquid  by  cooling. 


CHAPTER   XVIII 
CHANGE   OF  STATE— VAPORIZATION 


§  271.  Quite  unlike  Fusion,  the  Vaporization  of  a  substan^ 
goes  on  at  all  temperatures,  up  to  a  limiting  '  boiling  point,'  wIk 
quiet  Evaporation  suddenly  passes  into  turbulent  Ebullition. 

That  Evaporation  is  constantly  going  on  is  evidenced  by  tl 
smell  of  aromatic  substances,  many  of  which  disappear  so  slowj 
that  their  loss  of  weight  in  a  week  may  be  inappreciable.  Thj 
layers  of  ice  and  snow  gradually  disappear  from  the  paths  even 
the  hardest  frost,  and  wisps  of  mist  wreathe  over  a  sheltered  broc 
We  hang  things  out  to  dry  without  consulting  the  thermomete^l 
it  is  true  we  expect  them  to  dry  quicker  in  the  summer  (or  before" 
the  fire),  but  even  then  we  know  that  '  dampness  '  already  present 
in  the  air  will  hinder  evaporation  unless  there  is  a  wind  or  draught 
to  blow  the  moisture-laden  air  away  quickly. 

This  variability  of  temperature  of  vaporization  makes  it  less  eas\ 
to  fix  a  Latent  Heat  of  Vaporization,  e.g.  one  has  to  quote  tht 
Latent  Heat  of  Steam,  i.e.  the  number  of  calories  necessary  t<: 
convert  1  gm.  of  water  at  t°  into  vapour  without  rise  of  tempera- 
ture, as 

600  -  0-60  t°  C. 

At  its  normal  boiling  point,  100°  C,  this  amounts  to  540  cals. 
and  tabulated  Latent  Heats  usually  refer  to  boiling  away  at  tabu 
lated   Boiling  Points  under  normal   atmospheric   pressure. 

[The  '  total  heat '  necessary  to  convert  water  at  0°  into  steari 
at  f  is  600  +  0-4  t°  cals.  per  gm.] 

§  272.  How  rapid  quiet  evaporation  can  become  is  striking]] 
shown  in  the  Spheroidal  State. 

Drops  of  water  thrown  on  a  hot  plate,  e.g.  a  freshly  heated  flat-iron 
turned  up,  run  about  hastily,  but  only  gradually  shrink  up  an»i 
disappear,  without  the  least  noise.  A  bright-red-hot  iron,  plunge- 
into  water  and  held  still,  goes  on  glowing  for  many  seconds  withoii- 
producing  any  very  violent  disturbance  in  the  water.  Hot  molte 
metal  can  be  harmlessly  poured  over  damp  hands,  as  can  th 
volatile  liquid  air  over  dry  hands,  220°  warmer  than  itself. 

The  explanation  is  undoubtedly  that  the  vapour  of  the  volatil 
substance  is  being  distilled  so  fast  from  its  surface  that  it  blo\^ 
it  out  of  contact  with  the  hot  body  which  is  providing  the  hci- 
necessary  for  this  evaporation — ^partly  by  radiation,  mostly  b 
conduction  through  the  thin  layer  of  vapour. 

192 


§  274]  VAPORIZATION  108 

Carbonic-acid  *  snow  '  can  be  handled  lightly,  in  spite  of  its 
intense  cold  :  compressed  into  *  dry  ice  '  it  is  employed  by  Misa 
Waller  as  the  most  effective  means  of  ringing  tuning-forks,  and 
vibrating  metal  bars  and  plates  of  all  sorts.  The  point  of  a  stick 
pressed  on  the  warmer  metal  emits  a  puff  of  vapour,  which  blows 
them  apart,  the  continuing  pressure  of  the  hand  drives  them 
together  again,  and  thus  vibration  is  set  up,  and  grows  to  remarkable 
intensity. 

There  is  no  actual  contact ;  drops  of  sodium  sulphide  solution 
bounce  off  a  red-hot  half-crown  without  blackening  it  in  the  least. 

The  vapour  escaping  from  beneath,  unequally  in  different  direc- 
tions, drives  the  drop  about,  and  often  sets  a  large  drop  into  very 
pretty  vibration. 

The  drop  has  been  found  to  be  always  below  its  boiling  point — 
in  fact,  a  small  piece  of  ice  thrown  into  a  red-hot  bowl  runs  round 
for  three  or  four  seconds  before  entirely  melting. 

When  the  hot  surface  cools,  the  rush  of  vapour  slackens,  and 
presently  the  drop  sits  down  on  the  plate,  and  there  is  the  sudden 
splutter  one  has  been  expecting.  It  is  surprising,  however,  what  a 
length  of  time  a  spherule  of  water  will  remain  quiet  in  a  clean 
metal  bowl  after  the  gas  has  been  turned  out. 

§  273.  Sublimation.  It  is  not  every  substance  that  fuses. 
Ammonium  salts,  etc.,  volatilize  or  *  sublime  '  without  showing 
any  signs  of  melting ;  they  do  not  pass  through  the  usual  inter- 
mediate liquid  state.  And  substances  that  do,  differ  very  much 
in  the  length  of  it.  The  normal  boiling  point  of  argon  is  —  186°  C, 
and  it  freezes  only  3°  or  4°  lower,  water  has  normally  100°  range 
of  liquidity,  sulphur  330°,  mercury  400°,  iron  1000°,  etc. 

But  we  shall  see  presently  that  increased  pressure  so  increases 
the  difi&culty  of  vaporization  that  the  liquid  range  becomes  much 
longer,  and  under  pressure  camphor  melts  and  boils  in  the  usual 
way,  though  normally  *  its  melting  point  is  above  its  boiling  point.* 

§  274.  The  increase  of  volume  accompanying  vaporization  is 
very  great.  It  is  found  by  measuring  the  density  D  of  the  liquid, 
and  that,  d,  of  its  vapour  at  the  same  temperature.  This  change 
of  density  means  a  D/d-ioid  expansion.  Since  d  increases  fast  as 
the  temperature  of  vaporization  rises,  this  latter  must  be  specified. 
See  Table,  §  270. 

Note.-— <i  is  not  the  chemists'  'vapour  density,'  which  refers  to 
hydrogen  as  standard. 

A  great  deal  of  External  Work  must  therefore  be  done  by  the 
evaporating  liquid  in  lifting  the  atmosphere  to  make  room  for  ita 
vapour.  This  work,  however,  represents  on  the  average  only  one- 
eleventh  of  the  total  energy-value  of  the  latent  heat  of  vaporiza- 
tion, the  remainder  is  spent  in  disentangling  the  molecules  from 
their  mutual  Uquid  bondage.  But  it  shows  that  increiued  pressure 
vnHmise  the  boiling  point,  and  qreatlu. 


194 


HEAT 


[§274 


Methods  of  determination  of  the  density  of  a  vapour  are  detailed 
in  all  the  chemistry  books.  Here  we  are  conp^rned  more  with  the 
pressure  of  the  vapour,  which  depends  on  the  closeness  of  packing 
of  the  molecules  and  their  a  Wage  energy  of  motion,  and  not  on 
their  internal  constitution. 

§  275.  The  Pressure  of  a  Vapour.  The  straightfoi-ward  way  of 
finding  the  vapour  pressure  of  a  substance  is  to  introduce  it  intoj 
the  Torricellian  space  at  the  top  of  the  barometer,  where  the  vapour' 
forms  quickly,  unhindered  by  air,  and  drives  down  the  mercury 
for  a  distance  which  measures  its  pressure,  now  substituted  for  the  J 
dead  weight  per  square  centimetre  of  that  depth  of  mercury. 

The   process   is   demonstrated    as   in   Fig.    81.     Three   or   four] 


~~  -  -^ 

"~"-^ 

1      Qf 

1- 

\ 

UJ 

lii^ 

^^  ^ 

q: 

I 

a: 

Ul 

2 

o 
tr 
< 

I' 

D 







7^' 

(/ 

1 
1 

Li 

< 

q: 

1 

Fig.  81. 

barometer  tubes  stand  side  by  side.  The  first  is  kept  as  standard ; 
under  the  foot  of  the  second  a  few  drops  of  water  are  blown  from 
a  little  glass  '  filler  '  with  a  bent-up  point ;  and  under  the  third 
some  ether.  The  liquids  float  up  the  tubes,  the  water  drives  the- 
mercury  down  only  1  or  2  cm.,  but  the  ether  is  much  more  effective, 
having  evidently  a  much  greater  vapour  pressure  at  ordinary 
temperatures,  and  serves  better  for  demonstration  and  argument. 

The  first  drop  or  two  sent  up  break  into  long  bubbles  of  vapour 
half-way  up,  and  the  mercury,  after  being  thrown  about  violently, 
settles  perhaps  10 — 20  cm.  lower  than  it  was,  indicating  this  much 
pressure  in  the  perfectly  dry  vapour  above  it.     Another  drop  brings  - 


fi 


§277]  VAPORIZATION  196 

it  down   farther,   having   increased   both   the   quantity   and   the 
pressure  of  the  dry  vapour. 

But  continuing  drop  by  drop,  some  liquid  presently  remains 
unvaporized  at  the  top  of  the  mercury,  and  further  supplies  are 
now  quite  ineffective ;  the  vapour  has  evidently  reached  its 
maximum  elastic  pressure,  and  can  drive  the  mercury  no  lower. 

§  276.  For  distinction,  the  former  vapour,  into  which  more 
liquid  could  evaporate  and  increase  its  pressure,  is  spoken  of  aa 
unsaturated.  The  latter,  which  can  take  up  no  more  liquid,  is  a 
Saturated  Vapour;  it  remains  unchanged  in  contact  with  its  liquid, 
all  at  the  same  temperature.  Vapours  in  these  two  conditions 
behave  very  differently. 

The  hasty  evaporation  of  spilt  liquid  air  shows  that  for  the  present 
purpose  Air  may  he  regarded  as  the  unsaturated  vapour  of  this  liquid, 
and  therefore,  for  comparison  with  the  saturated  vapour,  some 
air  can  be  blown  into  a  fourth  barometer  tube,  until  it  brings  the 
mercury  down  to  the  same  level  as  in  the  other  tube. 

Now  incline  these  two  tubes ;  the  mercury  starts  running  along 
both  towards  their  closed  ends,  for  of  course  it  is  its  vertical  height 
that  measures  pressure.  In  the  air  (unsaturated  vapour)  tube, 
however,  its  level  falls,  for  the  compression  of  the  imprisoned  air 
by  the  advancing  mercury  has  raised  its  pressure,  according  to 
Boyle's  law.  But  in  the  saturated  vapour  tube  the  level  falls 
only  a  trifle,  the  liquid  above  the  mercury  increases  in  quantity, 
and  if  the  tube  is  left  to  itself  for  a  minute  or  two,  so  that  the 
heat  of  liquefaction  of  this  may  be  dissipated  by  cooling,  the 
mercury  returns  exactly  to  the  level  it  had  originally  in  the  vertical 
position.  Now  suddenly  lifting  to  the  vertical  again,  the  excess 
of  liquid  immediately  boils  off,  and  (after  a  minute  or  two  for 
warming  after  this  loss  of  latent  heat)  the  mercury  stands  again  at 
the  same  level. 

Evidently  the  saturated  vapour  has  no  characteristic  volume 
of  its  own,  so  long  as  there  is  enough  liquid  present  to  keep  it 
saturated.  Reduce  the  available  space  and  vapour  liquefies, 
simply  takes  the  intrusion  '  lying  down,'  so  to  speak  ;  increase  it, 
and  liquid  evaporates.  As  soon  as  equilibrium  is  reached  either 
way,  there  is  the  original  pressure  quite  unaltered.  At  a  fixed 
temperature  the  Saturated  Vapour  has  a  characteristic  pressure. 

§  277.  Rise  of  temperature  increases  this  pressure  very  rapidly. 
On  the  vapour  tube,  near  the  lower  end,  where  there  is  liquid  to 
evaporate  and  keep  up  the  saturation,  a  touch  of  a  flame  will 
send  the  mercury  down  with  a  rush.  Whereas  heating  the  top  of 
the  tube,  where  there  is  no  liquid  to  evaporate,  and  accordingly 
the  vapour  becomes  locally  expanded  ('  superheated  '),  and  there- 
fore unsaturated,  causes  only  a  very  trifling  motion  of  the  mercurj*, 
no  more  than  in  the  air  tube  after  a  similar  treatment. 

The  rise  of  Pressure  of  Saturated  Vapour  with  rise  of  Tempera- 


196 


HEAT 


[§27: 


ture  is  shown  in  Fig.  82,  wherein  the  portion  of  the  curve  froi 
—  10°  to   +  100°  can  be  obtained  from  a  barometer  tube,  con- 
taining water  as  the  volatile  substance,  and  jacketed  by  an  outei 
tube,  through  which  a  fluid  at  a  known  temperature  is  circulated" 
The  height  of  a  point  on  the  curve  shows  the  pressure  at  the  corre- 
sponding  temperature.     The   long   curve   is   on   a   vertical   scale 
graduated  in  centimetres  of  mercury.     Its  slope,  i.e.  the  rate  of  rise 
of    pressure    with    temperature,    changes    enormously ;     the    rise 
between  95°  and  100°  is  65  times  as  great  as  between  0°  and  5*^ 
The  lower  part  of  this  curve  has  therefore  been  shown  on  a  20  time 
magnified  vertical  scale. 

Scales  of  inches  of  mercury,  and  of  millibars,  applicable  to  the 
main  curve,  are  appended  on  the  right. 


' 

1 

~ 

7 

\i^     - 

I4«» 

u  . 
o  . 

z  ' 

fifi 

V 

A 

P 

0 

u 

R 

Ito 

CM 

/ 

j 

f)0 

P 

R 

E 

s 

s 

u 

R 

E 

lod 

1 

j 

AS" 

0 

F 

W 

A 

T 

E 

R 

•So 

IN 

c 

:N 

nr 

lei 

•R 

:s 

OF 

r^ 

:r 

;u 

^Y 

1 

/ 

^o 

\t\ 

^ 

It, 

1  1 

AZ 

TR 

ES 

1 

flo 

/ 

/ 

-L 

A.- 

LK 

OS 

pj 

EJ 

l£_ 

i_ 

?f> 

/ 

70 

/ 

^ 

MM 

60 

CM 

/ 

/ 

/ 

a!> 

V 

/ 

.To 

/ 

/ 

/ 

ftp 

// 

4o 

/ 

1 

/ 

/ 

15 

/) 

3<3 

/ 

t 

/ 

/ 

S.a* 

10 

\- 

// 



N 

9J> 

/ 

/ 

R 

~^ 

•• 

/ 

,T 

// 

R' 

to 

^ 

y 

T 

E 

M 

P 

E 

R 

A 

T 

u 

R 

E 

^: 

■}<$. 

^ 

>'c 

1 

0- 

Z 

0- 

3 

o" 

> 

5^ 

-^ 

o" 

6 

o" 

7 

o' 

8 

0° 

9 

o" 

to 

f 

n 

o" 

12 

o 

MM 

._. 

. 



— 

a; 

-"■ 

-2. 

CM, 

C 

E 

N 

T 

1 

G 

R. 

A 

k 

£j 

Fig.  82. 
The  curve  of  Fig.  82  continues  as  follows  : 

Pressuhe  and  Temperature  of  Saturated  Steam. 


Atmos. 

Atmos. 

Atmos. 

2 

120°  C. 

10 

180°  C. 

60  (880)    276° 

3 

133-5 

15  (220 

198 

90              304 

4 

144 

20 

213 

120  (1760)  326 

5 

152 

25 

225 

160               348 

6 

159 

30 

234 

210  (3080)  370 

7 

165 

35 

242 

220  (3230)  374 

8 

170 

40 

250 

Critical  Point 

The  (figures)  are  lbs.  per  sq.  in.  absolute,  1  atmo.  being  14-7. 


§279]  VAPORIZATION  197 

§  278.  But  we  seldom  go  to  the  trouble  of  removing  the  air 
from  the  space  in  which  a  vapour  is  to  be  produced.  (x)mmonly 
we  leave  the  air  in  and  let  the  vapour  mix  with  it,  or  blow  it  out 
as  it  can.  Does  a  mixture  of  vapour  and  air  obey  Dal  ton's  Law, 
that  Each  gas  in  a  mixture  exerts  its  own  *  partial  pressure  *  quite 
unchanged  by  the  presence  of  the  others :  does  the  vapour  attain 
the  same  '  partial '  pressure  as  if  there  were  no  air  present  ? 

This  can  be  tried  by  first  admitting  air  to  the  Torricellian  space, 
80  as  to  depress  the  mercury  permanently,  and  then  finding  if 
the  further  lowering  on  admitting  liquid  is  the  same  as  before. 
Or,  in  another  way  (chemical  hygrometer),  by  using  a  chemical 
to  absorb  all  the  saturated  vapour  which  filled  an  otherwise  vacuous 
space ;  and,  secondly,  all  the  vapour  which  formed  in  the  same 
space  already  occupied  by  air,  and  comparing  the  two  increases 
in  weight.  The  result  is  that,  as  nearly  as  one  can  tell,  a  liquid 
evaporates  to  the  same  ultimate  saturation  pressure  into  the 
presence  of  a  permanent  gas,  as  into  a  vacuum. 

Thus,  when  Ether  has  been  poured  from  its  bottle,  and  its  heavy 
vapour  has  visibly  poured  out  with  it,  and  a  lot  of  air  has  entered 
in  replacement,  evaporation  into  this  unsaturated  air  immediately 
begins,  and  raises  the  total  pressure  until  the  stopper  hops  out. 
Soon  an  equilibrium  is  reached  inside,  with  perhaps  2/5  atmos. 
due  to  ether  vapour  and  3/5  to  air,  and  the  stopper  put  back  shows 
no  further  desire  to  lift :  try  this  little  experiment,  but  don't 
smoke. 

Likewise  the  Barometric  Pressure  is  the  total  of  the  partial 
pressures  of  nitrogen,  oxygen,  COg,  aqueous  vapour,  argon,  etc. 

Ex.  1.  Calculate  the  weight  of  hydrogen  in  100  c.c.  of  electrolytic  sas 
(2H  +  O)  standing  over  water  which  rises  10  cm.  into  the  graduated  tube. 
The  gas  is  saturated  with  moisture,  at  17°  C,  barometer  76-6  cm.  1  c.c.  dry 
hydrogen  at  0°  and  76  cm.  weighs  00000895  gm. 

Of  total  pressure  in  tube,  which  =  75-5  -  (10  -f-  13-6)  =  74-75  cm.  the 
water  vapour  accounts  for  1-45  cm.  (Fig.  82),  leaving  73-3  cm.,  of  which  the 
hydrogen  causes  2/3rd8  =  48-9  cm.  pressure. 

Hence  Y,  =  100  x  *^\-  x  273^7  =  60-5  c.c. 
.-.  weight  =  60-5  x  00000895  =  000541  grm. 

But  Mixed  Vapours  of  Mutually  Soluble  Substances  obey  no  such 
rule,  e.g.  the  saturation  vapour  pressure  of  dilute  alcohol  is  far 
from  being  the  sum  of  those  of  alcohol  and  water. 

§  279.  Evaporation  and  boiling.  Observe  what  happens  as  wat45r 
is  warmed.  Bubbles  soon  begin  to  make  their  appearance  :  each 
consists  mainly  of  dissolved  air,  but  part  of  its  elastic  pressure  is 
due  to  the  vapour  which  has  evajwrated  into  it. 

As  the  temperature  rises,  the  '  partial  pressure  *  of  vapour  m 
the  bubble  (proportional  to  the  percentage  by  volume  of  vapour 
in  it ;    '  volume  '  and  '  pressure  '  are  not  opposed  to  each  other 


198  HEAT  [§279 

now)  increases  :  always  the  sum  of  the  two  has  to  equal  the  atmo- 
spheric pressure  above  the  hquid  (plus  a  trifle  of  hydrostatic  pres- 
sure due  to  depth  of  liquid  below  which  bubble  happens  to  form) 
or  else  the  bubble  could  not  hold  out  against  it.  The  Saturation 
Curve,  Fig.  82,  shows  what  the  relative  proportions  of  air  and 
vapour  are  :  taking  any  point  on  it,  its  height  above  the  base-line  is 
the  pressure  of  vapour  at  that  temperature,  and  then  the  rest  of 
the  height  up  to  the  Atmosphere  '  ceiling  '  is  the  pressure  of  air 
still  necessary  to  stiffen  out  the  bubble  ;  e.g.  at  50°  about  9  is  vapour 
and  67  air,  at  90°  52-5  vapour  and  23-5  air,  at  99°  73  vapour  and 
3  air.  Presently,  therefore,  it  takes  only  a  little  air  to  form  a  large 
bubble  at  full  atmospheric  pressure.  The  small  amount  of  air 
usually  dissolved  in  the  water  therefore  produces  an  increasing 
multitude  of  bubbles  as  the  temperature  rises,  and  these,  as  they 
gain  in  size  and  buoyancy,  float  up  to  the  surface.  All  taken 
together,  however,  they  have  not  carried  off  much  vapour. 


0006066889 


O'      10'     XC    30*    LO'    50'    60'     70'    80°    90'        95°      97°  99°,^  100 

Fig.  83. 

In  Fig.  83  the  proportions  of  air  and  vapour  have  been  taken 
from  the  Saturation  Curve,  as  described  above,  and  drawn  as  areas. 
The  lower  circle  represents  the  air  in  a  bubble ;  the  upper  one 
attached  to  it  represents,  on  the  same  area  scale,  the  proportion 
by  volume  of  aqueous  vapour  associated  with  it  at  that  tempera- 
ture, the  two  taken  together  constituting  the  actual  bubble. 

But  when  the  temperature  has  risen  so  that  the  vapour  pressure 
exceeds  in  the  least  the  hydrostatic  pressure  in  the  liquid  (made 
up  of  superjacent  liquid  +  atmosphere,  §  103),  then  bubbles /orme^i 
of  vapour  only  have  sufficient  strength  to  withstand  this  pressure, 
and  the  very  smallest  trace  of  air  will  suffice  to  start  a  bubble 
which  can  grow  to  any  extent :   the  last  arc  in  Fig.  83  is  straight. 

*  Singing.'     Bubbles   therefore   start   in   large   numbers   at   the 
hottest  parts,  but  rising  into  cooler  liquid,  collapse.     For  the  cooling 
of  the  vapour  lowers  its  pressure,  and  the  hydrostatic  pressure 
crushes  in  the  walls  of  the  bubble  with  an  audible  snap,  in  the' 
absence  of  any  residual  '  air  cushion  '  to  soften  the  shock. 

It  is  the  noise  of  numbers  of  such  collapses  in  its  resonant  interior 
that  makes  the  kettle  sing  :  the  bottom  layer  of  water  is  boiling 
hot,  though  the  main  bulk  is  far  from  it.  Near  the  boil  the  song 
is  softer,  the  bubbles  are  not  so  abruptly  condensed  by  the  warmer 
water.  You  see  them  rising,  in  tapering  spires,  almost  up  to  the 
surface. 


§280]  VAPORIZATION  I99 

The  bubbles  greatly  aid  the  convection  of  heat,  setting  up  a 
rapid  stream  by  their  buoyancy,  and  giving  up  heat  as  they 
liquefy. 

Boiling.  When  the  whole  bulk  of  liquid  has  thus  been  warmed 
to  this  temperature,  at  which  the  vapour  pressure  just  exceeds  the 
hydrostatic  pressure,  evaporation  continuously  goes  on  into  the 
bubbles,  they  grow  rapidly,  rise,  and  burst  in  abundance;  the 
liquid  boils.  Vaporization  suddenly  becomes  much  more  rapid, 
because  of  the  large  increase  of  available  evaporating  surface  afforded 
])y  the  growing  bubbles. 

Now,  any  attempt  to  heat  the  liquid  hotter  means  a  greatly 
increased  vapour  pressure,  much  faster  evaporation  at  any  surface 
that  presents  itself,  i.e.  faster  output  of  larger  bubbles — furious 
boiling — taking  away  latent  heat  so  rapidly  that  the  liquid  can 
never  rise  much  above  the  temperature  at  which  boiling  began. 

Hence  a  liquid  begins  to  boil  visibly  when  it  reaches  the  temperature 
at  which  its  saturated  vapour  pressure  is  equal  to  that  of  the  atmosphere 
on  its  surface,  and  thereafter  it  scarcely  rises  in  temperature. 

§  280.  This  statement  requires  a  little  qualification,  for  some- 
times a  liquid  can  be  *  overheated.'  It  was  suggested  above  that  a 
minute  amount  of  air  was  still  acting  as  nucleus  :  certainly  Nuclei 
of  bubble  formation  of  some  sort  have  to  be  present  for  steady 
boiling. 

Everyone  has  noticed  how  the  bubbles  in  a  beaker  of  boiling 
water  stream  up  from  invisible  specks  on  the  glass,  or  afloat. 
Very  similarly,  while  the  half-emptied  bottle  of  '  bubbly  '  is  gassing 
quietly  from  a  few  nuclear  points,  the  glass — up  till  then  exposed 
to  air,  dust,  cloth-fibres,  etc. — is  soon  quite  coated  with  hundreds 
of  bubbles,  and  effervesces  briskly ;   as  does  a  grape  dropped  in. 

The  long-continued  boiling  of  water  in  a  glass  vessel  gradually 
changes  from  a  free  continuous  ebulUtion  to  a  spasmodic  boiling 
with  bumping — and  all  the  sooner  if  there  is  present  a  trace  of 
caustic  alkali,  a  substance  which  assists  the  water  to  dissolve 
adherent  dirt  (and  glass  itself).  In  perfectly  quiet  intervals  a  ther- 
mometer in  the  liquid  will  rise  5°  or  10°  above  the  normal  boiling 
point,  to  fall  back  to  it  when  sullen  explosions  of  vapour  threaten 
to  burst  the  vessel.  Coke,  porous  potsherds,  etc.,  thrown  into 
the  bumping  liquid  (and  powdered  sugar  thrown  into  aerated 
waters)  originate  abundance  of  frothy  bubbles,  and  steady  boiling 
ensues  for  a  long  time.  All  are  things  on  which  air  persistently 
clings.  As  in  under-cooling,  this  over-heating  is  most  noticeable 
in  drops  of  liquid  entirely  surrounded  by  another  liquid,  e.g.  air- 
free  water  can  be  heated  in  oil  to  180°  C.  without  vaporizing. 

In  explanation  of  this,  it  will  be  shown  in  Chapter  XXIII  that 
Surface  Tension  in  the  bubble  walls  causes  an  added  pressure 
inside  it,  which  is  greater  the  sharper  the  curvature,  being  some- 
thing like  1  atmo.  for  a  sphere  000002  cm.  diam.,  2  for  OOOOOI, 
and  so  on.     A  very  minute  spherical  bubble  cannot  start  and  grow 


200 


HEAT 


[§280 


against  this  overwhelming  pressure.  But  if  there  is  a  microscopic 
crack,  say,  in  the  surface  of  the  glass,  and  air  has  got  in  and  sticks 
there  tenaciously,  as  it  will,  then  the  comparatively  large  and 
flat  end  of  this  air  wedge  will  form  the  starting-place  of  bubble 
after  bubble,  never  of  excessively  small  radius,  and  therefore  never 
crushed  by  the  surface  tension.  When  this  air  has  been  dislodged, 
by  gradual  solution  during  long  contact  as  in  the  soda-water 
bottle,  by  long  boiling,  or  by  pumping  down  the  pressure  above 
the  hot  liquid  for  a  short  time,  then  comes  about  the  scarcity  of 
possible  jumping-off  places  which  gives  time  for  over-heating, 
and  over-hasty  evaporation  into  any  bubble  that  does  chance  to 
form. 

Another  starting  place  is  the  large  '  vacuum '  bubble  which 
follows  the  contact  point  of  a  rolling  marble  or  glass  bead,  on  the 
principle  of  Fig.  109 ;  and  these  are  often  the  most  persistent,  and 
least  contaminating,  incentives  to  steady  ebullition. 

§  281.  The  visible  boiling  of  a 
liquid,  then,  is  a  useful  indication 
that  its,  saturated  vapour  pressure 
has  become  equal  to  that  of  the 
atmosphere  of  vapour,  or  air  and 
vapour,  above  it. 

This  can  be  experimentally 
shown  by  steam- jacketing  the  baro- 
meter tube  in  Fig.  81  which  contains 
water ;  when  steam  is  blowing 
freely  through  the  jacket,  the 
mercury  will  be  driven  down  just 
level  with  that  outside. 

Hence,  the  Temperature-Pressure 
of  Saturated  Vapour  Curve  may  also 
be  described  as  a  Boiling  Point- 
Pressure  of  superincumbent '  atmos- 
phere '  Curve,  and  we  are  relieved  of 
the  necessity  of  starting  in  a 
vacuum.  Accordingly,  the  curve  of 
Fig.  82  has  been  continued  by  ex- 
periments in  an  apparatus  of  which  Fig.  84  sufficiently  represents 
a  laboratory  specimen. 

Air  is  pumped  into  or  out  of  the  reservoir  E-  to  a  pressure  measured 
by  the  mercury  gauge  G,  and  the  liquid  in  the  flask  (containing  a 
potsherd  or  a  marble)  boils  steadily  at  the  temperature  corresponding 
to  this  pressure.  The  reflux  condenser  C  returns  the  boiled-away 
liquid,  and  keeps  R  and  G  comparatively  free  of  vapour.  The 
thermometer  is  put  in  the  Vapour,  as  in  Fig.  64  (which  is  a  particular 
case  of  this),  to  avoid  trouble  from  bumping  or  dissolved  impurities, 
and  the  flame  gases  must  not  play  on  the  steam  space. 

The  apparatus  works  best  by  starting  perhaps  1/3  above  the  atmo- 


FiG.  84. 


§  282]  VAPORIZATION  201 

<|)here  (the  corks  being  tied  in)  and  stepping  down  to  50**  or  less, 
w  hen,  of  course,  you  can  lay  your  hand  on  the  boiling  flask. 

For  high  temperatures  and  pressures  the  whole  apparatus  is 
])uilt  of  metal,  on  an  engineering  scale. 

This  boiling-off  at  low  pressures  is  frequently  utilized  by  the 
■  organic  '  chemist  to  concentrate  solutions  which  would  decompose 
at  the  normal  boiling  temperature  ;  notably  in  the  '  vacuum-pans  ' 
of  sugar  refineries,  which  produce  a  pure  crystallizable  concentrate 
instead  of  a  toffee.  They  are  worked  in  series,  the  steam  from  the 
hotter  stronger  solution  assisting  to  boil  the  weaker  at  a  lower 
temperature,  §  376. 

In  Yaryan  multiple-effect  Evaporators,  employed  for  distilling 
drinking-water  in  arid  places,  this  is  carried  to  the  limit.  High- 
pressure  boiler  steam  liquefies  under  pressure,  well  above  100°,  in 
the  pipes  of  a  first  evaporative  condenser.  Its  latent  heat  has 
boiled  off  water  from  the  outside  of  these  pipes  to  form  steam  of 
somewhat  lower  pressure.  This  passes  to  a  second  similar  '  con- 
denser-evaporator,' and  so  on,  stepping  down  in  pressure  and 
temperature,  until  in  the  sixth,  vapour  is  liquefying  at  hardly  more 
than  the  temperature  of  the  sea-water  circulated  outside  it.  Hot 
water  from  the  earlier  condensers  is  sent  through  pipes  in  the 
latter  to  help  evaporate  more  water.  Whereas  1  lb.  of  coal  evapor- 
ates only  a  gallon  of  water  in  a  boiler,  nearly  five  are  distilled  per  lb. 
in  this  apparatus. 

Per  contra,  every  cuisine  in  France  has  its  big  stock- pot  or  auto- 
clave, with  clamped-down  lid  and  heavy  plug  safety-valve,  wherein 
bones  and  gristle  and  other  culinary  horrors  are  made  to  yield  up 
their  gelatinous  goodness  to  high-boiling  water  under  pressure  : 
spoilt  grain,  cellulose,  sawdust,  etc.,  simply  hydrolyzed  for  an 
hour  by  weak  hydrochloric  acid,  under  3  or  4  atmos.  steam  pressure, 
are  the  present-day  source  of  Glucose  ;  and,  by  far  most  important 
to  us,  the  high-pressure  Steam  Sterilizer  is  in  constant  use  to 
destroy,  in  dressings  and  wherever  they  might  find  a  lodging,  the 
most  resistant  infective  germs,  and  their  spores  which  can  withstand 
ordinary  boiling  water. 

§  282.  The  general  shape  of  the  Saturated  Vapour  Pressure- 
Temperature  Curve  is  the  same  for  all  substances.  Indeed,  when 
A  and  B  are  '  chemically  similar  ' 

Boiling  point  "Absolute  of  A  x     x      u  x         au  ^ 

n;;iTT~-L;^.  oAu^^i„x^^^T>  =  constant,  whatever  the  pressure. 


Boiling  point  "Absolute  of  B 

I  That  is,  if  the  curve  for  Water  were  drawn  on  a  sheet  of  India- 
'  rubber,  Fig.  85,  fastened  along  the  Absolute  Zero  edge  ZZZ'  and 
stretched  horizontally,  the  curve  for  the  more  volatile,  lower-boiling- 
point  substance  A  would  be  obtained  by  letting  the  sheet  rela.x 
until  W  reached  A  ;  or  the  curve  for  the  less  volatile  M  by  stretching 
the  elastic  sheet  until  W  reached  M. 


HEAT 


[§282 


Even  when  one  compares,  as  in  the  diagram,  such  widely  different 
substances  as  oxygen,  the  normal  boiling  point  {i.e.  boiling  point 
at  76  cm.  barometric  pressure)  of  which  is  91°  A.,  Alcohol  351°  A., 


tT&I 


2^-0 


1 


Zoo' 


Fig.  85. 

Water  373°  A.,  Mercury  632°  A.,  and  Sulphur  718°  A.,  this  rule  stUl 
holds  as  a  rough  approximation.  Perhaps  its  failure  with  hydrogen, 
boiling  point  20°  A.,  is  excusable. 

§  283.  All  points  in  the  space  below  and  to  the 
right  of  the  Saturation  Curve,  Fig.  82,  refer  to 
Unsaturated  Vapour.  For  the  point  N  is  reached 
from  V  by  removing  part  of  the  vapour  which  was 
saturating  its  space  at  that  temperature,  as  by 
using  quicklime,  or  mixing  in  dry  air. 

Equally  this  is  Superheated  Vapour,  for  one 
reaches  N  from  H  simply  by  warming  the  vapour, 
without  letting  any  more  liquid  evaporate  into  it ; 
as  one  did  by  heating  the  top  of  the  ether  vapour- 
tube  in  §  275.  HN  is  the  natural  1/273  slope  of 
expanding  Gas  :   all  this  is  Gas  Space. 

Can  one  cross  the  Curve  into  the  region  above 
it,  can  vapour  be  more  than  saturated  ? 

§  284.  Yes,  the  following  experiment  will  show 
that  it  is  possible  to  break  through  into  the  Super- 
saturation  Space  to  the  left  of  it  (concave  side). 

A  flask  containing  a  little  lukewarm  water  is 
connected  by  a  long  flexible  siphon  to  a  further 
supply  in  a  vessel  on  the  table.  Fig.  86.  Lowering 
the  flask  to  the  floor,  water  siphons  in  and  com- 
presses the  air  a  trifle.  It  is  now  well  shaken,  to 
saturate  the  air,  and  suddenly  lifted  high  above 
the  table  ;  water  runs  out,  expanding  and  therefore 
cooling  the  air,  and  hence  condensing  some  of  its 
contained  vapour  into  a  mist,  or  cloud,  of  tiny 
drops.  A  similar  expansive  cooling  accounts  for  the 
mist  that  clings  in  the  neck  of  a  bottle  of  Bass 
when  the  cork  is  drawn. 

By  violent  splashing  you   partially  wash  away 

the  cloud,  but  it  clears  up  completely  as  soon  as 

the  flask  is  lowered  again  to  compress  and  warm  the  air.    Repeating 

the  process  half  a  dozen  times,  the  cloud  is  fainter  each  time,  and 


Fig.  86. 


§286]  VAPORIZATION  203 

ultimately  no  cloud  at  all  can  be  persuaded  to  form,  although  the 
supply  of  vapour  awaiting  condensation  is  as  great  as  ever. 

If  now  the  flask  be  held  at  table  level,  and  opened,  and  a  trace 
of  smoke  admitted,  the  lowering  and  raising  will  result  in  a  regular 
fog.  Evidently  it  was  for  want  of  nuclei  of  condensation  that  the 
vapour  had  remained  supersaturated.  These,  of  which  the  air  of 
the  room  probably  provided  several  thousand  per  cubic  centi- 
metre, had  been  gradually  washed  out  while  loaded  with  water : 
the  smoke  provided  them  in  abundance  :    see  §  313. 

In  the  absence  of  these  nuclei,  and  of  electrified  '  ions,'  water 
vapour  can  be  raised  to  an  eight- fold  supersaturation  before  visible 
precipitation  of  moisture  ensues. 

§  285.  The  cooling  effect  of  evaporation.  The  measurement  of 
the  Latent  Heat  of  Vapour  has  already  been  described  in  §§  225, 
226. 

If  a  liquid  is  induced  to  evaporate  without  supplying  it  with 
heat,  the  vapour  carries  ofif  large  quantities  of  latent  heat,  and 
hence  the  liquid  is  rapidly  cooled.  Water-coolers  of  thick  porous 
earthenware  (Sp. — alcarrazas)  used  in  hot  countries  from  time 
immemorial,  have  now  come  into  use  in  England  as  milk  and 
butter  coolers  :  the  water  percolates  and  evaporates  from  the 
surface,  cooling  the  contents  10°  or  more. 

The  chill  of  damp  clothes  is  due  to  removal  of  latent  heat  as  the 
warmth  of  the  body  dries  them  :  every  single  drop  you  can  wring 
out  of  a  bathing  costume,  or  an3rthing  you  have  washed,  means 
60  cals.  less  required  to  dry  it. 

Evaporation  of  this  sort  is  promoted  by  removing  the  vapour 
as  soon  as  formed,  e.g.  by  Wind.  Everyone  knows  the  intensely 
chilling  effect  of  wind  on  a  wet  skin ;  everyone  blows  on  hot  tea ; 
in  hot  damp  weather,  when  the  air  is  nearly  saturated,  everyone 
longs  for  a  breath  of  wind  to  blow  away  the  vapour  and  relieve 
the  insufferable  closeness,  by  once  more  permitting  the  natural 
evaporative  drying  of  the  perspiring  skin. 

With  a  more  volatile  liquid  the  cooling  is  exaggerated.  Hence 
the  use  of  eau-de-Cologne  to  bathe  an  aching  brow,  hence  the 
stinging  cold  of  petrol  spilt  on  the  hands,  hence  the  ease  with 
which  a  tin-box  lid  can  be  frozen  hard  to  a  wet  table  by  pouring 
a  little  ether  in  and  blowing  on  it  through  a  wide  paper  tul)e,  or 
with  the  bellows  ;  the  one  and  only  instance  of  evaporative  cooling 
which  carefully-crammed  candidates  dish  up  to  us,  always  with  a 
wealth  of  meticulous  detail  that  convinces  us  they  have  never 
!  done  it.  Hence,  too,  the  occasional  freezing  of  a  carburettor  where 
petrol  is  evaporating  rapidly,  and  the  hoar-frost  that  forms  on  a 
steel  bottle  of  nitrous  oxide  when  it  is  freshly  opened  for  use,  and 
the  liquid  is  boiling  away  into  anaesthetic  vapour  under  40  atnios. 
pressure. 

Or  the  removal  of  the  vapour  may  be  effected  by  liquefying 
it  elsewhere  in  a  colder  '  Condenser  *  as  in  a  steam  engine.  In 
DanielVs  hygrometer,  Fig.  93,  the  right-hand  bulb  ia  coole<l  by  the 


204 


HEAT 


[§285 


evaporation  of  ether  from  its  musliii-covered  exterior,  the  ether 
vapour  it  contains  is  condensed  (at  a  low  point  on  its  saturation 
curve),  vapour  flows  over  from  the  left-hand  bulb,  where  more 
vapour  forms  to  supply  the  deficiency,  and  the  contained  ether  is 
cooled. 

The  Cryophorus  is  a  toy  of  similar  construction  and  even  greater 
antiquity  ;  it  contains  water,  the  vapour  bulb  is  cooled  by  a  freezing 
mixture  and  the  other  freezes. 

Or  the  vapour  is  removed  by  an  absorbent,  strong  sulphuric 
acid.  Small  hand-worked  Freezing  Machines,  producing  a  pound 
or  two  of  ice,  on  this  principle,  have  long  been  in  use  in  hot  places 
where  ice  is  priceless.  There  is  a  small  vessel  for  the  water  and  a 
large  one  for  the  vitriol,  and  an  air-pump,  for  in  all  the  instruments 
of  this  paragraph  there  must  be  no  air.  Air  makes  their  action 
hopelessly  slow  simply  by  getting  in  the  way  of  the  vapour  molecules. 


!l 


K^ 


% 


^ 

i 

^ 


Fig.  87. 


§  286.  The  critical  state.  If  a  volatile  liquid,  such  as  ether  or 
liquid  sulphur  dioxide,  is  sealed  up  with  its  vapour  only,  in  a  little 

stout  glass  tube  which  it  half  fills,  it 
may  be  heated  high  above  its  normal 
boiling  point,  and  very  remarkable 
changes  presently  take  place.  Fig.  87. 

For  a  long  time  the  liquid  bubbles 
steadily,  and  a  compensating  trickling 
down  is  seen  on  the  walls  of  the  vapour- 
space.  The  liquid  expands  gradually  at 
first,  then  rapidly  to  nearly  double  its 
original  bulk,  and  bubbling  becomes  less 
active.  The  meniscus  separating  liquid 
and  vapour  becomes  fainter  and  flatter,  flickers,  breaks  up  into  a 
mist  of  visible  drops  in  rapid  motion,  this  melts  away  in  wreath- 
ing striae  and — the  tube's  contents  are  perfectly  clear  and  uniform. 
Looking  through  at  the  background,  the  tube  appears  rather  '  more 
refractive  '  than  if  empty,  and  that  is  all.  During  cooling  the  * 
same  events  occur  in  reverse  order. 

The  substance  has  ceased  to  exist  as  a  liquid,  it  spreads  uniformly 
over  the  whole  volume ;  it  has  the  properties  of  a  vapour,  in  that 
its  pressure  at  the  temperature  of  disappearance  does  not  depend 
on  the  relative  volumes  of  liquid  and  vapour  just  beforehand,  and 
the  temperature  itself  is  quite  fixed.  It  has,  however,  the  power 
of  retaining  in  solution  solid  matters,  e.g.  iodine,  which  were  dis- 
solved in  the  liquid,  but  are  either  insoluble  in  the  vapour,  or  of  a 
different  colour  when  mixed  with  it. 

It  is  said  to  be  in  the  Critical  State,  the  final  temperature  of 
disappearance  is  the  Critical  Temperature,  and  the  high  pressure 
that  must  then  be  employed  is  the  Critical  Pressure. 

Since  even  ten  times  the  critical  pressure  has  been  tried  in  a 
vain  attempt  to  obtain  liquid  above  the  critical  temperature,  this 
may  be  called  the  '  Ultimate  Boiling  Point ' ;  beyond  it  the  liquid 
cannot  exist. 


§287] 


VAPORIZATION 


206 


Thus  it  is  possible  to  smooth  away  the  customary  abrupt  transi- 
tion from  liquid  to  gas,  the  two  states  merge  gradually  into  each 
other. 

The  impossibility  of  liquefying  them  by  pressure  and  common 
freezing-mixtures,  which  long  ago  earned  for  half-a-dozen  gasee 
the  title  of  '  permanent  gases,'  means  that  their  critical  temperatures 
are  very  low. 

§  287.  Isothermal  curves.  The  sequence  of  Volume-Pressure 
changes  can  be  plotted  by  a  family  of  curves  as  in  Fig.  88.     Start- 


Fio.  88. 


ing  at  A  as  a  gas,  and  coming  slowly  backwards,  keeping  the  tem- 
perature constant  (hence  the  name  Isothermal),  reduction  of 
volume  is  caused  by  increasing  pressure,  and  the  curve  AB  riaea 
in  a  hyperbola  in  accordance  with  Boyle's  law,  Fig.  51.  Nearinff 
B,  the  gas  is  approaching  the  condition  of  saturated  vapour,  ana 
the  pressure-rise  may  falter. 

At  B  it  is  saturated,  and  further  reduction  of  volume  causes 
liquefaction,  without  any  change  of  pressure,  along  the  horizontal 

At  C  all  the  vapour  has  liquefied,  and  any  attempt  to  squeeze 
a  hquid  into  smaller  bulk  involves  an  enormous  increase  of  pressure, 
CD  is  almost  vertical. 

Then  assuming  the  substance  to  be  one  of  the  usual  type,  con- 
tracting on  soHdification,  heavy  pressure  will  crush  it  entirely  into 
solid  along  DE,  §  264.     EF  is  the  scarcely  compressible  solid. 

For  a  higher  temperature,  the  curve*  is  replaced  by  a  similar 
one  lying  wholly  above  it,  for  everything  is  expanded. 


206 


HEAT 


[§287 


Notice  how  the  liquefying  stage  BC  shortens  at  the  higher  tem- 
perature. This  evidently  accords  with  the  smaller  latent  heats 
at  higher  boiling  points  under  pressure,  §  271.  Ultimately  the  flat 
part  shortens  to  nothing,  at  K,  the  Critical  Point,  on  the  critical 
temperature  isothermal.  Above  this  temperature  the  curves  show 
less  and  less  inflexion,  and  are  soon  nearly  gas  hyperbolas  throughout. 

Areas  in  Fig.  88  represent  quantities  of  energy  PV. 

The  Boyle-Charles  equation  PV  =  RT  represents  a  family  of 
hyperbolas  (curves  like  Fig.  51),  AB,  A'B',  etc.,  as  successive 
values  are  chosen  for  T.  Van  der  Waals  (§  292)  produces  curves 
of  the  dotted  shapes,  A'B'SRC'D'  :  the  part  B'S  corresponds  to 
the  condition  of  supersaturation  of  a  vapour,  §  284,  and  C'R  repre- 
sents the  superheating  of  a  liquid  described  in  §  280  :  the  up-cast 
between  R  and  S  is  essentially  unstable. 


§290. 


Table 


Substance. 

Melt- 
ing 
point 

°A. 

Boiling 

point 

(1  atmos.) 

°A. 

Critical 
point 

°A. 

Critical 
pres- 
siu*e 

atmos. 

Density 

at 

b.pt. 

Vapour 
pres- 
sure at 
15°  C. 
atmos. 

Avail- 
able 

latent 
heat. 

Helium  . 

3 

4-2 

5 

2-3 

0-15 



Hydrogen 

14 

20-5 

35 

15 

0-07 

Above 

— . 

Nitrogen 

52-5 

77-5 

124 

27-5 

0-79 

critical 

— 

Argon     . 

85 

87 

155 

53 

1-21 

point. 

— 

Oxygen 

38 

90-0 

154 

58 

1-13 

— 

Ethylene 
Nitrous  oxide 

104 
170 

169-5 
183    - 

9°C. 
37°  C. 

58 

75 

0-57 

50 



Carbon  dioxid 

e     above 

b.  pt. 

194-5* 

31°  C. 

72 

1-53* 

52 

136 

Ammonia 

197-5 

-38-5°  C. 

131°  C. 

113 

0-67 

7 

300 

Freon  CClaFg 

— 

-29°  C. 

— 

— 

1-4 

5-0 

37 

Methyl  chloric 

e        — 

-25°  C. 





1-0 

4-2 

98 

Sulphur  dioxid 

e        — 

-10°  C. 

155°  C. 

79 

1-45 

2-6 

90 

For  less  volatile  liquids  see  Table  §  270. 
°A  -  273  =  °C. 


I 


*  Solid,  subliming,  '  dry  ice,'  two -thirds  as  heavy  again  as  ice,  subliming 
at  —  79°  C.  without  moisture,  and  absorbing  (including  the  cooling  effect  of 
the  cold  gas)  twice  as  much  latent  heat. 


VAPORIZATION  207 


EXAM   QUESTIONS,   CHAPTER   XVIII 

Many  of  these  little  experiments  you  have  done  :  do,  or  see  done,  as  many 
more  as  you  possibly  can.  The  three  next  chapters  depend  upon,  and  con- 
tinue, this  one  :  together  they  deal  with  Matter  in  its  most  mobile  form,  and 
might  be  called  the  Story  of  its  Struggle  for  Freedom. 

2.  A  small  quantity  of  volatile  liquid  is  sent  up  to  the  top  of  the  barometric 
(M,limm,  which  falls  to  a  lower  reading  More  mercury  is  forced  up  the  tube 
I  )y  raising  the  external  reservoir ;  how  is  the  change  in  level  inside  relato<l  to 
that  outside  (a)  if  the  raising  is  done  slowly ;    (6)  if  done  hastily  ? 

3.  A  wet  open  tube  is  dried  by  warming  its  outside,  while  held  upright. 
What  properties  of  tube,  liquid,  and  air  affect  the  rate  of  drying  ? 

4.  Distinguish  between  a  gas  and  a  vapour.  A  Boyle's  law  tube,  closed 
at  76  cm.  pressure,  contains  air  satm-ated  with  ether ;  if  an  increase  of  36  cm. 
pressure  halves  the  volmne,  calculate  the  vapoiu*  pressure  of  the  ether. 

5.  A  barometer  tube  contains  mixed  air  and  saturated  vapour  above  a 
70-cm.  column  of  mercm-y  (atmospheric  76).  What  is  height  of  mercury 
when  tube  is  depressed  to  halve  voliune  above  it,  pressure  of  saturated  vapour 
being  1-5  cm.  ? 

6.  Distinguish  carefully  between  saturated  and  unsaturated  vapours.  How 
would  you  determine  the  pressure  of  saturated  water  vapour  at  temperatures 
between  40°  C.  and  110°  C.  ?  Indicate  the  general  shape  of  the  p  t  cur\e. 
(X2) 

7.  By  what  distinctive  prox)erties  would  you  recognize  that  a  vapour  was 
'  saturated  '  ? 

Explain  how  it  is  that  equal  quantities  of  warm  and  cold  air,  both  saturated, 
will  produce  a  mist  when  mingled  together. 

Under  what  conditions  is  supersaturation  attainable  ?     (  X  3) 

8.  Describe  how  you  would  find  experimentally  the  relation  between  the 
pressure  and  temperature  of  a  saturated  vapour.  Show  on  a  diagram  the 
general  shape  of  the  curve  expressing  this  relation.  A  point  is  taken  in  the 
space  below  the  curve,  and  vertical  and  horizontal  lines  drawn  from  it  to 
meet  the  curve ;  what  do  this  point  and  these  lines  represent  ?  What  doea  a 
point  above  the  curve  represent  ? 

9.  Show  in  a  diagram  the  changes  of  volume  imdergone  by  water-subetanoe 
between  —  10°  C.  and  110°  C,  and  make  any  necessary  comments. 

10.  The  air  is  exhausted  from  a  tall  vessel  containing  two  saucers  of  water. 
Show  that,  when  in  equilibrium,  the  upper  saucer  is  slightly  the  cooler. 

11.  State  Dalton's  Law  of  the  pressvire  of  mixed  gases  and  vapours. 

60  CO.  of  mixed  gas  from  the  electrolysis  of  water  are  collected  in  a  tube 
inverted  over  water.  Calculate  the  weight  of  the  dry  hydrogen,  at  0°  and 
76  cm.,  given  that  the  water  stands  10  cm.  higher  inside  the  tube  than  outside, 
that  the  teraperatiu^  is  17°  and  the  btiro metric  height  74  cm. 

The  maximum  pressure  of  aqueous  vapour  at  17°  is  14-4  mm.     ( X  4) 

12.  Explain  the  repeated  jumping  of  the  stopper  of  an  ether  bottle  for  some 
time  after  the  bottle  has  been  opened  and  ether  (and  its  vapoiu-)  poured  out. 

13.  How  is  the  boiling  point  of  a  liquid  affected  by  pressure  ?  Give  some 
explanation  of  the  effect,  and  examples  of  its  practical  use. 

14.  What  is  the  spheroidal  state  of  a  liquid  ?  VVTiy  can  solid  CO,  be  held 
in  the  hand,  or  liquid  air  be  pomtjd  over  the  hand,  with  impunity  ? 

15.  Describe  the  vapour-jacket  method  of  maintaining  a  uniform  tem- 
perature, and  point  out  its  advantages  and  disadvantages  as  comp«ued  with 
circulating  liquid. 


208  HEAT 

16.  Why  does  a  liquid  vaporize  much  more  slowly  when  only  1°  below 
its  boiling  point  ? 

A  stout  flask  is  partly  filled  with  ice  and  is  sealed  up ;  what  is  the  pressure 
inside  at  100°  C.  ?     (  X  2) 

17.  A  gram  of  ice  is  dropped  into  a  stout  litre  flask  immersed  in  ice,  the 
flask  is  securely  stoppered  and  put  into  boiling  water;  find  the  pressure  of 
the  gaseous  mixture  inside,  and  its  mass. 

The  density  of  water-vapour  is  five-eighths  that  of  air,  or  nine  times  that 
of  hydrogen. 

18.  An  air-thermometer  bulb  of  200  c.c.  contains  by  accident  a  0-1  c.c. 
drop  of  water;  what  will  be  the  error  at  100°,  and  how  will  the  thermometer 
behave  afterwards  ? 

19.  A  bulb  half  full  of  water  is  sealed  up  and  put  into  an  oven  of  uniform 
temperature,  which  is  steadily  increased.     When  will  the  water  boil  ? 


CHAPTER  XIX 
HEAT  ENGINES  AND  COLD  ENGINES 


§  291.  The  Kinetic  Theory  will  help  us  a  good  deal  to  comprehend 
I  )oth  the  preceding  chapter  and  this.  We  have  used  it  in  Chapter  IX 
for  Boyle's  Law,  and  to  obtain  a  true  basis  for  Temperature  in  §  202  ; 
let  us  take  it  up  again. 

You  know  from  experience  that  pumping  up  your  bicycle  tyre, 
or  a  car  tyre,  makes  the  pump  hot  at  the  deUvery  end.  That  it  also 
makes  you  hot  has  nothing  to  do  with  us  here ;  the  physiologist 
will  give  you  a  new  theory  of  that  next  year.  The  Diesel  engine 
compresses  air  quickly  to  450  lb.  per  sq.  in. ;  crude  oil  sprayed  in 
immediately  catches  fire,  the  engine  has  no  artificial  ignition 
whatever. 

The  gas  in  a  cylinder  consists  of  a  swarm  of  molecules  possessing 
a  certain  average  velocity  :  one  wall,  the  piston,  now  moves  con- 
tinuously inwards,  every  molecule  that  hits  it  is  driven  back  fast«r 
than  it  would  otherwise  be,  this  increased  speed  is  disseminated  by 
the  frequent  colUsions,  and  so  long  as  the  inward  movement  con- 
tinues, the  average  speed  of  the  molecules  goes  on  increasing ;  i.e. 
the  gas  temperature,  which  is  equal  to  the  average  energy  |  mr*, 
goes  on  rising. 

If  the  piston  moves  at  only  J  the  speed,  the  molecules  get  only  J 
of  the  extra  knock,  but  it  goes  on  four  times  as  long,  and  four  times 
as  many  get  it,  so  that  fast  or  slow  makes  no  difference  to  the 
ultimate  heating,  except  if  heat  leaks  out  through  the  cylinder-walls 
meanwhile. 

The  '  adiabatic  '  (not-passing-through)  condition  when  it  does  not, 
is  to  be  dealt  with  in  §§  315,  414  :  the  other  extreme  is  the  *  iso- 
thermal '  (same-temperature)  condition,  when,  by  working  slowly  in 
narrow  tubes,  you  leave  abundant  time  for  cooling  to  the  original 
temperature  (energy  of  agitation),  in  which  case  you  get  Boyle's 
Law,  §  146. 

Conversely,  as  the  piston  moves  outwards,  the  molecules  hitting 
it  bounce  back  more  slowly  than  if  it  were  fixed,  i.e.  the  average 
molecular  speed  dies  down,  the  gas  cools.  The  piston  might  be  the 
cork  of  the  beer  bottle,  §  284 ;  or  the  sinkinc  water  surface  in  the 
flask,  or  there  may  be  no  visible  piston  at  all,  as  in  the  adiabatic 
convective  equiUbrium  of  the  atmosphere,  §  315. 

The  common  road-breaking  drill  plant  exemplifies  both  :  the 
reservoir  into  which  the  engine  is  compressing  air  gets  too  hot  to 
touch,  the  air  cools  considerably  as  it  passes  along  through  the  hose, 
and  the  working  cylinder  of  the  drill,  where  it  expands  and  expends 

209 


210  HEAT  [§  291 

its  energy,  is  cool.     All  the  same,  the  surrounding  air,  which  the| 
exhaust  puffs  into  and  agitates,  is  the  warmer  for  this  friction. 

Suppose  that,  as  in  any  compressor,  we  have  squeezed  molecules 
20  times  tighter  together,  and  torn  them  apart  again,  without  the 
slightest  regard  to  their  mutual  convenience.  Is  this  altogether 
negligible  ?  Have  they  no  idiosyncrasies  ?  What  about  Fig.  52 
and  §  200  ? 

§  292.  Van  der  Waals  modified  Boyle's  law  by  taking  into  account 
cohesion  in  the  gas.  Such  cohesion  is  strong  in  liquids,  for  a  great 
amount  of  heat  energy  has  to  be  supplied  to  tear  apart  their  mole- 
cules (latent  heat  of  vaporization,  §  271),  and  as  highly  compressed 
gas  and  liquid  can  on  occasion  become  indistinguishable  (critical 
point,  §  286),  it  is  not  absurd  to  assume  its  existence  in  gases.  This 
slight  mutual  attraction  holds  the  molecules  back  from  striking  a  full 
blow  on  the  walls,  i.e.  P  observed  is  less  than  the  true  pressure,  which 
may  be  written  (P  +  a/V^),  where  a  is  a  small  constant.  This 
correction,  while  very  small  at  ordinary  pressures,  becomes  rapidly 
larger  as  V  is  diminished  by  compression. 

Further,  the  molecules  are  not  mere  mathematical  points. 
Whether  one  thinks  of  them  as  hard  colliding  spheres,  or  as  centres 
of  strong  repulsion,  the  effect  is  that  each  occupies  a  certain  volume  of 
its  own,  into  which  no  other  can  penetrate.  So  that  the  space  actu- 
ally available  for  molecular  wanderings  is  the  measured  volume  V 
reduced  by  a  small  quantity  b. 

Van  der  Waals  therefore  wrote 


P  +  ^2  )(V  -b)  =  constant  =  RT,  §  203 


t 


and  with  a  proper  choice  of  a  and  b  {e.g.  for  COg  a  =  0-00874, 
b  =  0-0023)  this  equation  more  or  less  fits  the  experimental  curves. 
It  does  not  fit  the  curves  for  any  substance  too  well,  but  it*  has  held 
the  field  since  1870,  in  spite  of  a  new  competitive  theory  every  other 
year,  so  it  will  do  for  us. 

You  can  see  generally  how  mutual  attraction  helps  compression, 
and  accounts  for  the  initial  fall  of  PV  in  Fig.  52  ;   but  presently,  as 
the  squeeze  is  increased,   the  bulk  of  the  molecules  themselves  » 
begins  to  crowd  the  space  unduly,  and  the  curves  all  turn  up. 

Cohesive  attraction  and  abrupt  repulsion  are  not  irreconcilable ; 
think  of  shaking  up  a  bottleful  of  rather  sticky  sweets ;  or  of  a 
crowded  dance-floor. 

It  looks,  then,  as  if  a  compressed  gas  expanding  has  to  devote  a 
trifle  of  its  energy  to  tearing  itself  apart  against  the-shght  mutual 
attraction  a/V^  of  its  molecules  ;  they  part  more  slowly  than  if  this 
did  not  exist.  That  is,  even  if  the  gas  is  not  compelled  to  make  way 
for  itself  in  the  outside  world  by  pushing  back  pistons,  or  the 
atmosphere,  it  ought  to  cool. 

In  the  '  Porous  Plug  Experiment,'  gas  under  pressure  escaped 


§293]  HEAT  ENGINES  AND  COLD  ENGINES  211 

through  a  plug  of  cotton  wool :  thermometers  either  side  showed 
a  small  cooling  in  the  expanded  gas,  0-25°  C.  per  atmo.  fall  of  pressure 
for  air,  and  1-25°  for  COg-  But  hydrogen  warmed  1/20° ;  it  has  no 
down-drop  in  Fig.  52. 

Small  as  is  this  cooling  effect  it  was  the  starting  point  of  the 
modern  manufacture  of  liquid  air. 

§  293.  Kinetic  Theory  of  liquid-vapour  change. 

Condensation.  Compressing  a  gas  or  unsaturated  vapour  packs 
the  molecules  closer,  but  their  speed  is  too  great,  and  their  stay  in 
one  another's  proximity  too  short  for  mutual  attraction  to  overcome 
the  effects  of  '  collisions.'  But  at  a  lower  temperature  [speed],  or 
a  greater  pressure  [closeness  together],  this  may  happen,  and  the 
molecules  quickly  associate  in  twos  and  threes,  and  companies,  and 
drops  of  liquid,  as  soon  as  a  sharp  limit  has  been  over-stepped,  i.e. 
saturated  vapour — condenses  freely — as  scon  as  a  definite  pressure  is 
exceeded,  unless  above  the  limiting  critical  temperature.  And  that 
means  that  this  mutual  attraction  has  grown  to  be  no  small  thing,  for 
it  takes  200  ft. -lb.  of  work  to  pull  to  pieces  one  drop  of  water. 

Molecules  travelling  in  streams  side  by  side,  as  they  must  above  a 
small  flat  surface,  are  close  together  for  a  longer  time  than  those 
flying  in  all  directions  past  a  point ;  hence  one  would  expect  con- 
densation to  begin  on  nuclei,  of  comparatively  extensive  surface, 
§  280.  As  a  matter  of  fact,  however,  these  nuclei  have  to  be 
'  hygroscopic,'  i.e.  centres  of  definitely  increased  molecular  attrac- 
tion ;  how  we  don't  yet  know. 

Now  to  deal  with  the  reverse  process.  Vaporization.  In  a  vast  crowd 
(of  molecules)  possessing  a  definite  average  speed,  individuals  may 
at  any  moment  have  all  sorts  of  speeds  at  random — the  theory  of 
probability  suggests  that  of  1000  with  average  speed  S  there  will  be 
95  with  speeds  below  J  S  ;  167,  J— |  S  ;  417,  J— IJ  S  ;  153, 1  J— li  S ; 
and  168  above  this.  And  if  the  average  speed  is  reduced  by  removing 
the  momentarily  faster  individuals,  the  speeds  of  the  remainder  will 
re-distribute  themselves  '  by  colfision  '  in  the  same  proportions. 

Above  the  surface  of  a  glass  of  effervescent  liquid  may  be  seen  an 
active  cloud,  half  an  inch  or  more  thick,  of  droplets  flung  up  from 
the  bursting  bubbles  and  falling  back  under  the  pull  of  gravity. 
The  cloud  has  a  fairly  definite  flat  top,  i.e.  an  average  height  of  jump 
is  fairly  closely  kept  to  (as  above).  Kinetically,  the  surface  of  a 
liquid  more  or  less  resembles  the  top  of  this  cloud.  In  the  body  of 
the  liquid  the  mutual  attraction  acts  in  all  directions  on  a  molecule  ; 
near  the  edge  it  of  course  pulls  inwards  only.  The  average  mole- 
cule reaches  a  definite  range  before  being  pulled  back,  and  the  surface 
of  the  liquid  is  the  '  envelope  '  of  their  paths.  But  some  exception- 
ally fast  molecules  so  far  exceed  this  average  range  as  to  fly  clear  of 
the  restraining  attraction,  and  become  free  molecules  of  vapour. 

Since  it  is  the  faster  molecules  that  escape,  the  average  speed  of 
those  left  behind  in  the  liquid  is  dimmished,  and  the  energ>'  of 
travel  ^mv^  of  molecules  being  the  measure  of  temperature,  the 


212  HEAT  [§  293 

liquid  has  cooled.     The  escaping  molecules  have  taken  latent  heat  with\ 
Ihem  and  left  the  liquid  colder,  cf .  §  285. 

In  the  liquid  left  to  itself  there  will  always  be  some  molecules] 
chancing  to  approach  the  surface  exceptionally  fast,  and  escaping, 
but  the  general  falling-ofif  of  speed  diminishes  the  number  that 
come  into  possession  of  the  requisite  velocity.  Thus  Evaporation 
always  goes  on,  the  liquid  always  getting  colder,  but  slower  and 
slower  as  the  temperature  falls. 

Heat  continuously  supplied  from  without  goes  to  increase  speeds 
all  round.  If  the  average  speed  is  maintained,  so  also  is  the  number 
of  molecules  travelling  faster  and  escaping,  i.e.  Evaporation  goes  on 
at  a  constant  rate. 

As  the  temperature  rises,  the  increase  in  average  activity  of  the 
liquid  molecules  probably  makes  their  mutual  attraction  less 
e£Eective,  it  relaxes  their  liquid  bondage  [certainly,  one  of  its  indica- 
tions, the  surface  tension,  diminishes],  and  permits  a  larger  propor- 
tion of  the  more  rapid  molecules  to  escape.  Therefore  the  density 
and  crowd-pressure  of  the  vapour  increase  faster  than  in  mere 
proportion  to  the  molecular  energy  (absolute  temperature),  i.e. 
faster  than  that  of  a  gas  or  unsaturated  vapour.  Fig.  82. 

What  of  the  vapour-swarm  of  escaped  molecules  ?  Molecules 
travelling  near  the  liquid  surface  and  coming  within  range  of  the 
attractive  forces  will  be  constantly  falling  in  and  replacing  those  that 
fly  out.  Thus  at  any  particular  temperature  a  state  of  '  Statistical 
Equilibrium  '  is  reached,  when  as  many  molecules  are  dropping 
back  into  the  liquid  as  are  escaping — the  Saturated  Vapour  swarm- 
density,  and  therefore  Pressure,  is  constant ;  it  does  not  matter  what 
volume  it  spreads  through. 

Note  that  air  molecules  present  can  take  no  part  in  the  inter- 
change, therefore  the  saturation  pressure  of  the  vapour  is  reached 
ultimately  quite  independently  of  any  other  gas  pressure  present. 
But  the  neutral  gas  molecules,  of  course,  get  in  the  way  of  the 
vapour  molecules  ;  the  rate  of  evaporation  into  air  is  much  slower 
than  into  vacuum.  - 

§  294.  Heat  Engines.  In  the  working  cyHnder  of  a  Heat  Engim 
a  gas  or  gaseous  mixture  at  a  high  temperature  T°  A  expands  an( 
pushes  away  pistons  or  turbine-blades,  and  therefore  cools,  for  the 
molecules  are  bounced  back  ever  slower  from  the  retreating  walls, 
until  in  the  roomiest  apartment  that  can  be  afforded  it — the  end  of 
the  stroke  in  a  car-engine  or  a  locomotive,  or  the  low-pressure 
cylinder  in  a  multiple- expansion  marine  engine,  or  the  end  of  the 
low-pressure  turbine — ^it  reaches  a  low  temperature  t°  A.,  at  which  it 
is  '  exhausted  ' — into  the  air  from  a  car  or  locomotive,  or  to  the  cold 
wet  condenser  in  stationary  or  marine  steam  plant ;  for  all  of  it  must 
be  got  rid  of  somewhither. 

If  the  engine  were  a  perfect  one,  the  whole  of  the  kinetic  energy 
lost  by  the  slowing  molecules  would  be  transferred  to  the  retreating 
walls  on  which  they  beat,  and  would  thus  become  available  outside 
the  engine. 


§  294]  HEAT   ENGINES   AND   COLD   ENGINES  213 

Recollecting  from  §  76  that  the  Efficiency  of  a  machine  is 
Energy  obtained  from  machine  -f-  Energy  put  into  machine, 

Irt  us  calculate  the  Efficiency  of  a  Perfect  Heat  Engine,  taking  in  all 
1 1  s  heat  at  T°  A.  and  exhausting  at  i°  A. 
The  Absolute  Temperature  of  any  gas  measures  its  Energy,  §  202. 

.*.  Energy  put  into  engine  =  T 

,,        exhausted  from  engine  =  t 
„        available  for  use  =  T  —  < 

T  —  t 

.*.  Maximum  theoretical  Efficiency  = — =— 

It  has  taken  engineers  a  hundred  years  of  struggle  with  many 
difficulties  to  reaUze  that  what  essentially  matters  in  a  heat  engine 
is  not  the  nature  or  pressures  of  the  gases  employed,  but  simply  their 
temperature,  yet  you  see  that  it  follows  straightaway  as  soon  as  we 
have  a  true  scientific  definition  of  temperature.  Still,  don't  be  in  a 
hurry  to  decry  your  brain  if  some  of  the  ideas  in  this  book  take  you 
more  than  a  hundred  seconds  to  pick  up. 

Plainly,  no  engine  could  extract  the  whole  of  the  energy  from  a 
quantity  of  heat  unless  it  could  exhaust  at  the  absolute  zero  :  we 
must  be  content  with  a  fraction.  But  in  view  of  the  vast  demands 
being  made  for  mechanical  power  in  the  world,  and  of  the  really  very 
limited  supplies  of  fuel — coal  and  oil  being  used  up  in  our  spend- 
thrift generation  at  least  a  thousand  times  as  fast  as  the  sun  and  the 
plants  laid  them  down,  even  as  the  world's  phosphate  and  potash 
are  being  squandered  in  agriculture  (and  none  of  these  things  is 
being  or  can  be  replaced) — ^we  want  that  fraction  to  be  a  big  one. 
For  all  the  waterfall  power  in  use,  or  in  sight,  in  the  world,  is  no  great 
fraction  of  its  daily  demand,  and  no  other  great  source  promises, 
unless  it  be  the  unlocking  of  the  energy  of  the  atom ;  and  that,  as 
yet,  is  very  like  extracting  the  energy  of  a  spilt  box  of  matches  by 
bombarding  the  field  containing  them  with  the  guns  of  the  Fleet. 

The  engines  of  this  ship  have  been  introduced  already  in  §  110, 
Fig.  33  :  what  was  described  there  is  a  good  commercial  triple- 
expansion  engine,  taking  steam  at  240  lb.  pressure  and  100°  F. 
superheat  (total  535°  A.),  and  expanding  it  to  sixteen  times  its  boiler 
volume.  But  if  you  look  at  the  combined  indicator  diagram,  you 
see  that  the  long  flat  foot  is  not  touching  ground  :  even  at  that 
attenuation  the  steam  still  has  6  Ib./sq.  in.  of  kick  left  in  it  (350°  A.), 
but  this  is  thrown  away  because  there  is  no  more  room  for  it  to 
kick  in. 

So  the  great  exhaust  pipe  of  this  engine  is  led  into  an  *  exhaust 
turbine,'  and  there  the  steam  blows  through  wheel  after  wheel  of 
windmill  blades  before  it  liquefies  as  a  warm  breath  of  enormous 
volume  in  the  0-5  Ib./sq.  in.  vacuum  of  the  condenser.  If  this  wen* 
put  on  the  indicator  diagram,  it  would  appear  as  a  sole  underlying 
the  foot,  thin  indeed,  but  three  or  four  times  as  long  as  the  foot,  so 
that  its  area  is  even  greater  than  either  of  the  other  three. 


214  HEAT  [§  294 

Actually  the  turbine  drives  a  couple  of  dynamos,  the  one  usually 
absorbing  100  h.p.  or  so,  for  cargo-fans,  and  lighting ;    the  other 
machine  supplying  1750  amperes  at  575  volts  (=  1350  h.p.)  to  th( 
23-ton  rotor,  mounted  on  the  propeller  shaft,  of  a  great  drivu 
motor. 

Incidentally,  this  combination  proves  to  be  a  remarkably  effective 
check  on  '  engine-racing  '  in  a  sea  ;  but  the  point  is,  that  this  power 
costs  nothing  ;  and  the  result  is,  that  this  gleaming  white  cargo-ship, 
the  trimmest  of  her  type  sailing  from  the  Port  of  London — th( 
greatest  commercial  port  in  the  world,  one  you  might  remember  t( 
take  a  look  at  while  you  are  a  Student  in  London — can  show  hei 
heels  to  any  ship  in  her  trade,  and  do  it  on  a  dozen  tons  of  oil  a  day 
less  than  they. 

Here,  as  a  further  matter  of  possible  interest,  are  the  theoretical  i 
maximum  efficiencies  of  various  types  of  heat  engine  in  use  at  the 
present  day  : 

(1)  Portable  engine,  100/lb.  sq.  in.  pressure  =  170°  C,  puffing  into  the  air  =» 
100°  C. 

170  +  273  -  (100  +  273)  _  ,^^o, 
^-  170  +  273  -15  8/o. 

(2)  Railway  locomotive,  225  lb.  and  superheat  of  200°  F.  in  dry  fire  tubes 

_  315  +  273  -  (100  +  273)  _ 
^-  315  +  273 --365/0.  ^ 

(3)  Marine  engine,  quadruple  expansion,  steam  as  (2),  but  exhausting  into 
a  condenser  vacuiun  of  27|  in.  of  mercury  —  42°  C. 

(4)  Steam  turbine,  [ss.  *  Empress  of  Britain  ']  350  lb.  steam  superheated  to  i 
710°  F.,  exhausting  into  a  scrupulously  maintained  29-in.  vacuum  =  27°  C. 

„        377  +  273  -  (27  +  273) 


377  +  273 


54%. 


(5)  Mercury  turbine  [10,000  kwt.  set  at  Hartford,  Conn.]  80  lb.,  superheated 
to  880°  F.,  condensed  at  445°  F.  in  a  boiler  raising  steam  at  280  lb.,  super-  * 
heated  and  used  in  turbines 

472  +  273  -  (27  +  273)        „„„.  | 

^  ^ 472  +  273 =  ^^%- 

This  is  an  effort  to  evade  the  difficulty  of  finding  metal  to  withstand  high 
temperatures  and  high  pressure. 

(6)  Petrol  engine,   fiame  temperature   2150°   C,   exhaust    1250°   C.   (1934 
figures) 

E  =  900/2423  =  37% 

(7)  Diesel  oil  engines  expand  the  gases  more  completely  and  exhaust  as 
low  as  750°  C,  giving 

E  =  1400/2423  =  57%. 

Unfortunately,  in  practice,  steam  boilers  lose  at  least  15%  of  the 
heat  to  start  with,  and  practically  no  engine  exceeds  2/3rds  of  the 


§296]  HEAT   ENGINES   AND   COLD   ENGINES  215 

theoretical  efficiencies  calculated  above.  This  brings  the  Best 
Effort  of  the  Steam  Engine  to  30%,  and  of  the  Internal  Combustion 
Engine  to  38%.  The  Scott-Still  engine,  which  boils  the  Diesel 
jacket  water  by  the  waste  heat  of  the  exhaust  gases,  and  uses  the 
steam  in  the  lower  end  of  the  cylinder,  condensing  as  usual,  is 
actually  getting  another  4  or  5%  in  a  few  cases. 

§  295.  An  Endowment  in  the  City  provides  for  an  annual  Lecture 
in  which  the  Human  Body  is  compared  to  a  Steam  Engine. 

Anyone  can  see  points  of  resemblance,  but  there  it  ends.  The 
Body  is  not  a  Heat  Engine,  for,  as  it  is  practically  isothermal  through- 
out, its  maximum  possible  efficiency  as  such  would  be  zero. 

Digestion  and  assimilation  is  a  piecemeal  process,  and  while  in 
the  physiological  treatment  of  its  minutice  you  may  find  thermo- 
dynamical  reasoning  fitly  employed,  the  sum  total  is  not  open  to 
blind  statistical  generalization.  Who  should  have  known  better 
than  a  City  Father  the  difference  between  a  Commemoration 
Banquet  and  the  casting  of  everything  oxidizable  into  a  burning 
fiery  furnace  ? — though  really  our  present-day  civilization  seems 
almost  too  dependent  on  this  wasteful  practice. 

But,  Medical  Student  that  you  are,  learn  of  the  Engineer  to 
avoid  Shock. 

He  disowns  the  road-breaking  drill  and  the  motor-bike  '  silencer.' 
He  never  hits  twice  if  once  will  do,  and  not  once  if  he  can  help  it : 
the  steam  hammer  gives  place  to  the  h3'draulic  forging- press,  with 
its  silent  bear- like  squeeze.  He  cuts  the  teeth  of  his  wheels  with 
mathematics  and  a  microscope  ;  he  thinks  in  ten-thousandths  of  an 
inch,  and  pads  the  gap  with  appropriate  oil. 

Rigid  is  a  word  of  no  meaning  to  him — everything  springs ;  he 
counter-springs,  he  balances,  and  fifty  revolutions  become  five 
thousand. 

He  never  touches  the  baby  with  cold  hands,  never  lets  hot  play 
on  cold.  He  eliminates  the  stoker  who  dumps  cold  coal  into  his 
furnaces ;  and  dribbles  it  in  at  their  edge.  No  cold  surfaces  touch 
his  flame,  or  there  is  wasteful  smoke ;  he  pre-heats  the  air  for  his 
fires,  he  warms  his  feed-water  up  to  boiler  temperature,  he  super- 
heats his  steam — all  this  he  gets  from  cooling  his  chimneys — 85% 
boiler  efficiency.  He  nurses  the  near  end  of  his  engine  from  the 
least  exposure,  and  curses  if  the  far  end  is  too  hot  to  sit  on.  Even 
when  he  has  to  take  leave  of  the  steam,  he  contrives  that  it  shall 
meet  the  coldest  water  last — or  the  pipes  leak.  Some  different 
fault  every  time,  but  always  a  needless  Loss  of  Efficiency,  unless — 
you  avoid  Shock. 

§  296.  The  liquefaction  of  Air.  Most  gases  are  unceremoniously 
liquefied  by  a  strong  compression  pump,  aided  by  water-cooling  to  re- 
move the  heat  which  it  produces  ;  but  air,  coal-gas,  and  hydrogen  were 
at  one  time  called  '  permanent  gases  '  because  they  refused  to  liquefy 
at  any  pressure.     This  is  due  to  their  CYitical  Temperature,  §  286 ; 


216 


HEAT 


[§  296 


being  very  low  (see  Table),  it  is  useless  working  above  K  in  Fig.  88. 
All  are  now  liquefied  via  Liquid  Air,  which  is  produced  commercially 
on  a  large  scale. 

In  Fig.  89,  Air  is  drawn  in  through  layers  of  slaked  lime,  which 
abstracts  the  greater  part  of  its  CO2 ;  compressed  to  45  atmos. 
pressure,  =  675  Ib./sq.  in.,  which,  of  course,  makes  it  hot ;  and 
cooled  by  cold  water  to  perhaps  35°  C. 

It  is  saturated  with  moisture,  but  as  it  occupies  only  l/45th  its 
original  volume,  and  as  water  vapour  at  35°  cannot  exceed  40  mm. 
pressure.  Fig.  82,  or  l/20th  atmospheric  pressure,  it  drops  all  but 
about  l/30th  of  its  water  in  a  separator,  so  that  now  only  1/20  X 
1/45  X  5/9,  or  roughly  1/1600,  of  the  mass  is  water.  ; 


lEAT     EXCHANC  E  R 


Fig.  89. 

It  now  enters  a  heat -exchanging  tower,  where  it  traverses 
convolutions  of  copper  tubing  cooled  by  the  nitrogen  escaping  from 
the  liquefier  :  by  the  time  it  reaches  0°,  7/8ths  of  the  remaining  water 
will  have  drained  out ;  and  lower  down,  the  rest,  together  with 
remaining  COg,  NH3,  etc.,  gets  frozen  out  soUd. 

The  pure  dry  air,  much  shrunken  by  pressure  and  cold,  now  enters 
the  single  cylinder  of  a  small  '  expansion  engine,'  of  simple  steam- 
engine  type,  where  it  works  hard,  helping  to  drive  the  big  compressor, 
through  belting.  The  loss  of  so  much  energy  costs  the  air  much  of 
its  latent  heat,  and  it  exhausts  from  the  engine  as  a  very  wet  spray- 
laden  vapour,  at  4  atmos.  pressure,  into  a  tank.  From  this  the 
Liquid  Air  can  be  drawn  off  as  required,  at  about  —  190°  C,  for  use 
as  a  refrigerant,  etc.,  into  vacuum- jacketted  carboys,  which  are 
simply  5-gal.  steel '  vacuum  flasks,'  not  closely  corked. 

It  is  a  mixture  of  21%  oxygen,  b.pt.  90°  A.,  1%  argon,  b.pt.  87°  A., 
and  nitrogen,  b.pt.  77-5°  A.,  practically  four  times  as  volatile  as  the 
oxygen. 


f 


§  290]  HEAT   ENGINES   AND   COLD   ENGINES  217 

Accordingly,  the  tank  forms  a  Still,  the  compressed-air  supply 
ipipes  pass  through  it  just  before  they  reach  the  engine,  and  their 
[incoming  heat  distils  the  liquid  off  up  a  tall  '  rectifying  column,* 
iwhere  the  three  main  constituents  are  separated  by  '  fractionation,' 
[just  like  spirit  and  water  at  a  distillery,  or  many  other  mixtures 
^-ou  will  meet  with  in  organic  chemistry.  The  British  Oxygen 
'Company,  at  their  North  Wembley  compressing  station,  by  merely 
jletting  the  nitrogen  run  to  waste  (through  the  heat  exchanger) 
'obtain  all  the  necessary  chill  to  leave  Hquid  Argon  with  10% 
jnitrogen,  for  the  lamp  manufacturers ;  and  99-5%  Oxygen,  repre- 
senting nearly  1,000,000  cu.  ft.  a  week,  which  is  pumped  out  through 
a  set  of  pipes  in  the  heat  exchanger,  to  be  gasified  at  120  atmos. 
pressure  by  the  incoming  air,  and  filled  into  the  famiUar  steel 
cylinders,  from  the  1  cu.  ft.  issued  with  gas-helmets,  etc.,  to  the 
100  cu.  ft.  (reckoned  at  atmospheric  pressure)  for  oxy-gas  and  oxy- 
acetylene  metal -cutting,  welding,  etc.,  or  any  other  purpose  where 
an  extra  1000°  may  be  useful. 

A  liquefier  runs  for  6  weeks,  and  is  then  allowed  to  thaw  out  to 
get  rid  of  frozen  rubbish,  running  into  its  stride  again  6  hr.  after 
re-starting. 

By  more  elaborate  fractionation,  Nitrogen  99-8%  pure  is  obtain- 
able, for  condensation  with  hydrogen  (obtained  by  freezing  all  other 
gases  out  of  the  '  water-gas  '  from  a  coke,  air,  and  steam  gas-pro- 
ducer furnace)  at  a  ton  pressure,  with  a  nickel  catalyst,  to  form 
ammonia  ;  or  to  make  cyanamide  with  calcium  carbide,  both  for  use 
as  agricultural  fertilizers. 

At  other  stations,  not  where  a  whiff  of  it  from  the  air  might 

ultimately  find  itself  in  compressed  oxygen.  Acetylene  is  prepared 

from  calcium  carbide  and  water,  purified  from  phosphine  by  chromic 

or  ferric  oxidizers,  and  pumped  into  the  upright  purple  cylinders. 

These  are  rammed  tight  with  kapok,  and  half  the  remaining  80%  of 

free  space  is  filled  with  acetone,  which  dissolves  twenty-five  times  its 

volume  of  acetylene,  per  atmosphere,  up  to  15  atmos.,  and  from  this 

solution  it  re-distils  into  the  blowpipe,  into  the  reservoirs  of  gas- 

]  buoys,  etc.     Thus  these  cylinders  contain  150  vols,  of  the  gas, 

.  safely  in  solution  at  moderate  pressure ;    for  compressed  alone,  it 

:  polymerizes  into  benzene,  SCgHg  =  CgHg,  with  more  evolution  of 

j  heat  than  a  steel  cy finder  can  withstand. 

Hydrogen  is  liquefied,  after  cooling  in  liquid  air,  to  an  exceedingly 
!  light  colourless  fiquid,  at  20-5°  A.,  in  a  smaller  apparatus  in  which 
i  '  porous  plug '  cooling  at  a  nozzle  has  to  take  the  place  of  the  Expan- 
sion engine,  though  less  efficiently.     The  whole  apparatus  has  to  bo 
securely  vacuum-jacketted,  or  it  is  promptly  clogged  with  frozen 
air. 

In  a  bath  of  liquid  hydrogen,  evaporating  at  low  pressure  into 
large  exhausting-pumps,  Hefium  liquefies  at  4-2°  A.  By  the  utmost 
effort  of  pumping,  this  has  been  evaporated  at  an  estimated 
—  273-05°  C. ;  and  below  that  we  know  no  way  to  reach  the  cal- 
"ilated  Absolute  Zero  at  —  273-13°  C. 


218  HEAT  [§  297 

§  297.  Refrigeration.  But  these  are  the  fireworks  of  frost,  and 
Refrigeration  comes  closer  home  to  us  on  another  side.  Bar  bread 
and  groceries,  almost  everything  that  you,  as  a  London  medical 
student,  subsist  on,  has  come  to  you  through  cold  store  ;  even  beer. 

Our  mild  island  climate  exempts  us  from  setting  up  the  domestic 
refrigerator  as  a  household  god,  to  be  worshipped  with  copious 
libations  of  ice-water,  but  our  insular  position  has  led  to  the  develop- 
ment of  a  world-wide  trade  in  chilled  and  frozen  food-stufifs. 

.In  1860  an  experimental  cargo  of  meat  packed  in  artificial  ice  waai 
shipped  from  Australia,  but  had  to  be  thrown  overboard  when  the 
ice  failed  to  last  out  the  tropics,  and  only  a  small  import  of  meal; 
in  large  ice-tanks  grew  up  from  N.  America.  The  s.s.  '  Strathleven  ' 
was  the  first  ship  fitted  with  a  refrigerating  machine,  and  she  brought' 
34  tons  of  frozen  meat  from  Australia  in  1880,  whereupon  the  barque 
'  Dunedin,'  1248  tons,  was  fitted  out  with  similar  machinery,  and  in 
May  1882,  after  a  passage  of  100  days,  sold  5000  carcases  of  frozen 
New  Zealand  mutton  in  London  at  sixpence  a  pound  ;  and  from  this 
beginning  has  grown  an  import  trade  of  which  the  official  figures  for 
1932  were  : 

British  Imports  of  Refrigerated  Foodstuffs  for  1932  [in  thousands  oj 

tons) 

Chilled  {i.e.  kept  above  the  freezing  point)  :  Beef  440,  bacon  550 
eggs  27  (500  million),  cheese  96,  fish  66,  apples  112,  pears  13,  bananas 
335,  oranges  104,  grape  fruit  13. 

Frozen  :  Beef  123,  mutton  330,  pork  17,  poultry  11,  rabbits  27. 
liquid  eggs  from  China  40,  butter  385,  fish  3. 

Frozen  cargoes  are  a  comparatively  simple  matter,  demandin,;; 
nothing  more  than  a  regulated  rate  of  thawing  out,  steaks  hackee 
from  carcases  of  mammoth,  presumably  quickly  frozen  thousands  c> 
years  ago,  have  been  eaten  with  relish  ;  but  in  chilled  materials  thi 
vital  processes  are  merely  slowed  down  (in  meat  about  ten  times  a 
33°  F.  as  compared  with  63°),  especially  in  fruit,  which  is  quiti 
normally  alive,  and  respiring  COg. 

The  thick-skinned  banana  travels  green  and  wooden  at  45 — 50°  F, 
and  ripens,  in  warm  conditioning-rooms,  here,  every  bit  as  well  as  ii 
the  West  Indian  sun,  but  the  thin-skinned  papaw  does  not  ye 
survive  the  voyage,  and  the  local  green  orange  goes  spotty. 

Apples  once  allowed  to  freeze  are  unmarketable,  and  those  whicJ 
have  been  suffocated  in  their  own  output  of  COg  are  in  little  bettc 
plight,  being  full  of  brown  specks,  but  8%  of  COg  left  in  the  atmc 
sphere  is  their  best  accessory  protection  against  the  growth  c 
mould.  Oranges  are  saved  from  the  same  trouble  by  a  dip  in  sod 
bicarb.,  aldehyde  vapour  preserves  grapes,  ethylene  both  ripens  an 
brightens  the  colour  of  grape-fruit,  and  other  unexpected  gaseov 
preservatives  are  proving  their  worth. 

Still,  for  better  or  worse,  the  health  of  the  whole  community,  whic 


§297] 


HEAT   ENGINES   AND   COLD   ENGINES 


210 


it  will  be  your  privilege  to  help  to  guard,  hangs  increasingly  upon  cold 
storage. 

The  days  of  deep-thatched  ice-houses,  filled  with  cargoes  of 
Norway  lake  ice,  are  long  since  past,  and  it  was  even  a  shock  to  meet 
a  drifter  piled  with  boulders  of  it  steaming  up  the  Lyngenfjord,  for 
we  have  come  to  depend  on  heat-engines  to  cool  us,  through  the 
intermediary  of  a  great  variety  of  Refrigerating  Macliines. 

The  Refrigerating  Engineer  thinks  in  frigories,  which  are  Minus 
calories,  and  cost  at  least  4  times  as  much. 

Mostly  these  machines  employ  ammonia,  COg,  or  SOg,  though 
ethyl-  and  methyl-chlorides  and  the  non-toxic  dichlordifluor- 
methane  '  Freon,'  are  in  use  on  a  smaller  scale.    All  are  readily 


COMPRESSOR. 


Fig.  90. 


liquefied  by  moderate  compression  and  water-cooling ;  see  last 
column  of  the  Table,  §  290. 

Carbon  dioxide  is  of  most  interest.  On  a  really  hot  day  it  can 
no  longer  be  heard  splashing  about  as  you  shake  its  steel  bottle,  for 
it  has  passed  its  critical  temperature,  31°  C.  (and  anaesthetic 
nitrous  oxide  does  the  same  at  '  blood  heat ').  Liquid  COj  squirteil 
out  of  an  inverted  bottle  soHdifies  into  a  subliming  snow,  which  can 
be  handled  gently  because  it  '  assumes  the  spheroidal  state  '  on  tlie 
warm  hand ;  and  it  is  pressed  into  blocks  of  '  drj^  ice  '  (cf.  §  273) 
now  in  wide  commercial  use.  COg  is  administered  with  oxygen,  in 
asphyxiation,  to  stimulate  the  respiratory  centres,  and  we  all 
appreciate  its  presence  in  '  mineral  waters  '  and  other  such  imitations 
of  better  things. 

CO2  refrigerating  machines  have  to  work  at  750  lb.  pressure, 
but  are  compact,  and  the  gas  flows  away  hannlessly  in  case  of 
Ammonia   machines   work   at   more   familiar   '  steam 


220  HEAT  [§  21 

pressures,  and  often  exceed  a  thermodynamic  efficiency  of  25^/^ 
although  the  Hght  NHg  can  diffuse  away  dangerously  from  a  le 
No  water  is  used  with  the  ammonia  in  English  machines.  SOg  an^ 
the  other  substances  work  at  much  lower  pressures  in  domestil 
outfits. 

In  Fig.  90  is  sketched  the  lay-out  of  an  Ammonia  Plant,  which, 
running  dead  slow,  was  keeping  our  cargo  of  130,000  bunches  of 
bananas  in  the  verdant-green  condition  at  about  45°  F.  :  in  the 
corner  is  the  '  boiling  point  and  pressure  curve  '  for  NH3  (cf. 
Figs.  82,  85),  which  controls  everything. 

The  steam-driven  Compressor  receives  the  gas  from  circulation 
through  inlet  valves  which  are  thick  with  hoar-frost,  and  com- 
presses it  through  outlet  valves  too  hot  to  touch,  into  batteries  of 
inch  pipes  in  a  tank  of  sea -water,  sent  up  from  below  by  the  same 
circulating  pump  which  supplies  the  engine's  steam-condenser.  In 
these  pipes  the  hot  gas  is  cooled,  and  liquefies  at  some  pressure 
between  100  and  200  Ib./sq.  in.,  depending  on  the  temperature  of  the 
sea. 

The  liquid  NHg  crosses  to  a  rank  of  valves,  which  admit  it,  as  a  • 
blast  of  vapour  and  spray,  to  similar  batteries  of  pipes  in  the  Brine  1 
tank  :   by  regulating  these  '  centre-punch  '  hand- valves  the  pressure 
in  the  pipes  is  held  at  about  20  lb.,  and  the  brine  therefore  cooled 
to  10°  F. 

These  valves,  which  are  massed  with  ice,  form  at  least  part  of  the 
'  Porous  Plug  '  of  §  292,  the  rest  being  the  evaporation  of  the  spray 
in  the  pipes.  The  porous  plug  was  dealing  with  fluid  which  had  only 
just  begun  to  think  about  settling  down  out  of  the  gay  and  careless 
perfect-gas  condition ;  here  is  the  ultimate  stage  when  the  mole- 
cules are  clung  together  in  liquid  form,  and  their  tearing  apart 
demands  all  the  latent  heat  they  had  given  up  in  the  other  tank  : 
there  is  no  doubt  whatever  about  this  cooling  effect. 

By  Brine-pumps,  the  cold  liquid  is  circulated  through  four  screens 
of  tubing  in  the  deck-houses,  and  15-h.p.  fans  extract  the  air  from 
their  holds  and  blow  it  back  through  these  screens,  except  for  J  hr. 
daily,  when  they  draw  fresh  air  to  sweep  away  the  exhalations  of  the 
fruit.  Holds  and  deck-houses  are  insulated  by  a  lagging  of  6  in.  of 
slag  wool ;  and  this  45-h.p.  machine  would  be  perfectly  capable  of 
keeping  them  below  the  freezing  point  (as  is  the  ship's  larder),  for 
the  meat  trade. 

In  Ice  Factories,  fresh  water  is  frozen  in  slab  tanks  immersed  in 
the  brine,  so  that  there  is  just  a  chance  of  finding  accidental  traces  of 
salt  in  commercial  ice.  For  clear  ice  the  water  is  kept  stirred  by 
bubbling  air  through  :   clear  ice  is  completely  sterile. 

You  will  recognize  the  working  parts  described  above,  in  petto,  in 
most  domestic  refrigerating  machines ;  but  for  the  intricate  action 
of  the  motionless  liquid  absorption  refrigerators  you  must  consult  the 
makers'  diagrams. 


HEAT   ENGINES   AND   COLD   ENGINES  221 


EXAM   QUESTIONS,   CHAPTER   XIX 

If  the  exam,  the  exam,  and  only  the  exam  interests  (  ?)  you,  and  your  wits 
are  in  cold  store  over  such  matters  as  engines  and  the  power  supply  of  tho 
world;  oxygen  and  liquefied  gases;  and  your  food;  you  will  pass  this 
chapter  by,  without  the  relief  that  it  possibly  affords  to  others. 

1.  Describe  a  method  of  liquefying  air,  and  give  a  simplified  diagram  of 
the  apparatus  used.     (  X  2) 

2.  Distinguish  between  '  gas  '  and  '  vapour.'  Give  a  brief  account  of  the 
liquefaction  of  air ;  how  is  it  stored  ?  Mention  any  special  points  in  the  design 
of  the  vessel. 

3.  Describe  the  principle  of  a  practical  method  for  the  production  of  cold. 


CHAPTER   XX 


HYGROMETRY 


I 


§  301.  Hygrometry  is  the  study  of  the  dryness  or  dampness  of 
the  atmosphere. 

In  accordance  with  Dalton's  Law,  water  will  evaporate  until 
its  vapour  fills  the  space  above  it  to  the  same  partial  pressure, 
whether  any  other  gas  be  there  or  not.  But  the  presence  of  another 
gas  enormously  hinders  the  rate  of  evaporation,  for  the  escaping 
water  molecules  have  to  thread  their  way  through  a  crowd  of 
gas  molecules.  Hence  the  amount  of  water  vapour  present  in  the 
air  above  water  or  wet  soil  does  not  often  reach  its  saturation 
value ;  even  gentle  atmospheric  movements  sufiice  to  carry  it 
away  before  this.  Saturation  may  be  reached  on  subsequent 
cooling,  and  over-run,  and  mist  or  cloud  deposited,  §  284. 

The  further  the  contained  vapour  is  below  its  full  saturation 
pressure  the  more  water  can  the  atmosphere  still  take  up,  the 
quicker  wet  things  dry,  and  the  drier  the  air  feels.  Since  the 
maximum  vapour  pressure  increases  so  rapidly  with  temperature, 
Fig.  82,  summer  air  may  feel  very  dry,  and  yet  contain  more 
than  enough  water  to  saturate  it  in  the  cold  of  night.  On  a  cold 
winter  day  there  can  be  very  little  vapour  present  at  all,  and 
when  this  air  is  warmed  indoors,  without  any  addition  to  it« 
moisture,  it  feels  very  dry  indeed ;  your  hands  dry  up,  and  the 
cat's  back  crackles  and  sparkles  when  ruffled  up  the  wrong  way. 

Again,  assuming  a  half- saturated  state,  it  is  evident  that  thu 
vacant  10  mm.  or  so  in  summer  will  promote  a  faster  drying-uj: 
than  the  vacant  2  or  3  mm.  in  winter ;  compare  the  half -height 
for  25°  with  that  for  10°,  say,  N  and  R',  in  Fig.  82. 

This  rate  of  drying  interests  the  laundress,  or  the  farmer  who  \i 
getting  up  his  crops ;  on  the  other  hand,  the  gardener  is  ofter 
more  concerned  that  there  should  be  enough  moisture  in  his  houses 
As  a  sort  of  compromise  it  has  become  customary  to  specify  th( 
Hygrometric  State  of  the  Air,  its  Saturation  Fraction,  Relative 
Humidity,  or,  simply.  Humidity,  as  the  Ratio  of  the  mass  of  wate: 
vapour  actually  present  in  the  air,  to  the  mass  that  could  b< 
contained  in  the  same  bulk  at  the  same  temperature. 

Or  what  comes  to  practically  the  same  thing,  since  the  vapou 
obeys  Boyle's  law  almost  up  to  saturation 

TT      'rt't    —  V'"'^^'^''^^  ^f  ^«^6^  vapour  actually  present  in  air 
~  pressure  of  saturated  vapour  at  same  temperature 

e.g.  in  Fig.  82  the  Humidity  at  N  would  be  11/25,  or  44%. 

222 


I 


302] 


HYGROMETRY 


223 


The  dryness  or  dampness  of  a  local  climate  may  be  left  to  general 
observation  and  opinion;  but  the  Humidity  of  textile  mills, 
timber  seasoning-kihis,  cold-  and  perishable  food-stores,  public 
halls,  etc.,  calls  for  close  measurement  and  control. 

§  302.  Of  Hygrometers  for  measuring  Humidity,  the  *  chemical,' 
Fig.  91,  is  direct,  but  far  too  slow.  The  air  leaves  its  moisture  in 
weighed  '  drying  tubes  '  as  it  is  drawn  through  them  to  replace  the 
water  trickling  out  of  the  aspirator,  of  known  content.  The 
observed  increase  of  weight  is  then  divided  by  the  weight  of  the 
same  volume  of  saturated  vapour  at  the  same  temperature,  obtained 
either  from  the  tables,  or  by  a  similar  experiment  in  which  the 
air  would  be  first  passed  through  tubes  of  soaked  wool. 


Fig.  91. 

In  dew-point  hygrometers  a  cold  surface  cools  the  air  near  it, 
down  to  a  temperature  at  which  the  amount  of  vapour  present  suffices 
to  saturate  it,  and  thereafter  begins  to  precipitate  as  a  thin  *  dew  * 
on  the  cold  bright  surface.     You  travel  from  N  to  H,  Fig.  82. 

Then  the  Humidity  is  the  saturation  pressure  at  this  Dew-point 
divided  by  that  at  the  actual  air  temperature  (read  off  from  Fig.  82). 

A  clean  glass  of  water  kept  stirred  and  gnidually  cooled  by  a 
lump  of  ice.  Fig.  92,  will  serve  the  purpose,  in  a  way  familiar 
enough  on  summer  dinner- tables.  With  such,  of  course,  you 
cannot  reach  any  dew  point  of  very  low  temperature,  a  *  hoar- 
frost point.* 

The  old  Daniell's  hygrometer,  Fig.  93,  is  a  bent  double- bulb  tube 
containing  ether  and  its  vapour.  More  ether  is  poured  on  one 
mushned  bulb,  and  evaporating,  cools  and  condenses  the  vaixjur 
inside.  More  vapour  comes  over  from  the  ether  three  parts  filling 
the  lower  bulb,  bringing  its  latent  heat  with  it,  and  this  bulb 
gradually  cools  until  the  dew  appears  on  its  surface  (sometimes 
gilded).  The  instrument  must  be  kept  well  shaken  up  to  keep  the 
bnlb  at  the  same  temperature  throughout,  and  of  course,  as  with 


224 


HEAT 


[§302 


all  hygrometers,  neither  the  breath  nor  the  warm  perspiring  hand 
must  come  near  the  air-temperature-thermometer  (on  the  stand), 
nor  the  cold  surface ;  nor  need  success  be  expected  in  the  sun, 
or  in  a  draught. 

As  dew  enough  to  see  means  that  the  temperature  is  dropped 
a  little  too  far,  cooling  is  stopped  (grip  the  wet  bulb  for  a  few 
seconds)  and  a  rising  reading  taken  when  the  dew  just  dries  off, 
and  the  mean  of  both  is  the  Dew-point.     The  difficulty  with  all 


Fig.  92. 


Fig.  93. 


Fig.  94. 


these  hygrometers  is  to  glimpse  the  first  faint  dimming ;  work  in 
a  good  light,  and  touch  occasionally  with  a  strip  of  paper ;  with 
practice,  the  two  temperatures  are  within  a  degree. 

A  more  modern  instrument  has  a  little  box,  full  of  ether  or 
petrol,  soldered  to  the  back  of  a  thin  polished  metal  plate  which 
may  be  cut  to  prevent  the  cooling  from  spreading,  Fig.  94  :  the 
great  thing  is  to  do  away  with  badly- conducting  glass,  which 
causes  temperature-lag.     Evaporation  is  excited  by  a  bulb-bellows. 

In  its  latest  form,  due  to  E.  B. 
Moss,  this  is  a  plate  of  stainless 
steel,  optically  flat  and  highly 
polished,  P,  Fig.  95,  and  reflects 
light  coming  from  lens  L  to  form 
a  sharp  image  at  I  of  miniature 
lamp  S.  This  image  is  caught  on 
the  little  screen  D,  and  its  light 
prevented  from  entering  the  short 
Fig.  95.  wide   telescope  TE  through  which 

you  observe  the  mirror  :  as  there 
is  no  other  light  about,  this  remains  invisible.  But  as  soon  as 
dew  dims  P,  it  scatters  a  haze  of  light  which  overlaps  D,  al 
round,  as  shown  on  the  right,  and  enters  the  telescope,  and  I 
brightens  into  view. 

By  putting  a  photo-cell  (§  984)  at  E,  with  suitable  amplifiers 
the  hygrometer  can  be  made  completely  automatic,  and  be  giver 
control  of  all  sorts  of  moisture-regulating  machinery. 


§  302]  HYGROMETRY  225 

It  is  found  necessary  to  maintain  a  sharp  draught  over  the  cold 
plate,  cf .  §  303  and  Fig.  96. 

[This  apparatus  was  really  developed  from  one  which  records 
the  density  of  Smoke  by  the  light  it  scatters  back  to  the  photo- 
cell.] 

§  303.  In  the  wet  and  dry  bulb  hygrometer  (Psychrometer,  or  other 
names)  one  of  a  pair  of  thermometers  has  its  bulb  wrapped  in 
old  washed  linen  kept  wet,  like  a  wick,  by  distilled  water.  The 
drier  the  air,  the  faster  the  moisture  evaporates  and  abstracts 
latent  heat  from  the  bulb,  which  therefore  cools  until  the  influx 
of  heat  by  convection  balances  the  rate  of  loss. 

The  accurate  instrument  is  the  '  sling  psychrometer,'  sketched 
down  on  the  right  of  the  Chart,  Fig.  96.  You  dip  the  covered 
bulb  in  water  and  then  whirl  the  pair  round  until  steady  readings 
aie  obtained.  This  convective  cooling  in  a  strong  draught  is  most 
<lesirable,  and  I  have  drawn  the  Chart  from  the  best  Tables  for 
tliat  use  (barometer  about  30  in.). 

If  only  the  older  stationary  wall  pattern,  sketched  on  the  left, 
is  available,  the  pecked  lines  on  the  Chart  must  be  used,  and  both 
I  range  and  accuracy  are  more  limited. 

On  this  chart,  which  is  graduated  for  use  either  C.  or  F.,  take  the 
dry-bulb  air  temperature  as  ordinate,  and  go  along  the  horizontal 
to  reach  the  difference  between  thermometers  as  abscissa.     Your 
position  on  or  between  the  more  upright  family  of  lines  gives  the 
(Humidity,  the  more  slanting  lighter  lines  give  the  corresponding 
Dew-point.     No   difference,   of  course,  means   saturation ;    a   big 
j  difference,  very  dry  air,  being  proportional  to  NV,  Fig.  82,  the 
I  washer-woman's  '  Ih-y  '  (in  the  dew-point  hygrometer  you  found 
NO).     That  is  all  she  would  need  look  for,  but  as  you  want  the 
relative  Humidity,  NO/VO,  you  must  find  VO  also,  and  that  is 
why  the  dry-bulb  temperature  must  be  referred  to  as  well.     Gar- 
dener and  fruit-grower  look  for  a  high  Dew-point  in  the  afternoon, 
because   they   expect   that   ample  condensation,   as   night   comes 
down,  will  stave  off  risk  of  frost. 

Do  not  fall  into  the  common  delusion  that  a  dry  wind  causes 

evaporation,  that  dry  air  sucks  up  moisture  like  a  sponge.     Water 

evaporates  at  a  rate  which  depends  solely  on  the  temperature  of 

its  surface.  Fig.  82  :  what  does  differ  is,  how  much  of  the  evaporated 

vapour  is  put  back  again.     None,  in  a  dry  wind  which  blows  it  away  ; 

all,  from  calm  saturated  air ;    more  or  less,  according  as  the  sur- 

I  rounding  air  is  wetter  or  drier,  slower  or  faster  moving. 

!      When  your  mother  finds  a  room  cold  and  damp,  say  R,  10°, 

j  Fig.  82,  she  opens  the  windows,  and  wind  mechanically  removes 

I  the  vapour  and  dries  the  room,  without  change  of  temperature, 

down  to  R'. 

'      Or,  if  the  outer  air  is  also  saturated,  and  brings  in  as  much 
1  moisture  as  it  blows  out,  she  must  light  a  fire,  and  warm  the  room 
j  to  R",  20°,  when  again  it  feels  thoroughly  dry.     But  do  what  she 
I 


226 


HEAT 


[§303 


Fig.  96. 


§  305]  HYGROMETRY  227 

will,  she  can  never  reduce  the  actual  moisture-content  below  that 
of  the  outer  air. 

Damp  rooms,  of  course,  are  a  curse.  In  old  houses  one  can 
relay  tile  floors  in  concrete  surfaced  with  2  :  1  sand  and  cement, 
and  run  a  dado  of  oilcloth  round  the  walls,  and  then  depend  on 
open  windows,  and  fires.  In  the  absence  of  a  waterproof  damp- 
course  it  is  useless  cementing  the  walls  outside,  for  the  ground 
moisture  rising  by  capillary  action  in  the  w^ll,  and  now  denied  a 
chance  of  evaporating  outwards,  will  simply  climb  higher  until  it 
does  find  adequate  evaporating  surface.  Wooden  floors  must  be 
ventilated  underneath,  or  Merulius  lacrymans  will  creep  and  weep 
over  them,  and  leave  them  dry-rotten. 

Modem  houses,  with  efficient  slate  damp-courses  bearing  in- 
expensive double  walls,  the  outer  shell  hard  and  weather- proofed, 
an  air-space,  and  the  inner  walls  porous  and  absorbent  (a  combina- 
tion like  your  own  clothing),  should  bo  proof  against  all  intrusive 
damp  and  chill,  and,  further,  free  from  unpleasant  internal  *  con- 
densation.* 

§  304.  There  are  many  instruments  regarded  as  less  reliable,  and 
more  properly  called  Hygroscopes,  depending  on  the  hygroscopic 
(*  moisture  showing  *)  nature  of  fibrous  materials  (§  350)  or  chemicals. 
Such  is  the  Hair  Hygroscope,  in  which  a  few  long  strands  of  hair, 
freed  from  grease  by  ether  extraction,  stretch  in  moist  air,  and 
permit  their  tension  spring  to  move  a  pointer.  It  has  the  advantage 
of  being  entirely  self-acting,  and  unaffected  by  frost.  The  '  seed  * 
(achene)  of  the  feather-grass  (Stipa  pennata),  or  of  the  wild  geranium, 
can  be  stuck  upright  on  a  card  with  a  drop  of  wax,  and  the  twiste*! 
hygroscopic  awn  waves  round  its  natural  pointer  (Fig.  91,  leaning 
against  the  aspirator).  Twisted  catgut  suspends  the  weather- wijMj 
old  couple  in  their  hut,  perched  on  the  shelf  in  many  a  country 
cottage ;  while  every  one  of  3'our  young  imtients  will  have  a  salt 
seaw^d  trophy  of  tlie  summer  holiday,  going  limp  before  rain. 

§  305.  Changes  of  Humidity  by  day  and  night  are  familiar  to  us 
all,  but  are  so  dependent  on  temperature  as  to  have. little  value  in 
scientific  weather-forecasting. 

The  east  coast  of  England  is  hotter  and  colder  than  the  south, 
but  drier,  just  as  bearable  to  a  healthy  man  summer  and  winter, 
and  definitely  more  bracing.  Sea  air  is  moister  than  coastal  air; 
:  voyage  is  not  so  invigorating  as  a  seaside  stav. 

Why  is  this?  And  why,  when  the  humidity  is  high,  do  we 
leel  most  horribly  the  bitter  chill  of  winter,  or  most  oppresmvely 
the  close  heat  of  summer  ?  It  is  not  due  to  inoreaaed  conductivity 
in  the  atmosphere,  for  water  vapour  is  less  conductive  than  air. 
A  simple  electrical  test  (|§  787,  901)  settles  it,  for  thermal  and 
electrical  conductivities  often  go  hand  in  hand  :  the  Skin  is  a 
much  better  conductor  of  botn  in  moist  air.  The  sweat  rises 
higber  through  it  before  evaporating,  it  becomes  a  wet  garment 


228  HEAT  [§  305 

instead  of  a  dry  shield,  and  the  water  conducts.  The  loss  of  heat 
becomes  painful  in  winter  :  the  failure  to  cool  causes  discomfort 
in  summer,  for  if  there  is  but  Uttle  difference  between  blood-heat 
and  air  temperature,  the  saturated  skin  at  S,  Fig.  82,  has  but  little 
'  dry  '  into  saturated  air  at  A.  After  a  swim  from  a  tropic  island 
beach,  after  tea,  it  is  ridiculously  impossible  to  get  dry,  in  spite 
of  the  heat ;  a  saturated  breeze  helps  not  at  all,  and  the  towel 
remains  a  soaked  rag  until  the  sun  catches  it  out  on  the  line  in 
the  morning,  and  dries  it  in  a  matter  of  minutes. 

Really,  the  Wet  Bulb  gives  the  closer  idea  of  the  sensible  tem- 
perature of  the  day,  but  the  instrument  can  be  used  in  a  better 
way  as  a  '  comfort -measurer.'  Warm  both  bulbs  in  steamy  heat 
to  '  blood-heat,'  then  expose  them  and  take  their  times  of  fall 
through  the  same  10°  or  15°  ;  the  difference  between  them  practically 
represents  the  differential  effect,  on  your  skin  and  your  lungs,  of 
the  humidity  and  the  movement  of  the  air.  (The  latter  is  quite 
20  times  as  much  outdoors  in  calm  weather,  as  it  is  indoors.)  A 
special  larger  form  of  the  instrument,  better  adapted  to  this 
test,  is  sold  complete  with  explanatory  tables,  as  the  '  Kata- 
thermometer.' 

One  hears  it  said,  when  weary  of  town  in  the  summer,  '  Oh, 
the  other  place  may  be  as  hot,  but  the  heat  is  different.'  But 
when  allowance  has  been  made  for  less  and  more  porous  clothing, 
for  change  of  scene  and  diet  and  occupation,  for  more  air,  and 
more  perfect  elimination  of  toxins ;  Heat  and  Humidity  still  hold 
you  in  their  grip,  and  can  be  just  as  trying,  in  a  cleaner,  greener 
land,  as  in  the  sombre  streets  of  Urbs  Augusta. 

§  306.  Following  measurement,  control.  Fires  of  all  sorts  have 
long  saved  our  skins  from  cold  and  damp,  the  two  disappearing 
together,  and  then,  if  the  humidity  falls  too  low — below  50%  or 
40% — steam  or  fine  water  spray  can  be  blown  into  the  ventilating 
ducts  to  make  up  the  deficiency. 

Damp  heat  is  more  difficult  to  cope  with  ;  cooling  the  air  by 
brine-pipes  from  refrigerating  machines  may  leave  it  intolerably 
saturated,  or  even  misty.  To  deal  with  this,  half  the  air  can  be  chilled 
almost  to  freezing  point,  when  all  but  5  mm.  of  its  moisture  settles 
out,  and  then  the  dried  air  is  partly  re-warmed  as  required  in  a 
pipe  heater  through  which  passes  the  warm  water  discharge  from  the 
refrigerating  machinery.  Fig.  90,  right-hand  tank,  and  mixed  with 
the  untreated  air,  under  the  automatic  control  of  thermostat  and 
hygrostat  such  as  Fig.  95. 

Installations  of  this  type  afford  escape  from  the  fierce  and  often 
humid  heat  of  the  great  cities  of  the  eastern  U.S.  ;  and  have  spread 
to  this  country,  but  threepence  spent  on  viewing  Mantegna's 
masterpiece  of  1460  in  the  old  Orangery  at  Hampton  Court  will 
introduce  you  to  the  activity  of  a  much  simpler  one  set  up  there 
in  1934.  These  nine  tempera  cartoons  of  the  Triumph  of  Caesar 
have  been  restored  by  glueing  the  drought-loosened  fiakes  of  colour 


§  306]  HYGROMETRY  229 

back  on  to  the  canvas  (and  subsequently  waxing  the  surface  to 
bring  out  the  colours,  §  561),  and  it  is  intended  to  keep  them  there 
by  maintaining  a  fairly  constant  60%  humidity. 

The  artist  has  adopted  the  singularly  common-sense  expedient 
of  stretching  1800  lb.  of  old  canvas  fire-hose,  anti-mildew  doped 
with  a  little  thymol,  in  lockers  through  which  blows  the  incoming 
air,  damped  if  need  be  by  spray  from  a  pump.  The  canvas  takes 
up  9%  of  its  weight  of  water  from  55%  saturated  air,  and  13% 
from  75%  saturated,  and  modulates  sudden  variations  in  supply  of 
moisture  very  successfully,  while  the  20  tons  of  woodwork  lining 
the  walls  act  as  the  final  slow  absorbent  regulator. 

A  foot-long  vertical  hank  of  hair  has  been  put  in  charge,  and  is 
holding  the  humidity  steady  within  2%  ;  it  pulls  on  a  light  balance 
beam  only  a  few  mm.  from  its  fulcrum,  and  its  swelling  or  shrinking 
dips  a  n  wire  connector  at  one  or  other  end  of  the  beam  into  mer- 
cury cups,  passing  current  enough  to  tip  over  the  main  mercury- 
switch  of  the  machinery.  At  the  little  royal  private  view,  H.M. 
amused  himself  alternately  breathing  on  the  hair  and  fanning  it 
with  his  hat,  and  listening  to  the  dutiful  stopping  and  starting  of 
the  deluded  spray  pump ;  so  that  the  long-looked-down-upon  Hair 
Hygrometer  has  at  length  come  into  some  measure  of  honour 
again. 


EXAM   QUESTIONS,   CHAPTER   XX 

All  the  theory  of  this  chapter  is  in  Chap  XVIII ;  it  is  now  merely  largely 
diluted  with  air,  and  studied  over  that  very  limited  range  in  which  humanity 
reckons  life  endurable.  The  chapter  is  descriptive  :  handle  and  use  as  many 
of  the  instruments  in  it  as  you  can,  and  don't  bother  about  the  rest. 

1.  When  do  your  spectacles  dew  over  and  blind  you;  entering  a  warm 
greenhouse,  or  leaving  it,  on  a  cold  damp  day  ? 

2.  In  what  sort  of  weather  do  tanks,  walls,  etc.,  inside  the  house  run  and 
drip  with  condensed  moisture  ? 

3.  What  is  meant  by  the  Dew  Point  ?  Describe  a  method  of  determining 
it.  Mention  any  refinements  you  think  are  necessary  in  the  experiment. 
From  a  knowledge  of  the  dew  point,  how  can  you  ascertain  the  percentage 
dampness  of  the  air  ?     (  X  4) 

4.  Discuss  the  desirability,  or  otherwise,  of  sprinkling  water  about,  on  a 
very  hot  day,  '  to  relieve  the  oppressiveness  of  the  atmosphere.' 

5.  Define  Dew  Point,  and  the  Humidity  or  Hygrometric  State  of  the  air. 
Why  does  it  vary  diu-ing  the  day  ? 

Describe  the  action  of  the  '  wet  and  dry  bulb  thermometers,'  and  explain 
how  it  is  that  the  humidity  is  not  simply  proportional  to  the  diflFerence  of 
temperatures.     (  X  3) 

6.  How  are  dew  point  and  humidity  determined  ?  If  the  air  temperature 
be  18°  and  the  dew  pt.  14°,  find  the  degree  to  which  the  air  is  saturated. 


230  HEAT 

7.  The  dew  point  of  the  air  in  a  greenhouse  rises  from  9*5*'  C.  to  20'2''  C. ; 
calculate  the  proportion  in  which  the  vapour  is  increased. 

8.  Air  is  half  saturated  at  15";  calculate  its  dew  point.  If  it  were  cooled 
to  0°,  what  fraction  of  its  moisture  would  condense  to  mist  ? 

9.  Describe  and  explain  the  transfer  of  cold  in  the  Daniel!  hygrometer. 
How  do  you  hold  the  instrument  to  drive  condensed  ether  out  of  the  covered 
bulb  ? 

10.  Show  how  to  find,  from  the  dew  point,  the  mass  of  aqueous  vapour 
present  in  a  cubic  metre  of  air.  What  effect  has  the  temperature  of  the  air 
on  the  calculation  ?     (  X  3) 

11.  Calculate  the  amount  of  aqueous  vapour  present  in  1000  litres  of  air 
satiu-ated  at  18°,  its  pressiue  being  15-46  mm. 

12.  How  would  you  keep  watch  over  the  humidity  of  a  room  ? 
Calculate  the  actual  mass  of  water  present  as  vapour  in  an  ordinary  room 

5  X  4  X  3  m.,  0-6  saturated  at  18°  C. 


PRACTICAL  QUESTION 

Determine  the  dew  point  (usually  by  a  flask  of  water  cooled  by  ice)  and  deduce 
the  humidity  of  the  air. 


I 


CHAPTER   XXI 


METEOROLOGY  AND  THE  WEATHER 


§  311.  The  Weather  touches  us,  all  of  us,  all  the  time. 

We  know  it  all  depends  on  the  radiant  heat  of  the  Sun,  who  sends 
us,  perpetually,  nearly  2  cals.  per  minute  per  sq.  cm.  of  surface 
facing  him.  Half  this  is  estimated  to  be  reflected  back  at  once  into 
outer  space,  from  the  atmosphere  itself  (which  forever  hides  all 
terrestrial  topography  from  any  inhabitants  of  Mars),  from  Clouds, 
and  sea ;  the  other  calorie  is  retained  for  the  time  being  to  warm 
us,  and,  literally,  to  get  up  steam  in  our  Atmosphere  and  involve  us 
in  Weather. 

How  is  the  sun's  heat  retained  ?  You  know  how  after  days  of  mist 
and  rain,  sunshine  bursting  through  between  clouds  in  the  dry  N.W. 
wind  bums  and  tans  with  a  fury  beyond  that  of  an  average  unclouded 
summer  day  :  evidently  clean  dry  air  does  not  intercept  nor  hold 
much  solar  radiation.  But  a  pocket  spectroscope,  turned  on  the 
sun,  shows  '  atmospheric  '  absorption  lines  in  the  red  which  darken 
greatly  as  he  sinks  and  slants  through  multiplied  mileage  of  air,  and 
before  Rain  a  broad  dark  band  becomes  conspicuous  to  the  eye,  in 
the  yellow,  Fig.  223,  over  D,  and  others  to  radiation  measurers  in 
the  infra-red.  Chapter  LVI :  it  is  WATER  VAPOUR  that  absorbs 
radiation  in  the  atmosphere,  and  it  absorbs  one-tenth,  on  the 
average  ;  leaving  the  rest  to  reach  the  earth's  surface. 

§  312.  There  is  another  absorbent.  Haze,  which  sometimes  causes 
considerable  stuffiness,  though,  fortunately,  intense  heat  can  dry 
its  particles  into  iavisible  smallness,  and  London  in  really  hot 
summer  sunshine  can  be  brilHantly  clear  beyond  belief.  Haze  may 
exceptionally  be  due  to  dry  solid  particles,  as  above  a  dusty  road,  or 
the  sand  dunes,  or  volcanic  dust  particles  pervading  the  high 
atmosphere — one-fifth  loss  of  heat  has  been  traced  to  this  after 
eruption,  and  protracted  volcanic  activity  may  very  likely  have  been 
the  chief  cause  of  the  Glacial  Periods — but  more  commonly  it 
consists  of  exceedingly  minute  Hygroscopic  Particles,  which,  we  shall 
see  later,  act  as  Nuclei  for  the  condensation  of  moisture  (and  even 
to  a  small  extent,  from  air  only  3/4  saturated) .  These  seldom  number 
less  than  100  per  c.c,  more  usually  thousands  in  open  country, 
and  may  reach  tens  of  thousands  in  cities.  They  are  formed  in 
abundance  by  the  evaporated  salt  spray  of  the  sea  (recollect  how 
persistently  the  looking-glass  in  your  beach  hut  hazes  over  at  a 
breath),  it  is  said  also  by  solar  ultra-violet  acting  directly  on  moist 
air,  and  in  enormous  bulk  by  all  processes  of  combustion,  especially 

231 


232  HEAT  [§  312 

by  the  5000  tons  of  sulphur  in  coal  burnt  daily  in  England.  One 
can  watch  the  white  fume  from  the  cement-kilns  of  Northfleet 
and  the  oil -refineries  of  Thames  Haven  drifting  up  north-eastwards 
into  the  grey  sky,  growing  rapidly,  and  becoming,  by  deposit  of 
moisture  upon  them,  sizeable  clouds ;  such  as  might  later  make  a 
curious  contribution  to  the  soft-water-butts  of  Suffolk. 

Incidentally,  a  London  fog  may  contain  20,000  particles  per  c.c, 
amounting  to  2  mg.  of  solid  matter  per  cubic  metre  (or  exceptionally, 
more) ;  the  non-foggy  maximum  being  0-8  mg.  On  a  five  years 
average  in  the  City,  50  gm.  of  dust  and  ash  were  deposited  per  sq. 
metre  per  annum,  with  30  gm.  of  soot,  and  25  gm.  of  '  SO3.' 

§  313.  But  the  1  cal.  reaches  and  warms  land  or  sea,  and  evapor- 
ates a  good  deal  of  Moisture  from  them.  There  will  be  great  differ- 
ences in  the  amount  of  this ;  we  may  be  anywhere  underneath  the 
left-hand  curve  of  Fig.  82. 

Let  us  deal  with  this  moisture  before  it  is  carried  aloft. 

After  sunset,  if  the  sky  is  clear,  radiation  towards  the  utter  cold 
of  cosmic  space,  from  ground  and  leaves,  cools  them,  and  the 
adjoining  air  and  vapour,  and  the  latter  condenses  on  the  hygro- 
scopic nuclei  as  droplets  forming  Mist.  Haze  particles  are  exceed- 
ingly small  and  ultramicroscopic  ;  mist  drops  (which  are  all  water, 
not  hollow  bubbles)  may  perhaps  vary  from  a  very  few  to  20  microns 
(1/50  mm.)  diameter.  All  are  falling  through  the  air  all  the  time, 
a  Uttle  8  /A  drop  at  about  2  mm. /sec,  the  20  /x  drop  at  14  mm. /sec. 
In  clean  air  they  are  about  2  mm.  apart,  the  mist  or  cloud  containing 
from  0-1  to  2  gm.  of  them  per  cubic  metre  =  500  drops  per  c.c.  The 
city  fog  above  mentioned,  with  40  times  as  many  nuclei,  might  be  no 
wetter,  but  would  be  much  more  opaque  :  but  a  Mist  anywhere, 
high  and  dense  enough  to  impede  traffic,  constitutes  Fog. 

The  mist  lies  close  to  the  cooling  surface,  and  the  heavy  chilled 
air  containing  it,  of  course,  flows  down  into  hollows  and  fills  them. 
Although  droplets  are  falling,  the  '  Mist  rises  in  the  meadows,'  for 
the  thin  leaves  of  the  herbage  soon  radiate  away  the  heat  their  little 
mass  contains — ^watch  how  the  clover  leaflets  fold  and  '  go  to  sleep  ' 
and  reduce  radiating  surface — and  this  increasing  chill  spreads 
slowly  upwards  into  the  air,  and  cools  it  below  dew  point  at  greater 
and  greater  heights. 

If  the  mist  lies  undisturbed  all  night,  the  morning  sun,  shining  on 
its  droplets,  warms  and  evaporates  them,  and  they  lift  and  vanish 
into  thinnest  haze  again. 

But  if,  as  is  very  likely  in  a  dead  calm,  the  air  lying  above  the  cold 
fog  gets  warm,  either  by  the  sun  or  by  general  drift  from  elsewhere, 
constituting  what  is  called  a  '  temperature  inversion,'  the  heavy 
cold  air  has  no  chance  whatever  of  rising,  and  the  fog  has  to  lie 
there  until  wind  comes  along  to  shift  it.  In  winter,  unless  200  miles 
of  air  flow  over  Greenwich  Hill  in  the  24  hr.,  there  is  Likelihood  of  fog 
in  the  streets  of  London,  lying  there  to  thicken  on  the  nuclei  of 
combustion  from  millions  of  chimneys — ^until  in  2  days  all  of  it  has 


§  314]  METEOROLOGY  233 

had  a  trip  up  somebody's  flue,  and  got  well  tinted  and  flavoured  to 
the  '  real  thing,'  a  pall  foul  and  dark  enough  if  only  100  ft.  thick,  but 
exceptionally  piling  up  to  half  a  mile. 

Sea  Fog  usually  results  from  the  moist  air  from  warmer  waters 
flowing  over  cold  currents  (instance  the  persistent  fog  of  the  Grand 
Banks,  where  southerly  winds  that  were  laden  2  days  before  with 
vapour  from  a  sea  at  25°  C,  flow  over  the  Labrador  current  at 
5 — 10°)  ;  it  is  low,  clinging  to  the  water,  and  may  often  be  seen  over 
from  the  foretop  ;   floes  of  it  may  blow  far  under  cloudy  skies. 

Another  variety  forms  when  cold  puffs  of  air  descend  on  a  warm 
sea,  chilling  the  damp  atmosphere  below  its  dew  point.  This  is 
patchy,  and  the  warmth  of  the  water  melts  it  below ;  it  lies  high, 
and  may  be  seen  under  from  the  fore-deck.  Occasionally  this 
happens  on  a  large  scale  over  London,  causing  a  midday 
darkness,  without  fog  in  the  streets,  and  the  town  lights  up  and 
carries  on  in  warmth  and  comfort. 

The  Western  Isles  have  a  reputation  for  being  enfolded  in  mist 
while  the  intervening  sounds  are  clear.  In  June,  however,  under  the 
high  sun,  they  enjoy  the  best  of  weather,  for  the  quickly  absorbing, 
quickly  radiating,  land  is  then  warmer  than  the  sea,  and  melts  the 
mists.  With  the  lower  sun  and  longer  nights  of  August,  the  land 
has  begun  to  cool,  while  the  sea  has  attained  its  maximum  temper- 
ature, and  its  vapours  then  condense  over  the  land. 

§  314.  Dew.  If,  however,  the  evening  mist  is  heavy  enough  to 
fall  during  the  night,  we  awake  to  a  '  Fall  of  Dew.'  The  bigger  the 
drops  the  quicker  they  settle,  but  the  fewer  they  are,  and  the  less 
they  obscure  the  sky  :  the  star-gazer  finds  everything  smothered  with 
moisture  in  a  few  minutes,  without  warning. 

We  shall  see  in  §  315  that  air  must  cool  as  it  is  lifted  to  greater 
heights ;  consequently  this  variety  of  dew  is  to  be  expected  most 
abundantly  on  hilltops,  near  a  warm  sea.  It  fills  the  Dewponds 
strung  along  the  South  Downs,  popularly  regarded  as  such  ancient  and 
inexhaustible  mysteries  :  in  a  West  Indian  island  bay  one  slept  warm 
and  dry  enough  beneath  the  tropic  moon,  while  on  the  spur,  3000  ft. 
above,  everything  turned  damp  before  sundown,  and  by  bedtime 
the  eaves-gutters  were  merry  trickling  rills.  But  the  CaUfomian 
coast  is  washed  by  a  cold  current,  and  the  hills  stand  parched. 

Even  if  the  day  were  very  dry  there  may  be  Dew  in  the  morning. 
For  the  warm  earth  goes  on  distilling  out  the  moisture  brought  up 
by  capillary  action  from  below,  and  this  vapour  rises  but  few  inches 
above  the  now  chilled  surface  before  it  condenses  and  drops  back ; 
and  the  quickest  radiator  collects  the  best  fall,  draining  the  neigh- 
bouring air  by  its  persistent  cold.  This  sort  of  dew,  indeed,  you  are 
quite  likely  to  find  on  the  under  side  of  a  tin  lid  thrown  down  on  the 
soil. 

Grass-blades  radiate  well,  but  for  the  abundant  formation  of  dew 
on  grass  there  is  a  more  potent  cause.  It  is  not  a  fall  at  all ;  it  is  a 
rise^  through  the  vessels  of  the  leaf,  of  transpiration  water — crude 


234  HEAT  [§  314 

sap — from  the  roots.  By  day,  this  evaporates  from  the  stomata, 
but  it  camiot  do  so  in  the  saturated  air  at  night,  and  the  root- 
pressure  continuing  forces  it  out  in  drops.  On  many  plants  there 
are  specially  large  water-pores  through  which  this  water  can  exude ; 
there  is  one  at  the  tip  of  each  tooth  on  a  fuchsia  leaf.  In  late 
summer  the  earth  is  thoroughly  warm,  and  keeps  the  roots  active, 
hence  the  grass  dews  are  heavy  at  that  season. 

However  formed,  dew  does  preserve  the  crop  from  drought,  for 
'  a  penny  saved  is  a  penny  earned.' 

Everyone  knows  that  clouds  prevent  dew,  acting  as  blankets  to 
check  radiation  out  into  space.  In  the  absence  of  cloud  the  careful 
gardener  will  read  his  wet-  and  dry-bulb  hygrometer  in  the  after- 
noon, and  find  by  a  Fig.  96  where  the  Dew  Point  is.  If  it  is  well 
above  the  Freezing  Point,  the  blanket  of  water-vapour,  which  is 
nine  times  as  absorbent  of  cool  terrestrial  radiation  as  it  is  of  hot 
solar,  will  be  thick  enough  to  ward  off  Frost ;  but  if  low,  radiation 
will  escape  twice  as  fast,  and  he  must  look  to  his  greenhouse  fires, 
and  do  all  that  he  can  to  protect  his  outdoor  crop — e.g.  clouds  of  smoke 
from  green  or  greasy  smudge  fires  in  orchards  when  the  fruit  is 
setting.  It  is  no  small  matter  when  a  late  frost  may  wipe  out  a 
season's  profits. 

If  cooling  be  very  rapid,  the  vapour  or  the  mist  goes  to  build  up 
crystals  of  Hoar-lrost  instead  of  liquid  drops.  Observation  of  a 
morning's  hoar-frost  ought  to  give  a  pretty  good  idea  of  the  relative 
efficiency  of  various  objects  in  usually  condensing  dew. 

Incidentally,  Kadiation  all  night  long  ensures  that  '  the  coldest 
hour  is  just  before  the  dawn,'  a  fact  which  his  night  work  impresses 
on  the  G.P.,  both  as  regards  his  patients'  welfare  and  his  own 
comfort.  Cloudy  nights,  with  little  radiation,  are  more  likely  to 
keep  a  steady  temperature  throughout.  The  impression  one  some- 
times gets,  of  increasing  warmth  before  the  dawn,  has  no  meteoro- 
logical foundation,  and  appears  to  be  purely  physiological.  , 

§  315.  Altitude  and  temperature.  We  live  in  the  depths  of  a  great 
ocean  of  air,  and  on  every  square  inch  rests  a  column  of  that  elastic 
fiuid  nearly  15  lb.  in  weight.  Climbing  a  liill,  we  climb  above  the 
lower  layers  and  are  relieved  of  their  weight ;  the  atmospheric 
pressure  is  less  at  the  height.  As  an  ordinary  partly-filled  balloon 
rises,  the  gas  expands  and  fills  it.  Liliewise,  if  a  quantity  of  air  is 
rising,  it  expands  ;  a  little  square-inch  column  of  it,  1  ft.  high  by  the 
loch  shore,  would  be  14  in.  high  on  the  top  of  Ben  Nevis,  §  119,  Fig.  40. 
As  it  expands  it  does  work ;  for  imagine  it  enclosed  in  a  tube,  it 
would  drive  a  sliding  cork  outwards,  against  the  remaining  atmo- 
spheric pressure  ;  the  little  column  would  have  forced  back  an  average 
of  13 J  lb.  through  2  in.,  2 J  ft. -lb.  of  work — ^no  mean  effort  for  a  little 
fellow  weighing  4  grains.     Hence  it  cools,   §  291. 

The  ultimate  result  is  that  the  Atmosphere  '  in  convective  equili- 
brium '  shows  a  decrease  in  temperature  upwards,  a  Lapse  Bate;, 
averaging  0-6  C.  in  100  m.  or  1°  F.  in  100  yd.  of  ascent. 


I 


§316] 


METEOROLOGY 


235 


Thus  the  transparent  air  is  really  stratified  by  its  temperature  into 
layers  which  are  quite  impervious  to  air  rising  from  below,  unless 
it  has  the  temperature  necessary  to  furnish  the  key  to  get  through. 
But  the  air  from  the  sun-warmed  moist  land  or  sea,  Hghtened  by  its 
warmth  and  moisture  (for  water  vapour  is  only  5/8  as  dense  as  air), 
manages  it  quite  easily ;  it  climbs  to  higher  things  on  stepping-stones 
of  its  own  dead  water  vapour.  For  presently  it  passes  its  dew 
point,  condenses  part  of  its  moisture,  and  takes  over  the  heat  of 
vaporization  previously  latent,  but  now  set  free.  Fortified  there- 
with, it  surges  upward,  sometimes  with  rush  enough  to  carry  up 
huge  hailstones,  until  it  meets  its  match  in  an  environment  which 
makes  nothing  of  its  unusually  slow  fall  of  temperature — being  quite 
dry  and  having  no  fall  at  all  with  height.  This  is  the  Stratosphere, 
8  km.  high  above  the  poles,  about  10  above  us,  rising  to  17  km.  in 
the  tropics  :  into  it  not  even  the 
wettest  air  can  penetrate,  it  is  im- 
possible and  impassable ;  it  has 
nothing  directly  to  do  with  weather, 
we  shall  stay  below  in  the  Tropo- 
sphere. 

§  316.  Above  the  dew  point,  the 
unsaturated  vapour  in  the  ascending 
column  is  just  as  good  a  gas  as  the 
surrounding  air,  consequently  con- 
densation occurs,  on  the  nuclei,  at 
a  definite  level  agreed  upon  by  all 
the  atmosphere  ;  the  column  wears  a 
cap  of  Cloud,  perfectly  flat  beneath, 
but  puffed  out  aloft,  the  little 
Cumulus  that  floats  so  gaily  through 
the  noonday  azure  of  true  English 
June. 

In  the  Horse  Latitudes  and  the  Doldrums,  with  a  fiery  tropic  sun 
distilhng  the  upper  layers  of  the  calm  sea  at  25°  C,  these  cumuU 
swarm,  not  now  little  puffs,  but  towering  pillars  of  cloud.  Fig.  97, 
their  upcast  engined  by  the  latent  heat  of  thrice  the  vapour  of 
latitude  50°.  Some  of  them  stand  on  a  dark  stalk  of  Rain,  many  of 
them  light  up  at  night  with  quiet  flashes,  electrical  puppies,  mostly 
too  little  to  growl,  but  broadcasting  the  most  part  of  the  atmospherics 
that  trouble  our  wireless  in  England. 

In  §  350  it  will  be  shown  that  the  smaller  drops  of  a  cloud  evaporate 
and  deposit  on  the  larger  ones,  and  for  the  same  reason  fresh  supplies 
of  vapour  condense  on  existing  droplets  rather  than  form  new  ones  : 
from  this  and  other  causes,  possibly  electrical,  the  drop  size  in  the 
cloud  naturally  increases,  the  whole  cloud  containing  from  2  to 
even  5  gm.  of  water  per  cubic  metre.  A  fresh-formed  cloud  droplet 
has  a  diameter  of  about  0-02  mm.  (1/1250  in.),  and  falls  through 
13  mm.,  half  an  inch,  per  second ;  0-04  mm.  drops  fall  at  100  mm./ 


Fig.  97. 


236  HEAT  [§  316 

sec.,  0-15  mm.  at  1  m./sec,  a  1  mm.  Raindrop  falls  at  4m. /sec,  and 
they  rapidly  approach  a  Maximum  Speed  of  Fall  of  8m. /sec.  and  a 
Maximum  Size  of  5-5  mm.  diameter.  Beyond  this  the  spherical  drop 
flattens  and  shatters,  as  can  be  seen  happening  on  the  face  of  a  high 
waterfall ;  not  even  Niagara  can  fall  faster  through  the  surrounding 
air  than  this. 

This  means  that,  sooner  or  later,  a  well-fed  cloud  falls  faster  than 
the  upcast  current,  in  Rain.  Unless,  that  is,  the  uprush  is  faster 
than  8  m./sec.  All  water  condensed  in  such  an  upheaval — and  it 
will  be  abundance — must  be  carried  up  until  the  velocity  falls  below 
8  m./sec.,  as  it  is  bound  to  do  at  some  height,  by  lateral  spreading. 
Here  the  water  accumulates  in  large  amounts,  and  when  the  ascend- 
ing current  slackens,  or  is  deflected,  down  comes  the  lot — a  '  Cloud- 
burst.' 

On  the  other  hand,  a  general  slow  updrift  of  moist  air  over 
hundreds  of  square  miles  produces  those  thin  layers  of  cloud  which 
may  drizzle  hour  after  hour.  Turbulence,  or  else  higher  cross 
current,  breaks  larger  clouds  into  rolls  and  flocks. 

The  thin  dark  rain-cloud  (Nimbus)  is  often  only  J  mile  high,  the 
quiet  streaks  of  Stratus  in  the  sunset  may  be  lower  still.  Rolled 
Strato-cumulus  and  puffed-up  Cumulus  may  be  a  mile  up,  the  great 
Stormcloud  is  piled  from  there  2  or  3  miles  upward.  The  flocky 
Alto-cumulus  and  hazy  Alto-stratus  lie  about  2J  miles,  speckly 
Cirro-cumulus,  halo-producing  Cirro-stratus,  and  highest  feathery 
Cirrus,  from  4  to  7  miles,  all  three  being  a  dust  of  ice  crystals. 

§  317.  Apparently  the  tearing  to  pieces  of  water-drops  by  violently 
uprushing  air  causes  the  wide  separation  of  electrical  charges,  and 
the  result  is  a  thunder-cloud,  a  discussion  of  the  electrical  aspect  of 
which  must  be  deferred  until  §  897. 

With  us,  the  great  towering  storm-cloud,  a  mile  overhead  and  3  or 
even  6  miles  high,  is  the  product  of  hot  weather  and  plenty  of 
surface  moisture,  especially  in  early  summer,  when  the  upper  air 
still  remains  cold  and  heavy,  and  the  light  warm  wet  mixture  surges 
up  through  it  at  great  speed  :  in  dry  seasons  thunder-storms  fail  for 
lack  of  water. 

Atmospheric  upcasts,  sometimes  not  cloud-capped  because  not 
wet  enough,  are  the  support  of  the  Glider,  who,  of  course,  rejoices 
greatly  in  a  hefty  stormcloud.  Often  it  comes  up  against  the  wind  ; 
really  the  warm  wind  is  rushing  into  and  up  the  near  side  of  it, 
building,  from  its  contained  vapour,  new  cloud  closer  to  us  :  under 
its  tread  the  weathercocks  go  crazy,  as  the  air  drifts  from  all  sides 
to  reach  the  mighty  funnel. 

After  all,  the  biggest  of  clouds  is  only  a  symptom  of  some  goings-on 
in  the  atmosphere. 

Heavy  rain  tells  of  rapid  updraught ;  much  more  so  does  the 
Hail  on  the  skirts  of  the  storm.  Hailstones  are  formed  when  water- 
drops  are  blown  up  to  heights  where  the  temperature  is  below 
—  20°  C,  at  which  the  supercooled  water  which  forms  the  upper 


§319] 


METEOROLOGY 


237 


part  of  tall  storm-clouds  most  certainly  freezes.  Flung  out  at  the 
top,  they  fall  round  the  funnel,  and  are  sometimes  dragged  in  and 
blown  up  again,  gathering  up  the  supercooled  drops  as  clear  ice,  and 
gaining  a  concentric  structure  and  an  astonishing  size,  so  that  their 
free  fall  at  last,  possibly  a  mile  or  more  from  the  centre,  may  be  the 
worst  catastrophe  of  the  storm. 

The  finer  material  flung  out  at  the  top,  which  congeals  to  snow- 
dust,  often  forms  a  fibrous  cloud  crest,  of  '  anvil '  shape,  which  may 
persist  for  some  time  after  the  storm  has  died  out. 

§  318.  Soft  Hail  is  a  feeble  imitation  produced  by  the  cooler 
quieter  updraughts  of  winter  :   it  lacks  the  clear  hard  envelope. 

Snow  probably  forms  in  the  ascending  current  by  condensation 
below  the  freezing  point  direct  into  the  solid  form,  and  grows  into 
hexagonal  skeleton-crystals  of  most  varied  patterns.  In  extreme 
cold  the  growth  is  little,  for  lack  of  material,  and  the  snow  remains 
almost  dusty ;  when  formed  near  the  freezing  point,  the  crystals 
grow  large  and  straggling,  collide  in  the  air,  get  entangled  and  welded 
together  by  regelation,  and  form  the  familiar  large  snowflakes. 

Roughly,  a  foot  of  fresh  snow  is  equivalent  to  an  inch  of  rain. 

Snow  is  a  good  blanket  for  vegetation,  but  it  reflects  the  greater 
part  of  direct  solar  heat,  and  at  the  same  time  radiates  low-temper- 
ature heat  freely  ;  thus  being  hard  to  get  rid  of. 


^2 

-  -  -~"z 

— 

■~c 

%  ,        ^"^ 

- — 

— 



^ri^f',: 

— 

\  :^ 

«v    — 

-^"~ 

si; 

S 



—           

_^ 

■^ 

=^^^ 

^^^ 



Fig.  98. 


§  319.  Fig.  98  shows  another  familiar  way  in  which  moist  air  is 
lifted  to  form  cloud.  You  are  looking  south  on  the  hills  of  Hoy, 
in  the  Orkneys,  Aug.  1st,  6  p.m.  ;  the  warm  west  wind  blowing  in 
from  the  Atlantic,  and  compelled  to  rise  up  the  hillside,  formed  the 
caps  of  cloud  seen  on  the  right,  perfectly  fixed  in  shape,  though 
vapour  was  wreathing  up  through  them  at  a  great  pace.  In  the 
gap  the  air  could  sink  again,  and  the  clouds  thinned  out,  to  form 
again  in  rolling  masses  on  the  second  hill.  Doubtless  some  of  its 
moisture  was  precipitated  in  Scotch  mist  on  the  first  ascent  :  the 
Selkirks  rob  the  Pacific  winds  of  so  much  of  their  moisture  that  the 
Rockies,  over  which  they  lift  again,  are  clear  and  dry  and  rocky ; 
and  the  prairie  beyond  goes  very  short  of  rain. 


238 


HEAT 


[§319 


The  stretch  of  water  in  front  has  since  become  famous  as  Scapa 
Flow.  In  Fig.  99  the  Rock  of  more  ancient  fame  is  withstanding  the 
Levanter  (part  of  which  probably  found  its  way  up  the  Adriatic  as  a 


Fig.  99. 

sticky  Sirocco),  blowing  warm  and  wet  the  length  of  the  Mediter- 
ranean, flung  aloft  over  the  high  brow,  and  trailing  a  magnificent 
plume  of  cloud  away  to  the  west. 

§  320.  Wind. 

*  Come  from  the  four  winds,  O  breath,  and  breathe  upon  these, 
that  they  may  Kve.' 


The  air  at  the  top  of  these  hills  was  warmer  than  is  proper  at  that 
altitude,  by  reason  of  the  latent  heat  given  up  to  it  by  the  condensed 
moisture.  If,  now,  this  latter  falls  out  on  the  hilltops  as  rain  or 
snow,  the  descending  wind  does  not  have  to  re-evaporate  it  on  the 
way  down,  while  the  reverse  process  of  §  315  is  superheating  the  air ; 
so  that  it  blows  now  as  a  warm  dry  wind,  the  Fohn  of  the  Swiss 
valleys,  or  the  sudden  dry  Chinook  which  rolls  down  from  the  Rockies, 
melts  the  snow  from  the  face  of  the  prairie,  and  is  hailed  as  harbinger 
of  Spring. 

On  the  other  hand,  masses  of  air,  chilled  by  long  sojourn  on  the 
height  of  land  among  the  summits,  sooner  or  later  topple  off,  and 
roll  down  the  valleys  ;  the  sudden  storms  that  break  from  the  gorges 
on  the  eastern  shore  of  Gahlee ;  the  bitter  N.E.  Bora  which  keeps 
northern  Dalmatia  a  stony  wilderness  and  shrivels  Venice  with  its 
icy  breath  at  New  Year's ;  the  winter  Mistral  singing  down  the 
Rhone  valley  and  '  strewing  the  Riviera  with  the  bones  of  EngUsh- 
men  seeking  sunshine.' 

Winds  like  these  are,  of  course,  very  local ;  less  so  are  Land  and  Sea 
Breezes.  The  sea  is  usually  disturbed  and  mixed  to  a  depth  of  several 
feet,  consequently  the  sun  does  not  heat  its  surface  by  day  as  hot  as 
it  does  the  land,  and  by  night  the  sea  remains  warmer  than  the 
rapidly  radiating  land.  This  gives  rise  to  a  Sea  breeze  by  day, 
flowing  in  to  supply  the  place  of  rising  hot  air  over  the  land,  and  to  a 
Land  breeze  by  might,  off  the  cool  land  on  to  the  warmer  sea. 


§  321]  METEOROLOGY  239 

Though  seldom  noticeable  in  this  country,  these  are  of  regular 
occurrence  in  the  tropics,  where  radiation  is  intense  and  barometric 
changes  are  trifling. 

The  Doctor  springs  up  and  ruffles  the  water  of  the  harbour  and 
makes  the  town  bearable  of  an  afternoon,  and  at  night  the  Under- 
taker rolls  down  from  the  Blue  Mountain,  through  your  bedroom 
window  set  open  to  face  it,  and  brings  you  the  mercy  of  sleep. 

More  important  to  us  is  the  larger  slower  action  of  the  sea  on  our 
island  climate.  The  supply  of  solar  heat  is  at  its  maximum  at  the 
summer  solstice,  when  the  sun  is  highest  and  shines  on  us  longest, 
but  the  turbulent  sea,  with  its  great  capacity  for  heat,  holds  down 
the  land  heat  of  June,  while  it  itself  goes  on  accumulating  heat,  and 
reaches  its  maximum  temperature  in  mid-August,  the  modem 
inducement  to  school  holiday- making  then  (the  older  being  helping  in 
the  harvest-field) .  The  climate  compromises,  in  English  fashion,  and 
has  its  land  maximum  temperature  about  July  20th.  Thus  the  Sea 
moderates  our  island  seasonal  temperatures,  and  causes  about  a 
month's  lag  behind  the  Sun,  all  the  year  round. 

A  still  larger  effect  analogous  to  land  and  sea  breezes  is  this. 
As  the  sun  rapidly  goes  south  at  the  equinox,  continental  land  masses 
cool  faster  than  the  sea,  which  has  been  cooler  than  they  all  summer, 
but  is  now  warmer.  Therefore  a  wide- spreading  land  breeze  may  be 
expected — the  Equinoctial  Gales.  This,  however,  may  be  taken 
with  a  grain  of  salt :  the  fact  is,  that  our  winter  half  of  the  year  is 
abruptly  and  immensely  windier  than  the  summer,  the  first  warning 
of  its  coming  being  winds  which  refuse  to  sink  at  night. 

The  Indian  Monsoons  are  the  resultants  of  gigantic  seasonal 
land  and  sea  breezes  from  the  land  masses  around  the  Indian  Ocean, 
as  the  sun  moves  from  tropic  to  tropic,  and  of  the  Trade  Winds. 

The  Trade  Winds  open  a  mightier  story — that  of  the  general 
circulation  of  the  Atmosphere.  That  we  will  roughly  outline,  and 
then  confine  ourselves  to  the  Atlantic. 

§  321.  Four-fifths  round  the  Equator — the  rest  is  land — stretches 
that  belt  of  calms,  some  6°  wide,  the  Doldrums,  to  which  the 
Ancient  Mariner  so  much  objected.  Forty  North  Seas  in  area, 
at  28°  C.  instead  of  8°  (think  of  swimming  all  the  year  round  at 
80°  F.),  i.e.  distilling  vapour  3  times  as  fast,  the  Sun  beats  down  upon 
it  with  perpetual  power  (which  you  can  calculate  from  §  976) 
5  X  10^2  kilowatts,  or  7  billion  h.p.,  raising  vapour  which,  only  5/8  as 
dense  as  air,  heaves  all  up  together.  Presently  condensing  into  cloud, 
then  falling  as  the  equatorial  rains,  this  vapour  hands  all  its  latent 
heat  and  power  to  the  circumambient  air,  such  a  leg-up  that  the  inert 
Stratosphere  is  pushed  17  km.  aloft,  though  the  great  mass  of  the 
air  rises  only  4  or  5  km.,  and  then  divides  and  drifts  off  N.  and  S., 
driven  away  by  the  continued  upheaval,  Fig.  100. 

Arrived  at  about  30°  N.  or  S.  latitude — the  actual  figure  is  a  good 


240  HEAT  [§  321 

deal  pushed  about  by  the  sun  in  his  annual  migration,  getting  less  as 
he  goes  away — it  sinks  again,  from  a  dry  blue  sky  (because  it  warms 
by  compression  as  it  comes  down,  §  320,  and  Fig.  86),  and  forms  those 
belts  of  calms  where  horses  had  to  be  thrown  overboard  when  the 
regimental  water-supply  ran  short.  Its  warmth,  and  the  clear 
sunshine,  keep  the  sea-surface  at  25°  C,  and  rapid  distillation  goes 
on,  raising  vapour  which  partly  condenses  into  columns  of  clouds 
like  Fig.  97,  of  which  dozens  dot  the  sky — shaggy  spaniels  sitting  up 
to  beg — you  see  their  flat  bases  fading  into  distance  in  the  figure. 

From  the  southern  edge  of  the 
belt  (in  the  north  hemisphere) 
these  drift  off  south,  borne  by  the 
wind  which  is  going  to  supply 
the  equatorial  upheaval :  from  the 
northern  edge  they  march  to  the 
colder  north. 

Not  due  S.  and  N.,  however,  for 
here  comes  in  the  Rotation  of  the 
Earth,  Fig.  100,  which  carries 
latitude  30°  towards  the  E.  at  910 
m.p.h.,  but  drives  the  equator  round 
at  1040  m.p.h.,  i.e.  the  equator 
runs  into  the  N.  wind  from  lat. 
Fig.  100.  30°  N.,  at  130  m.p.h.  to  the  east : 

a  lot  of  this  difference  is  lost  on 
the  way  by  friction,  but  it  results  that  the  wind  arrives  from  the  N.E. 

Strong  and  steady  and  warm  and  soft,  bringing  no  chill  to  a  bare 
skin,  yet  always  coming  somewhere  warmer,  with  nothing  cold  to 
weep  over,  flecking  the  indigo  sea  with  brave  white  horses,  or  chant- 
ing a  hearty  antiphon  through  the  aisles  of  the  church  that  is  set  on 
a  hill,  these  are  the  Trade  Winds,  les  Vents  AUzes,  the  Winds  of 
Dehght. 

§  322.  The  South  wind  is  headed  for  our  latitudes,  where  London  is 
trundling  along  at  650  m.p.h.  east,  or  to  60°  N.,  where  the  speed  is 
half  the  equatorial  (1040  cosine  lat.),  and  it  therefore  over-runs  the 
earth,  and  blows  with  an  increasing  westerly  component,  the 
South-westerly  and  Westerly  winds  of  the  N.  Atlantic. 

Warm  and  moist,  they  have  come  to  cooler  regions,  and  condensa- 
tion increases,  the  clouds  grow  and  spread,  until,  too  soon,  they  cover 
the  sky  ;  there  may  be  local  fog,  §  313.  The  Trades  had  no  opposi- 
tion, and  blew  steady ;  but  the  South-westerUes  have  a  hand-to- 
hand  struggle  with  the  heavy  horsemen  of  the  frozen  North,  the 
cold  dry  winds  which  cannot  lift  from  the  surface.  Hence  the  Vari- 
able South-westerly  and  Westerly  winds  and  cloudy  skies  which  we 
dub  Atlantic  Weather.  On  the  whole,  they  have  their  way,  for, 
literally,  they  have  steam  up,  and  they  can  over-ride  opposition  with 
their  200 — 400  m.p.h.  excess  eastward  momentum,  as  their  path 
drifts  farther  north. 


§  323]  METEOROLOGY  241 

§  323.  Consider  the  Atlantic  circuit  which  starts  about  Ber- 
muda :  get  your  atlas  and  trace  it,  for  to  us  it  is  one  of  the  most 
important  things  in  the  world. 

Beneath  it  spreads  the  Gulf  Stream,  laden,  they  say,  with 
0-4  X  1020  calories  per  day,  tracking  N.E.  for  the  same  geo- 
dynamical  reason ;  but  already  sadly  slowed  down,  for  water- 
friction  is  serious,  it  is  going  to  take  a  year  or  more  to  get  across. 
But  let  its  15°  warmer  water  take  wings  of  vapour,  and  1  gm.  shall 
carry  the  calories  of  40,  and  then  travel  50  times  as  fast,  in  the 
wind.  So  the  South-westerUes  pick  up  that  contribution  as  well ; 
and  we  can  regard  with  equanimity  the  proposal  of  the  anglophobe 
American  to  dam  the  Florida  Channel  and  freeze  us  out ;  it  might 
make  some  difference  to  us,  but  it  would  be  a  direct  invitation  to 
every  hurricane  in  the  Caribbean  to  go  ashore  in  the  southern  States. 

The  track  sweeps  on,  12°  wide  or  so,  its  mid-hne  crossing  these 
islands  at  their  thinnest,  56°  N.,  entering  the  Skager  Rak  (a  dis- 
tributary passing  up  the  length  of  the  high  Norwegian  coast),  then 
over  rough  country  and  E.  across  the  Baltic.  By  this  time  the 
bulk  of  the  moisture  has  been  dropped,  Fig.  98,  §  319,  any  remainder 
is  left  on  the  cold  plains  of  Russia. 

Now  here  comes  in  another  point  :  this  circulation  of  air  is  shallow, 
it  came  dovm  in  lat.  -30°,  seldom  is  it  2  miles  thick :  the  Daily 
Weather  Map  is  not  half  as  thick  as  the  paper  it  is  printed  on. 

Opposed  in  the  N.  and  E.,  it  turns  S.,  and  sweeps  down  across 
the  Black  Sea  :  forced  to  drift  westerly  as  it  goes  south,  as  were 
the  Trades,  unable  to  drop  on  hot  plains  the  little  moisture  picked 
up  crossing  the  narrow  Levant,  it  sweeps  on  towards  the  S.W.  and 
W.,  over  the  Libyan  desert  and  the  Sahara,  maintaining  their 
aridity,  until  it  blends  in  the  trade  winds  off  the  coast. 

Do  not  think  the  wind  blows  at  its  ease  in  this  track ;  in  all  the 
northern  part  it  has  to  struggle  with  the  polar  air,  in  those  counter- 
clockwise whirls,  running  like  balls  in  a  bearing,  the  *  cyclonic 
depressions,'  to  be  described  shortly. 

The  importance  of  this  circulation  is  this  :  We  live  here  10°  and 
more  north  of  a  respectable  winter  sun.  We  grumble  that  we  see 
him  so  seldom,  but  how  strong  is  he  when  we  do  ?  He  sneaks 
low  down  across  the  southern  sky,  and  leaves  us  in  darkness  two- 
thirds  of  the  time.  When  he  alone  takes  charge,  as  in  clear  calm 
winter  weather,  we  freeze  up  soUd.  So  Nature  has  laid  on  for  us 
her  very  finest  steam  heating  system,  and  on  it  we  depend  for  half 
our  warmth  in  winter.  Vapour  brings  us  sun's  heat  caught  in  sub- 
tropical seas.  To  give  us  its  great  latent  heat  it  must  condense, 
its  clouds  are  our  thermogenic  blankets ;  if  our  local  sun  hasn't 
strength  to  Hft  them,  let  us  keep  tucked  in ;  if  we  want  more 
blankets  we  have  a  complete  river-system  for  carrying  away  the 
used-up  ones. 

And  in  Sujnmer,  at  convenient  times,  we  want  Rain. 

We  have  seen  why  it  rains  on  barren  hills.  Fig.  98,  but  what 
brings  rain  down  on  the  crops  in  our  flat  countryside  ?  Answer, 
the  Conflicts  of  the  South -westerlies  on  the  Polar  Front. 


242  HEAT  [§  324 

§  324.  In  the  cold  Polar  Regions  air  lies  heavy,  over  a  larger 
area,  of  course,  in  the  long  sunless  winter,  but  it  cannot  lie  at  rest. 
For  look  at  Fig.  100,  or,  better,  at  a  globe,  the  Arctic  Circle  rings 
round  a  turntable  not  very  far  from  flat,  and  off  that  turntable 
centrifugal  force  is  perpetually  flinging  the  air.  If  in  doubt  about 
it,  try  one  of  them  at  an  amusement  park. 

Tromso,  70°  N.  on  Lyngenfjord,  is  a  delightful  little  town,  but 
not  for  a  Christmas  holiday  :  incessant  north  wind,  say  the  B.  & 
N.  Line.  Working  out  mv'^jr  for  this  20°  radius  from  the  pole 
[strictly  {mv^jr)  x  sine  latitude,  for  the  tangential  force]  gives  a 
centrifugal  force  I'l  djrnes  per  gm.  :  against  30  miles  of  such  a 
wind  coming  across  the  flats  you  could  not  stand.  At  60°  N.  the 
force  is  1-5  dynes. 

The  result  is  a  jagged  '  coast  line  of  cold  air,'  called  the  Polar 
Front,  from  which  long  promontories  must  trail  away  down  south, 
like  your  hand  outspread  on  top  of  the  globe.  Or  like  custard- 
sauce  streaking  down  over  a  rotating  plum-pudding ;  and  as  the 
earth  under-runs  them  E.  as  they  travel  S.,  they  all  blow  as  N.E. 
winds. 

Into  the  bays  of  this  front  surge  the  South- westerlies,  and  the 
struggle  of  Fig.  101  is  joined.  They  cannot  flght  level,  for  the 
heavy  cold  air  clings  to  the  ground,  and  the  light  warm  wet  air 
glides  up  over  it,  like  water  washing  at  sand,  at  a  slope  greatly 
exaggerated  in  the  Warm  Front  Section  below  on  the  right,  really 
about  1  in  100  :   these  things  are  flat  and  thin. 

As  it  lifts  it  must  cool,  §§  119,  291,  315,  and  its  moisture  con- 
denses in  cloud.  Three  usual  layers  appear  in  the  diagram,  the 
high  flecks  and  streaks  of  cirrus,  which  herald  the  coming  change 
the  day  before,  the  middle  alto-stratus,  screening  the  sun  or  moon 
and  perhaps  beginning  a  drizzle,  and  the  low  mass  of  nimbus  with 
steady  and  abundant  rain. 

The  foot  of  the  slope,  where  the  wind  that  has  come  from  so  far 
is  just  sweeping  the  cold  away,  is  the  Warm  Front  (ground  lines 
are  solid  in  the  plan). 

The  uplifting  of  the  hght  air  means  a  removal  of  pressure,  a 
cyclonic  depression,  or  Low,  of  the  Barometer,  a  partial  vacuum, 
and  the  polar  wind  on  the  west  draws  round  from  its  south-westerly 
course  and  joins  in  the  scrum.  It  is  heavy,  and  rolls  along  the 
ground  as  a  steep  Cold  Front  (left-hand  Section),  tossing  high 
overhead  the  light  wet  S.W.  air,  the  moisture  of  which  condenses 
now  suddenly  in  broken  lumps  of  cloud,  driven  along  from  the 
N.W.  in  a  dry  blue  sky,  and  coming  down  in  heavy  Clearing  Showers, 
while  the  sheer  weight  of  cold  air  behind  this  '  Squall  Line  '  drives 
up  the  barometer,  and  deceives  the  townsman  who  trusts  to  it  too 
blindly. 

Ultimately  it  joins  its  own  tail,  encircles  and  '  secludes  '  the  warm 
air.  Fig.  102,  plan  and  section,  and  finally  lifts  it  clear  off  the 
ground,  '  occludes  '  it ;  so  that  the  whole  is  like  the  section  below 
Fig.  103,  and  lifts,  and  dries  out ;    '  the  depression  fills  up.'     Many 


§324] 


METEOROLOGY 


243 


depressions  are  already  in  the  '  secluded  '  condition  when  they 
reach  our  shores,  and  then  they  resemble  Fig.  103,  which  was  all 
we  knew  of  them  in  1910.  You  may  still  use  it  as  a  fair  guide  in 
forecasting  weather  for  yourself  :    these  depressions  (the  Isobars 


Fig.  101. 


^ 

^— 

WARn 

y 

.., 

-lir^ 

" 

.^_ _^ 

^        °        ^       " 

OCC  LU  0  6  D 


Fig.  102. 


Fig.  103. 


of  equal  pressure  of  this  are  spaced  in  tenths  of  an  inch  of  mercury) 
last  from  1  to  5  days,  and  are  usually  blown  along  by  the  dominant 
south- westerlies,  towards  the  E.  or  N.E.,  at  speeds  up  to  20  m.p.h. 
The  sequence  of  events,  when  one  happens  to  pass  just  N.  of  you, 
going  E.,  you  can  figure  out  by  supposing  it  fixed,  and  walking  W. 


244  HEAT  [§  324 

from  a  to  e  across  it  (the  line  a  e  is  similarly  lettered  in  the  more 
modem  diagram,  Fig.  101)  : 

(a)  High  cirrus  appearing  from  S.W.,  barometer  beginning  to 
fall,  calm  or  light  southerly  airs,  air  often  very  clear,  under 
growing  cloud. 

(6)  Barometer  falling  rapidly,  wind  stronger  S.E.  or  S.,  warm 
in  winter,  cold  in  summer  ;   cloudy  and  wet. 

(c)  Wind  veering  toward  S.W.,  strong ;    continuous  rain. 

{d)  Veering  rapidly  N.W.,  strong;  rain  breaking  into  smart 
showers  at  increasmg  intervals.  Barometer  rising,  but  take 
your  mac. 

(e)  Clear  atmosphere,  barometer  rising  briskly,  cold  N.W.  wind 
gradually  dying  down,  driving  small  clouds  in  a  blue  sky ;  hot 
sunshine. 

The  number  of  feathers  on  the  wind-arrows  indicates  the  strength 
of  the  wind  on  the  '  Beaufort  scale.' 

When  a  large  depression  passes  centrally  overhead,  an  easterly 
gale  may  be  succeeded  by  a  calm  12  hr. — wet,  or  fine  and  warm 
— and  then  follows  an  equally  strong  westerly  gale.  For  North  of 
the  centre,  walk  westward  along  the  line  of  latitude  in  Fig.  103. 

Fig.  104  shows  a  deep  Depression,  Nov.  10th,  1931.  It  was  a 
warm  day  of  cloud,  except  for  brief  glimpses  of  sunshine  in  the 
Eastern  Counties.  Scotland,  in  the  centre  of  the  depression, 
remained  dry,  dull,  and  almost  calm  ;  drizzle  and  rain  increased 
southward,  the  south  coasts  of  Wales  and  England,  the  Channel, 
and  the  north  of  France  being  swept  by  gales,  accompanied  by 
thunderstorms  and  an  inch  of  rain.  The  depression  continuously 
moved  off  N.E.  and  filled  up,  and  the  strong  winds  moderated. 

Depressions  may  come  singly,  or  in  successions  up  to  four,  giving 
us  a  fortnight's  '  unsettled  weather.' 

As  a  spinning-top  on  the  pavement,  while  intensely  energetic, 
has  no  particular  '  local  habitation,'  and  is  easily  drifted  about ; 
so  these  thin  flat  whirligigs  are  sensitive  to  terrestrial  obstructions 
of  no  stupendous  height,  and  the  centres  of  all  but  the  largest  prefer 
paths  of  least  resistance ;  up  past  the  Hebrides  and  along  the 
Norway  coast,  unable  to  go  ashore,  is  a  favourite  summer  route. 
In  winter,  the  advance  of  the  polar  front  and  the  retreat  of  the  sun 
drive  them  lower  down,  across  the  Scottish  lowlands,  or  up  St. 
George's  Channel,  or  the  EngHsh  Channel  into  Flanders,  or  into 
the  Bay  and  over  France,  when  London,  being  N.  of  them,  doesn't 
know  what  to  make  of  the  sequence  of  weather.  If  they  take  these 
tracks  in  summer,  the  southerner,  unused  to  Hebridean  skies,  #► 
considers  himself  very  badly  treated. 

Coming  to  extremes,  the  ScilUes  suffer  many  little  depressions 
that  are  refused  a  landing  by  the  cliffs  of  Cornwall ;  and  the  intense 
cyclonic  hurricanes  of  the  Caribbean  are  seldom  able  to  approach 
the  sizeable  turtle-back  mass  of  Jamaica. 


§  324]  METEOROLOGY  245 

A  frequent  complication  occurs  when  the  N.W.  current  drives 
quickly  across  the  S.W.  and  cuts  off  a  tail,  which  forms  a  '  Secon- 
dary,' and  always  weeps  many  gusty  tears  for  its  mother. 

Very  occasionally  the  turbulent  clouds  over  the  Cold  Front  join 
in  one  long  roll  of  cloud,  which  may  stretch  from  horizon  to  horizon, 
and  beyond,  and  may  last  12  hr. ;  the  Squall  Line  has  bred  a 
Line  Squall,  the  ace  of  spades  and  joker  of  the  meteorological  pack 
rolled  into  one.  Driving  uphill  near  Lyme  Regis,  I  have  had  to 
relinquish  the  wheel  with  numbed  hands,  while,  250  miles  N.E., 
farmers  attending  market  at  Lynn  Regis  spent  the  night  there  for 
the  first  time  in  their  lives  ;  and  I  have  dodged  one  in  the  narrow 
seas,  only  to  land  and  find  ducks  swimming  in  the  stackyard,  and 
furniture  floating  in  the  chapel. 

The  Weather  Office  calls  this  squall  line  region  a  Trough,  and 
when  that  expression  appears  in  the  Forecast  you  may  look  out  for 
the  massive  clouds  of  §  317,  sometimes  with  '  anvils  '  spreading 
high  above  them  like  the  horsehair  crest  of  a  Homeric  helmet. 
Such  clouds,  drifting  away  east  at  the  close  of  day,  dark  themselves, 
but  with  this  vast  upflung  explosion  flaming  in  the  rays  of  the 
setting  sun,  are  a  gorgeous  spectacle ;  but  if  you  find  one  bearing 
down  on  you,  do  not  emulate  the  defiance  of  Ajax  the  son  of  Oileus, 
but  reflect  rather  that  here  comes  a  potential  million  horse-power 
of  mischief,  and — be  near  to  cover. 

Figures  computed  from  an  ordinary  Depression,  800  miles 
diameter,  and  only  a  third  of  a  barometric  inch  deep,  which  hap- 
pened to  form  over  the  North  Sea,  and  stay  there  and  work  itself 
out,  filling  up  in  a  week,  may  be  of  interest :  70,000  million  tons  of 
air  were  removed  to  make  it ;  and  700  million  tons  of  water  vapour 
provided,  by  their  condensation,  an  average  4000  million  h.p. 
for  the  week,  the  kinetic  energy  of  the  wind  at  any  time  being  about 
a  tenth  of  this. 

This  Polar  Front  Theory  comes  to  us  from  Norway,  and  has 
found  acceptance  as  explaining  more  than  did  the  older  upcast 
theory  (below)  on  which  Fig.  103  may  be  regarded  as  based. 
Probably,  however,  it  is  a  case  for  the  comment  of  §  38,  for  quite 
likely  it  over-emphasizes  the  part  played  by  cold,  just  as  James 
Watt  was  so  preoccupied  with  the  condenser  of  his  steam-engines 
that  he  would  never  allow  more  than  7  lb.  steam  pressure  :  after 
all,  it  is  the  heat  that  does  the  work. 

It  cannot  imaginably  have  anything  to  do  with  the  cyclonic 
hurricanes,  tornadoes,  etc.,  that  arise  in  the  tropics,  within  the 
high-pressure  dividing  wall  of  downcast  air  about  30°,  and  only 
occasionally  break  through  it.  Undoubtedly,  here,  the  lower  levels 
of  air  near  the  warm  sea  are  full  of  water  vapour,  and  therefore  very 
absorbent  of  solar  radiation,  §  311,  whereby  they  become  unduly 
heated,  expanded  and  lightened,  so  that  the  drier,  and  therefore 
colder  and  comparatively  heavier,  upper  layers,  lie  quite  unstably 
on   top   of  them.     Somewhere   the   arrangement  capsizes,   and   a 


246  HEAT  [§  324 

great  local  upcast  of  warm  wet  air  ensues,  an  enlarged  storm-cloud 
system,  and  the  '  draught '  rushing  into  it  from  all  directions  takes 
on  the  circular  swirl,  anti- clockwise  in  the  northern  hemisphere, 
for  the  reasons  already  gone  into. 

If  these  seem  to  you  too  small  a  cause  to  initiate  a  circulation 
which  goes  on  to  gain  enormous  energy  from  the  power  system, 
try  to  get  the  water  in  a  round  basin  so  still  that  it  will  all  run  out 
without  starting  a  whirlpool  when  you  pull  out  the  plug. 

Hurricanes  are  objectionable  not  only  on  account  of  the  wind, 
which  dies  down  and  then  starts  again  from  the  opposite  direction, 
but  also  from  the  obscurity  and  persistent  heavy  rain.  1933, 
which  bred  Fig.  97  and  his  many  brethren,  was  a  heavy  hurricane 
year. 

§  325.  The  air  that  goes  up  in  Cyclones  must  come  down  some- 
where, and  it  does  so  in  Anticyclones,  great  quiet  areas  of  High 
Pressure,  Fig.  105,  over  which  air,  already  dry  because  high  and 
cold,  is  sinking,  at  any  speed  from  300  ft.  in  large  systems  to  1500  ft. 
in  small,  per  day,  and  being  heated  by  compression.  Round  them 
it  flows  out,  with  again  a  rotary  drift,  on  account  of  the  earth's 
motion,  and  now  of  course  reversed,  or  clockwise^  as  dry  and  usually 
gentle  winds. 

Anticy clonic  weather  is  '  settled,'  clear  to  hazy,  sunny  and  hot 
in  summer,  when  the  vast  quasi-permanent  Anti- cyclone  of  the 
Azores  reaches  out  to  us  and  gives  us  the  weather  we  all  want  for 
holidays.  Kound  this  latter,  like  balls  running  in  their  ball-race 
round  a  rotating  axle,  whirl  the  cyclones  of  the  Atlantic  circulation 
described  in  §  323,  so,  if  it  doesn't  reach  right  out  to  us,  we  may  get 
a  fringe  of  unsettled  wet  weather  instead.  Between  it  and  the 
Pacific  anticyclone,  spreading  over  Panama  until  they  meet  in  the 
end  of  summer,  are  ground  out  the  West  Indian  hurricanes. 

This  great  anticyclone  sits  on  the  N.  Atlantic  in  Summer  because 
that  is  the  only  cool  place.  Round  it  are  N.  and  S.  America  and 
W.  Africa  and  Europe,  all  being  heated  by  the  summer  sun,  their 
hot  air  rising  and  drifting  overhead  to  the  common  cooler  centre, 
where  it  gradually  sinks. 

In  Winter  it  loses  strength,  but  prevents  any  cold  spells  reaching 
us  from  New  York ;  and  as  it  contends  with  the  colder  westerlies 
below  the  great  depression  which  forms  between  Greenland  and 
Iceland,  we  get  our  mixed  Atlantic  weather,  wet  and  warm  or 
merely  dreary  and  damp. 

A  great  winter  settlement  of  air  takes  place  over  the  vast  dry 
radiating  plains  of  Siberia,  and  reaches  out  towards  us  as  the 
Scandinavian  anticyclone.  When  this  takes  charge,  we  get  either 
fine  clear  quiet  frosty  weather,  with  foggy  mornings  and  sunny 
days,  ideal  for  skating;  or,  more  usually,  the  cold  dry  easterly 
and  north-easterly  arctic  air  flows  continuously  under  a  thin  layer 
of  cloud,  gloomy  in  smoky  towns,  condensed  out  of  the  warmer 
oceanic  air  lying  above. 


§  325]  METEOROLOGY  247 

Fig.  104.  Fig.  106. 


Fig.  105. 


248  HEAT  [§  325 

Winter  in  due  course  brightens  into  our  long  halting  English 
Spring,  with  its  many  chilly  setbacks.  The  sun  coming  back 
northwards  melts  the  snows  and  warms  the  soil  of  Europe,  and  lifts 
its  load  of  cold  air  in  a  tangle  of  depressions,  which  make  its  weather- 
map  almost  as  complicated  as  its  political  map,  and  this  air  migrates 
and  settles  on  the  still  cold  sea. 

Fig.  105  illustrates  this  rather  well :  The  drought  of  the  winter 
of  1933-4  continued  through  an  unclouded  spring,  so  that  the  sun 
could  have  its  full  effect,  and  the  great  settlement  of  air  over  the 
N.  Atlantic,  still  almost  wintry-cold,  has  brought  the  mass-centre 
of  the  anticyclone  of  the  Azores  unusually  far  north.  Down  its 
eastern  side  had  curled  a  milder-than-usual  succession  of  the  little 
depressions  commonly  responsible  for  our  '  March  many- weathers  ' 
and  '  April  showers,'  and  May  12th  was  a  day  of  light  airs  and 
sunshine,  almost  a  record  warm  day.  See  what  happened  in 
24  hr. ;  look  up  at  Fig.  106  :  under  the  resultant  drive  of  the 
westerly  winds  along  the  top  of  the  anticyclone,  and  pressure 
from  the  north,  the  Low  lying  north  of  Scotland  drifted  towards 
the  Baltic,  and  left  open  the  long  polar  corridor  of  sea,  stretching 
from  the  Arctic  past  Spitzbergen  (just  off  at  the  N.E.),  and  down 
this,  almost  along  the  isobars,  rolled  unhindered  the  great  stream  of 
polar  air. 

Deflected  and  tempered  a  little  by  the  westerly  of  the  anticyclone, 
which  it  pushed  away  down  south  (see  the  1024  isobar),  it  swept 
the  length  of  Britain  as  a  cutting  north-wester,  mocking  the  sun- 
shine, making  Stonehenge  the  poorest  of  shelters,  bringing  repentance 
to  all  who  had  '  cast  a  clout,'  and  ushering  in  the  Three  Ice-Men 
of  May — days  the  persistent  recurrence  of  which  is  still  an  unex- 
plained puzzle,  and  has  made  the  French  maintain  that  their  cold 
is  due  to  the  screening  off  of  solar  radiation  from  the  Earth,  by 
the  intervention  of  that  same  dense  stream  of  meteorites  the  orbit 
of  which,  they  say,  lies  just  on  the  shadow  side  of  our  path  six 
months  later,  reflects  back  to  us  then  the  heat  that  makes  the 
brief  '  St.  Martin's  summer,'  and  sprinkles  the  sky  for  a  night  or 
two  with  the  brightest  shooting- stars  of  the  year. 

Be  that  as  it  may,  Fig.  106  helps  to  make  our  final  point — that 
the  feel  of  English  Wind  depends  on  where  it  came  from.  It  can 
come  down  the  polar  corridor  and  curl  round  a  Low  as  in  this  case, 
or  it  can  come  straight  down  in  the  steadier  conditions  of  winter,  a 
snowy  north  wind,  or  it  can  come  via  Russia  round  the  Scandinavian 
anticyclone,  as  a  north-easter  or  a  biting  east  wind,  and  only  when 
the  plains  of  Russia  are  heated  by  summer  sun  can  its  teeth  be 
disregarded.  Winds  from  E.  to  S.  with  high  barometer  are  clock-  . 
wise  anticyclonic  winds  from  Europe,  a  spring  south  wind  quite 
likely  off  the  Alps ;  later,  as  pressures  even  up  and  circulation 
broadens,  south  winds  reach  us  from  the  Sahara,  often  late  in 
summer ;  and  these,  with  the  warm  wet  winds  of  the  ocean,  round 
the  circle  of  our  English  Weather. 


METEOROLOGY  249 


EXAM   QUESTIONS,   CHAPTER   XXI 

A  chapter  for  a  rainy  day  :  it  comprises  the  very  finest  and  largest  examples 
of  the  various  mechanical  and  thermal  actions  we  have  been  discussing  in 
the  book. 

1.  Discuss  the  formation  of  fog,  cloud,  and  rain. 

2.  Discuss  the  formation  of  dew,  mist,  and  hoar-frost.  Explain  why,  on 
a  clear  still  evening,  after  a  wet  simuner  day,  mist  seems  to  rise  from  the  grass 
of  a  lawn.     (X  2) 

3.  Describe  the  formation  of  Dew,  and  consider  the  conditions  for  a  copious 
deposit.     What  is  the  '  dew  point,'  and  how  can  you  utilize  it  ? 

4.  Why  on  a  frosty  night  is  it  often  colder  in  the  valley  than  on  the  neigh- 
bouring hillsides  ? 


MOLECULAR   PHYSICS 

CHAPTER   XXII 
VISCOSITY 


§  331.  'If  the  paint  be  too  thick,  thin  it  by  the  addition  of 
turps.'  So  runs  the  amateur  painter's  instruction,  making  use 
of  one  of  the  many  meanings  of  '  thick  '  and  '  thin.'  Physically 
we  say  '.  .  .  too  viscous,  reduce  its  viscosity.  .  .  .'  In  spreading 
paint,  the  bottom  layer  of  a  thick  smear  adheres  to  the  wood,  and 
the  upper  '  layers  '  are  dragged  over  it  by  the  brush.  The  force 
necessary  for  this,  the  drag  felt  by  the  brush,  is  due  to  the  friction 
between  '  layer  and  layer  '  of  liquid.  It  is  to  this  Internal  Friction, 
between  contiguous  portions  of  fluid  moving  at  different  speeds, 
that  the  name  Viscosity  is  appHed. 

Experiments  have  shown  that  wherever  a  liquid  is  flowing  past 
a  soHd  surface,  the  two  surfaces  adhere  without  any  slipping  at  all, 
so  that  viscosity  always  comes  into  play  to  hinder  the  flow  of  the 
upper  layers. 

In  what  follows,  only  smooth  quiet  motion,  '  stream-line  flow,"  at 
slow  speeds,  is  intended.  Motion  producing  eddies,  turbulence,  and 
noise  is  deferred  to  §  336. 

§  332.  First  take  a  case  like  that  of  honey  flowing  off  a  flat 
blade.  Magnify  it,  in  Fig.  107  (A),  and  divide  into  strata  of  equal 
thickness  and  weight.  One  side  of  the  first  adheres  to  the  blade, 
the  outer  side  moves  at  speed  v,  the  average  speed  of  fall  of  the 
whole  stratum  is  ^v.  The  second  layer  moves  on  one  side  at  v, 
and  on  the  other  at  2v,  being  subject  to  forces  exactly  like  the 
first,  but  attached  to  an  already  moving  surface.  And  so  on, 
the  speed  increasing  proportionally  to  the  distance  from  the 
solid,  and  the  average  speed  being  half  that  of  the  outside  layer. 

The  area  of  cross -section  of  the  stream  is  its  depth  X  its  width. 
The  total  flow  =  average  speed  X  area  of  cross-section  is  therefore 
proportional  to  J  {square  of  depth)  of  stream ;  the  discharge  from 
a  tenth-inch  layer  is  100  times  as  fast  as  from  a  hundredth-inch 
film  left  to  drain. 

Next  take  an  equal  stream,  but  left-handed,  and  bring  it  up 
to  the  first,  as  in  B.  The  two  free  surfaces  are  moving  at  the 
same  speed,  therefore  nothing  alters  if  we  let  them  touch,  and 
we  then  have  a  stream  flowing  between  two  parallel  walls,  C. 

250 


332] 


VISCOSITY 


261 


The  discharge  is  twice  that  of  either,  it  is  in  the  same  proportion 
as  before  to 

2  X  J  (i  dist.  apart  of  walUY  i.e.  to  -J  {dist.  apart  of  walUY. 

Finally,  suppose  the  stream  confined  by  front  and  back  walls 
as  well  as  by  the  side  walls.     The  outflow  now 

oc  J  {depth  between  walls)^  and  also  ex  J  {vndth  between  other  walUY 
i.e.  oc  their  product  y\  {width  x  depth)^ 

oc  ^^  {sectional  areaf 

and  the  same  argument  holds  good  for  a  circular  tube. 


A 

B 

\ 

V. 

2V. 

3v. 

4v. 

y^ 

/ 

\ 

■^  AVCRAOE. 

ooo 

B 

^ 

7^ 

^ 

^ 

^ 

•  ooc 

S* 

Al^ 

^ 

r> 

^ 

/ 

i/ 

/- 

y 

•oo< 

ZO 

/ 

^ 

/ 

y 

CO 

wo 

6^ 

V 

/" 

y 

•ooo 

IS 

^^ 

^ 

-^ 

'" 

HYI 

iRO 

;Eh 

oo< 

08 

^ 

'et 

^EF 

V> 

P: 

•ooo 

M- 

6< 

c 

la 

JC 

If 

o'c 

* 

o-o 

«0> 

LIQUIDS. 
VARIATION        OF       COEFF:      VISCOSITY 

Fig.  107. 


GASES. 
WITH       TEMPERATURE. 


Thus  the  carrying  or  discharging  power  of  small  tubes,  e.g.  the 
arterioles  and  capillaries,  increases  very  fast  with  increased  size, 
viz.  as  the  square  of  the  area,  or  the  fourth  power  of  the  dmmeter  : 
so  that  a  1%  relaxation  of  the  arteriole  wall  passes  4%  more  blood  ; 


262  MOLECULAR   PHYSICS  [§332 

this  is  how   local   blood-supply  is   controlled   in  the   body,   the 
simplest  instance  being  blushing. 

It  is  only  in  larger  pipes,  such  as  the  aorta,  when  friction  in  the 
pipe  itself  is  only  a  small  part  of  the  resistance  to  be  overcome  in 
pumping,  that  the  carrying  power  becomes  proportional  to  the 
area,  as  one  expects.  The  water  engineer  pumping  to  a  high 
reservoir  ignores  viscosity. 

§  333.  The  Coefficient  of  Viscosity.  On  the  honey  spoon  the 
driving  force  is  the  weight  of  the  honey,  but  if  we  suppose  one  centi- 
metre cube  to  be  set  in  motion  by  a  force  applied  all  over,  and  in 
the  plane  of,  its  outer  face,  causing  that  face  to  move  at  1  cm.  per 
sec.  faster  than  the  inner  face,  this  force  (in  dynes)  is  equal  to  the 
coefficient  of  viscosity  of  the  fluid.  With  this  definition  it  can  be 
shown  that 

the  Total  Quantity  discharged  through  a  narrow  pipe 
{areaY       total  pressure  fall,  dynes /cm.^       time  of  outflow,       2 
~      16  length  of  tube  X  Viscosity  in  seconds  n 

The  first  term  we  have  already  obtained,  the  second  is  the  speed 
of  flow,  being  evidently  just  how  many  times  the  driving  force 
available  per  cm.  exceeds  the  coefficient  of  viscosity,  which  would 
push  along  at  1  cm. /sec,  the  third  factor  is  straightforward,  but  the 
fourth,  though  simple  enough,  takes  a  lot  of  mathematics  to  discover. 

§  334.  The  (coefficient  of)  Viscosity  is  measured,  in  an  apparatus 

resembling  Fig.  108,  by  running  the  fluid  through  a  capillary  tube 

of  known  length   and   area,   under  known 

^    ^,.       difference  of  level.     Nobody  wants  you  to 

'  ^     memorize  the  foregoing  definition  or  formula, 

but  in  the  laboratory  or  the  exam  room  you 

may  be  called  upon  to  compare  the  viscosities 

g    >^/       of  two  liquids. 

To  do  this,  use  the  same  narrow  tube, 
and  run  out  the  same  bulk  of  both  liquids 
(from  mark  A  to  mark  B),  keeping  the  height 
BC  the  same  in  both  cases.  Little  simple 
Viscosimeters  are  made  for  the  purpose. 
Then,  simply  enough,  the  viscosities  are 
Fig.  108.  directly  proportional  to  the  times  of  efflux, 

excepting  that,  as  the  liquids  are  probably  of 
different  densities,  and  the  denser  one  is  being  driven  out  in  a  time 
unduly  shortened  by  the  greater  force  (hydrostatic  pressure  = 
h  X  d,  ^  103),  you  must  correct  for  this  by  multiplying  each  time  by 
the  density  of  its  liquid. 

Thick  oils,  glycerine,  etc.,  perhaps  5000  times  more  viscous,  when 
cold,  than  water,  are  filled  into  a  tall  cylinder,  and  a  very  small  steel 
bearing  ball  is  let  fall  through  them  ;   viscosities  are  proportional  to 


J 


§  335]  VISCOSITY  263 

times  of  fall,  nearly  enough  :  haul  up  with  a  magnet.  This  lends 
itself  easily  to  measurements  at  higher  temperatures,  and  the 
results  are  usually  startling,  in  face  of  advertisers'  claims  as  to  their 
engine  oils  retaining  their  body  at  high  temperatures.  Another 
Viscometer,  much  used  in  the  oil  trade,  takes  the  running-out  time 
through  a  hole  (drilled  through  an  agate)  in  the  bottom  of  a  little 
bucket  kept  at  a  fixed  temperature. 

Coefficients  for  various  liquids  and  gases  are  given  in  Fig.  107. 
Liquids  become  less  viscous — more  mobile — as  temperature  rises. 
Glycerine  gets  quite  '  watery  '  at  100°.  Substances  like  candle- 
wax  and  pitch  can  be  called  very  viscous  liquids,  candles  hardly 
bend  in  winter,  but  collapse  in  summer,  a  particular  pound  square 
of  pitch  in  the  laboratory  cupboard  stands  apparently  changeless 
all  the  session,  but  has  subsided  perceptibly  more  after  each  long 
vacation.     '  Blood  is  thicker  than  water,'  five  times,  at  blood  heat. 

Gases  are  somewhere  about  100  times  less  viscous  than  liquids  : 
their  viscosity  increases  with  temperature ;  but  is  independent  of 
pressure,  down  to  a  very  low  limit,  when  it  suddenly  all  but  vanishes. 

It  is  now  specified  that  Dough,  to  be  suitable  for  making  good 
bread,  must  knead  plastically  as  a  fluid  10^  times  as  viscous  as 
water,  and  cling  as  a  solid  with  10"^  times  the  elastic  strength  of 
steel. 

§  335.  Dry  friction  and  fluid  friction.  The  formula,  by  a  good 
deal  of  twisting,  will  disclose  the  Laws  of  Fluid  Friction,  but  it  is 
pleasanter  to  reflect  on  the  slow  movement  of  a  light  boat  on  a 
calm  pool.  Hitching  the  painter  round  your  finger,  you  find  that 
the  boat  responds  slowly  to  the  very  smallest  pull,  accelerating 
gently  up  to  a  steady  slow  speed.  If  you  want  it  to  move  faster, 
you  pull  harder,  and  it  gains  and  holds  a  higher  speed. 

That  is,  there  is  no  lower  limit  of  force ;  and  the  frictional  drag 
depends  on  the  speed  (is  proportional  to  it,  at  low  speeds) ;  whereas 
a  beached  boat  won't  move  until  you  exert  a  definite  force,  and  then 
can  be  rushed  down  quickly  with  but  little  more. 

If  a  lot  of  floating  weed  gathers  round,  and  virtually  increases 
the  viscosity  of  the  water,  you  have  to  pull  harder  for  the  same 
speed. 

Again,  you  find  that  so  long  as  you  paddle  very  gently,  pro- 
ducing no  appreciable  wave,  there  is  not  so  much  difference  as  one 
might  expect  between  an  empty  and  a  loaded  boat.  The  deeper 
sunlien  boat  wets  more  of  its  surface,  and  drags  proportionally  to 
that :  this  you  see  particularly  well  in  a  '  speed  boat,'  which  travels 
very  sluggishly  until  it  sits  up  on  its  tail,  when  the  wetted  surface 
of  contact  falls  to  a  fraction,  and  the  speed  leaps  up. 

Further,  there  is  no  more  skin-friction  per  sq.  ft.  on  the  bottom 
of  a  big  ship,  30  ft.  under  water,  than  at  her  water-line  where  you 
can  watch  the  wriggling  eddies  of  her  narrow  '  friction  belt,'  although 
the  pressure  is  a  ton  per  sq.  ft.  greater. 


264  MOLECULAR   PHYSICS  [§335 

Solid  friction 

prevents  movement  below  a  limiting  force. 
Its  drag  is  : 

proportional  to  a  coefficient  of  friction, 
independent  of  Area, 
independent  of  Speed, 
proportional  to  Weight  carried. 

Fluid  friction 

permits  movement  in  response  to  any  force,  however  slight. 
Its  drag  is  : 

proportional  to  a  coefficient  of  Viscosity, 
proportional  to  Area, 
at  least  proportional  to  Speed, 
independent  of  Weight  carried. 

A  ship  is  exposed  to  the  fluid  frictional  drag  of  wind  and  stream 
and  tide  :  she  drops  her  anchor,  1/1500  of  her  weight,  and  it  lies 
on  the  solid  ground  with  a  coefficient  of  friction  which  can  seldom 
exceed  1/1  :   a  ship  at  anchor  is  the  symbol  of  security  and  rest. 

§  336.  High  speeds  of  flow.  At  higher  speeds  (depending  on 
dimensions  and  viscosities,  and  in  bearings  and  everywhere  else), 
the  fluid  moves  with  eddies  producing  turbulence  and,  if  air  gets 
drawn  in,  noise.  The  water-tap  begins  to  splutter,  the  gas-jet 
roars,  the  bullet  sings,  the  boat  leaves  waves  and  busy  little  whirl- 
pools in  its  wake. 

The  flow  through  a  pipe  is  less  than  expected ;  the  resistance  of 
the  air  far  more,  the  top  speed  of  fall  of  a  big  raindrop  is  only  a 
hundredth  of  that  calculated  by  the  viscosity  formula  from 
observation  of  a  minute  mist-drop ;  '  skin-friction  '  hardly  counts 
with  a  ship. 

Quantities  of  fluid  are  set  into  varying  violent  motions,  and  the 
friction  among  them  largely  exceeds  that  at  the  measurable  surfaces. 
Empirical  laws  are  obtained  suiting  special  cases.  Resistances 
increase  as  the  square  of  the  speed,  at  least,  and  although  in  the 
end  viscosity  quiets  it  all,  they  often  appear  independent  of  it, 
diminished  viscosity  being  counterbalanced  by  increased  bulk  of 
disturbance. 

§  337.  Lubrication.  The  ample  slow-moving  bearing  surfaces  of 
the  animal  framework  are  constantly  lubricated  by  the  synovial 
fluid,  and  probably  obey  fluid  laws.  Slow-moving  bearings  we 
grease,  but  quick-moving  lubricated  machinery  is  a  strange  new 
story.  An  average  engine-piston  has  a  surface-area  almost  as  big 
as  your  hand  :  take  a  thin  board,  swish  it  through  water  edge- 
wise, and  reflect  that  cold  engine-oil  is  thousands  of  times  as  viscous 
as  water  :   how  can  you  ever  start  her  up  at  all  ? 


§337] 


VISCOSITY 


255 


Try  this  little  experiment  :  put  some  oil  on  a  glass  slide,  lay  a 
convex  lens  on  it,  and  rock  or  slide  it ;  behind  the  point  of  contact 
you  will  see  a  vacuum  bubble  open,  like  a  '  pulsating  vacuole,' 
Fig.  109.  You  have  taken  the  bulge  of  glass  away,  and  the 
atmospheric  pressure  has  not  yet  been  able  to  drive  the  viscous 
oil  into  the  place  it  left :  the  thicker  the  oil,  the  easier  the  experiment. 

Quick-moving  surfaces  *  turbulently '  tear 
the  oil  film  to  a  net- work  of  shreds  and  fibrils, 
and  these  act  as  rollers,  like  threads  of  cotton 
under  your  hand,  or  the  little  fibrils  you  get 
when  rubbing  out  with  indiarubber.  You 
glimpse  the  network  when  you  'lift  a  brass.' 
The  more  viscous  the  oil,  the  more  complete 
the  tearing,  and  this  counteracts  differences  in 
viscosity  to  some  extent. 

If  this  is  so,  and  as  in  §  334,  oils  do  not  differ 

luch  in  losing  their  viscosity  when  hot,  why 

ron't  any  old  oil  do  for  the  engine  ? 
If  metal  surfaces  meet,  within  atomic  distance,  they  cohere  as  the 

letal  itself  does,  §  145  ;   the  bearing  '  seizes.'     They  must  be  kept 

)art,  not  by  a  straying  film  of  oil,  but  by  one  that  hangs  on  grimly. 

[ere  we  must  introduce  a  little  organic  chemistry. 
Castor  oil  is  a  fatty- acid  salt  of  glycerin,  and  upon  the  slightest 

)rovocation,  e.g.  heat  and  moisture,  splits  up  with  the  production 

"  free  ricinoleic  acid 


Fig.  109. 


H     H       H    H    H  HO 

H— C— (C)5— C— C— C=C— (C)^— C— 0— H 
H    H     OH  H  H     H 


At  the  right-hand  end  the  O  sheds  off  the  H  unceremoniously, 
[and  attaches  the  long  molecule  to  iron,  or  copper,  or  zinc,  to  form 
'le  ricinoleate  of  the  metal.  As  many  molecules  do  this  as  can, 
they  all  get  their  jaws  down  to  the  metal  and  pack  tight  side 
side,  like  pigs  at  a  trough,  completely  covering  the  metal  with  a 
ily  adherent  velvety  pUe,  about  0-004  micron  thick.  The  free 
face  of  the  velvet  consists  of  the  CHg  ends,  presenting  simply 
hydrogen  atoms,  which  have  no  more  bite  about  them  than  the 
rounded  hinder  ends  of  the  pigs,  no  notable  chemical  affinity, 
certainly  not  for  more  hydrogen.  Hydrogen  is  hard  to  persuade, 
§  296,  to  cling  together  to  form  even  the  lightest  of  liquids — there  is 
no  fear  of  its  atoms  packing  together  as  do  metalhc  atoms  to  form 
j  the  strongest  cohering  soUds.  Consequently,  in  your  bearing,  you 
now  have  velvety  surfaces  gliding  over  each  other  without  the 
slightest  tendency  to  seize. 

That  is  pretty  much  what  is  happening  in  light  machinery 
running  *  dry,'  like  most  people's  watches,  unoiled  for  years ;  or 
Harrison's  two  timepieces,  in  the   Science  Museum,  where  metal 


266  MOLECULAR    PHYSICS  [§  337 

runs  in  lignum  vitse,  the  wood  of  which  '  bowls  '  are  made,  hard 
and  dense,  but  so  full  of  wax  as  to  bum  famously. 

But  heavy  pressure  may  conceivably  force  one  pile  down  among 
the  other.  Look  at  the  double  bond  in  the  middle  of  the  molecule, 
the  sign  of  an  '  unsaturated  '  fatty  acid  :  one  of  these  bonds  is 
notoriously  willing  to  break,  and  they  probably  seize  hold  of  neigh- 
bouring molecules  and  weave  the  whole  together  into  a  tougher 
layer  ;  unsaturated  fatty  acids  are  better  lubricants  than  saturated, 
which  lack  this  double  bond.  If,  even  in  spite  of  this,  the  molecules 
'  come  to  grips,'  they  snap  as  they  are  rent  apart,  or  tear  off  the 
metal.  Understand,  please,  that  there  is  nothing  sacrosanct  about 
a  '  chemical '  bond  :  it  is  electrical  in  origin,  and  it  represents  a 
mechanical  force  which  can  be  overcome  by  a  greater.  The 
chemists  have  their  ways  of  applying  the  greater,  and  the  physicists 
have  others ;  quite  likely  your  mother  snapping  a  cotton  thread 
snaps  across  some  long  molecules,  just  as  you  tear  atom  from  atom 
when  you  break  a  wire. 

Therefore  pad  out  your  system  with  some  inactive  viscous  liquid 
which  will  form  the  maze  of  '  rollers  '  mentioned  above,  between 
the  two  '  velvets.'  Long  splines  of  that  same  lignum  vitse  carry 
the  bronze-sheathed  tail-shaft  and  all  the  weight  of  the  ship's 
propeller,  and  sea-water  is  the  padding — for  a  sea-water  padding 
can  be  elastic,  as  in  ducks  and  drakes — but  a  new  patent  succeeds 
in  shutting  out  the  sea  from  the  stem  tube,  and  this  unique  type  of 
bearing  is  being  superseded. 

In  the  engine,  which  is  hot  and  corrodible,  one  prefers  to  pad  with 
mineral  hydrocarbon  oils 

CH3'(CH2)„*CH3 

which  have  no  grip  either  end — ^you  see  the  danger  of  using  them 
alone — but  you  must  have  1%  of  castor  or  other  fatty  acid,  or  some 
factitious  active  gripping  oil,  in  the  mixture,  which  ensures  keeping 
your  '  velvets  '  in  good  repair. 

§  338.  Vaseline  is  a  mixture  of  these  inert  '  paraffins  ' ;  vaseline 
your  tools  and  leave  them  in  the  rain,  and  see  them  rust.  The  mild 
emollient  lanoline  is  a  complex  of  fatty  acids  and  higher  alcohols ; 
smear  it  on  your  bright  steel,  or  thin  with  a  little  petrol  and  paint 
on,  and  the  invisible  film  defies  city  rain  and  sea-water.  In  cutane- 
ous lesions,  vaseline  has  no  grip  on  the  skin,  which  dries  and 
hardens ;  lanoline  lays  hold,  and  is  an  incomparably  better  base 
for  ointments ;  try  it  on  your  own  scratches,  and  don't  forget  it 
when  you  have  patients  worth  propitiating.  '  Medicinal  parafiin  ' 
is  futile  stuff  by  itself  ;  with  anything  from  a  trace  to  50%  of  castor 
oil  it  acts  as  a  useful  and  merciful  diluent.  This  is  not  pharma- 
cological mystery,  it  is  plain  physics. 

In  shoe-poUshing,  as  the  trace  of  solvent  evaporates,  you  jostle 
the  long  complex  wax  molecules  about  with  the  brush,  and  they 


§  338]  VISCOSITY  257 

bite  down,  layer  on  layer  (for  oxygen  can  bite  on  anything),  and 
present  to  you  their  glossy  inert  hydrogen  ends.  This  is  no  idle 
conjecture  ;  the  X-ray  spectrograph  discloses  the  structure,  and 
distinguishes  the  waxes.  Paraffin  wax  has  both  ends  CHg ;  it  is  a 
useless  adulterant,  does  not  polish,  unsettles  at  a  touch,  and 
finger-marks. 

See,  further,  §  351. 


EXAM   QUESTIONS,   CHAPTER   XXII 

These  are  of  limited  scope  and  interest.  The  last  three  chapters  dealt 
ith  vapours;    these  three  with  liquids,  gradually  prying  deeper. 

Dry  friction  came  in  §  41,  but  as  every  moving  thing  in  the  world  is  lubri- 
ated  by  fluid,  excepting  brakes  intended  to  stop  movement,  the  discoveries 
f  recent  years  are  offered  you  here. 

1.  Explain  the  meaning  of  the  term  Viscosity. 

How  are  the  viscosities  of  water  and  a  saline  solution  compared  experi- 
aentally,  and  what  results  would  you  expect  ?     (  X  2) 
\    2.  What  factors  influence  the  flow  of  liquids  through  narrow  tubes  ?     How 
(eould  you  compare  the  viscosities  of  two  liquids  by  a  flow  through  tube 
method  ? 

3.  Explain  Viscosity.  Give  methods  of  investigating  its  variation  with 
lemperature  ?    (  X  3) 

4.  Contrast  the  Laws  of  Solid  and  Fluid  Friction  (  X  3) 
Questions  on  lubrication  are  unlikely. 


PRACTICAL    QUESTIONS 


r 


Compare  the  viscosities  of  two  liquids. 
Compare  viscosities  at  different  temperatures. 


CHAPTER  XXIII 
THE  LIQUID  SURFACE 


§  341.  Surface  Tension.  Down  on  the  margin  of  the  pond  in 
summer  you  have  watched  the  '  pond- skaters  '  darting  over  the 
surface,  which  only  their  long  legs  touch.  And,  less  conspicuous, 
the  '  water-boatmen  '  resting  or  sculling  on  the  underside,  like  a 
fly  on  the  ceiling.  By  a  lucky  chance,  you  may  have  lit  upon  a 
flock  of  water-fleas  beneath,  and  frightened  one  or  two  into  jumping 
right  through  the  surface  ;  and  then  they  can't  get  back,  and  drown 
dismally  in  unaccustomed  air. 

To  these  wee  beasties  the  surface  is  a  stretched  sheet,  smooth  and 
tense,  sustaining  all  the  force  they  exert  upon  it.  There  is  no  sheet 
(unless  there  be  one  of  scum) — a  sheet  would  have  two  surfaces. 
There  is,  however,  a  boundary,  a  superficies ;  with  a  stretch  in  it, 
a  surface  in  tension. 

Light  weights  depress  the  plane  surface  into  little  dimples,  the 
skater  rests  in  half  a  dozen  miniature  hammocks.  If  the  season  for 
pond-life  is  past,  raid  your  mother's  needles,  borrow  the  smallest, 
and  grease  it  ever  so  slightly  in  your  fingers,  and  with  care  and  a 
bent  slip  of  paper  you  can  lay  it  on  clean  water,  and  emulate  the 
borrowed  axe  of  the  sons  of  the  prophets. 

Before  we  go  on  to  make  measurements  of  Surface  Tension,  let 
us  get  better  acquainted  with  it. 

For  one  thing,  all  surface  tensions  diminish  as  the  liquid  is  heated, 
and  ultimately  vanish  at  the  Critical  Temperature,  §  286,  when  the 
surface  dividing  '  liquid  and  vapour  '  disappears.  This  diminution 
you  can  show  by  putting  a  very  little  water  in  a  tin  dish,  and  grating 
down  some  cork-dust  over  it,  to  show  its  movements  ;  then  heating 
one  spot  underneath  with  a  flame.  The  heated  spot  spreads  open — 
it  has  become  a  weak  place,  and  the  stronger  pull  of  the  colder  sur- 
face all  round  tears  it. 

Water,  we  shall  see,  has  a  remarkably  high  surface  tension. 
That  of  soapy  water  is  only  one-third  as  much  ;  from  a  spot  of 
shaving-soap  froth  dropped  on  to  clean  warm  water  there  is  a 
rushing  outwards  in  all  directions  as  the  stronger  surface  tears  at 
the  weaker.  In  soft  water  this  stops  as  the  whole  becomes  soapy, 
but  goes  on  longer  in  hard  water,  which  continuously  destroys  the 
soap. 

The  high  surface  tension  of  water  makes  it  difficult  to  keep  its 
surface  clean  ;  it  is  always  pulling  sheets  of  every  sort  of  contamina- 
tion over  itself.  Nature  cleans  the  pond  by  wind  ;  but  you  had 
better  not  breathe  on  your  dish  of  water  nor  touch  it  :  keep  it  con- 
tinuously overflowing  the  brim  if  you  can. 

258 


§  342]  THE   LIQUID   SURFACE  269 

Chips  of  Camphor  dropped  on  it  then  begin  to  perform  little  erratic 
movements  reminding  one  of  *  whirhgig  beetles.'  As  the  chip  very 
slowly  dissolves,  one  side  is  for  the  moment  dissolving  faster,  the 
tension  of  the  more  contaminated  surface  there  is  weakest,  and  the 
chip  is  dragged  the  other  way.  The  toy-shops  sell  chip-battleships 
propelled  by  a  bit  of  camphor  at  the  stern ;  but  for  real  agihty, 
watch  a  tiny  crumb  under  your  pocket-lens.  Movements  cease 
when  the  surface  layer  is  saturated  with  camphor,  or  instantly  if 
you  touch  the  water  with  a  greasy  finger. 

Try  sprinkling  methylated  spirit  into  a  wet  sink,  and  watch  the 
violent  commotion  in  the  thin  layer  of  water,  as  each  drop  of  spirit 
is  left  with  a  nearly  dry  halo  round  it ;  the  strong  water  surface  has 
shrivelled  up  and  pulled  out  the  weak  methylated  spirit  surface ; 
presently  the  liquids  dissolve  each  other.  Wine  creeps  up  the  side 
of  the  glass  ;  there  the  alcohol  evaporates  the  faster,  and  a  watery 
residue  pulls  itself  together  into  '  tears,'  which  trickle  down  through 
the  spirituous  film. 

The  best  known  of  all  these  effects  is  the  rapid  spreading  of  oil 
dropped  on  water,  to  form  the  familiar  iridescent  film  ;  the  water 
surface  pulls  out  the  weak  oil.  Conversely,  on  a  greasy  surface, 
water  pulls  itself  together  into  drops  :  try  ordinary  Glass,  and  then 
after  you  have  scrubbed  it  in  hot  soap  and  water ;  when  clean, 
water  does  not  collect  in  drops  as  it  dries  off. 

Touch  your  cork-dusted  water-surface  with  a  match  flame  : 
the  greasy  hydrocarbon  contamination  clears  a  space  instantly  and 
permanently. 

Ask  your  nurseryman  friend  to  throw  two  leaves  of  Schinus  molle 
on  the  water-tank  in  the  sunshine  :  their  oil-cells  will  keep  firing  off 
drops  of  oil,  which  blue  and  weaken  the  water  surface,  and  you  see  a 
most  gallant  engagement  between  fast  cruisers. 

§  342.  The  wetting  of  surfaces.  Why  some  liquids  wet,  i.e.  adhere 
to  and  spread  on,  some  sohds,  and  not  others,  we  don't  know.  A 
familiar  difficulty  gives  some  clue,  however  :  melted  solder  will  not 
stick  to  a  tarnished  copper  bit,  but  adheres  instantly  if  a  corrosive 
chloride  is  present.  Wiped  off,  it  contains  traces  of  copper,  i.e. 
the  adhesion  is  probably  due  to  the  same  forces  as  are  concerned  in 
solution  or  chemical  action.  Mercury  adheres  to  iron  very  reluct- 
antly, but  seizes  on  silver  and  gold,  and  eats  them  away. 

Probably  the  wetting  of  most  surfaces  by  ordinary  liquids  depends 
on  their  being  covered  already  with  an  imperceptible  film  of 
moisture.  Glass  collects  a  particularly  thick  one  out  of  the  atmos- 
phere, sodium  amalgam  kept  hot  on  it  develops  a  layer  of  hydrogen 
bubbles  which  I  found  to  correspond  to  00001  mm.  of  water.  Visible 
wetting  will  ensue  with  any  liquid  which  can  mix  with  or  '  dissolve  ' 
this  film.  Another  common  film,  grease,  which  hinders  wetting  by 
water  (so  that  you  must  reduce  its  surface  tension  by  soap  in  order 
to  wash  your  hands),  is  an  encouragement  to  the  well-known 
creeping  of  paraffin  oils. 


260  MOLECULAR   PHYSICS  [§  342 

One  of  the  commonest  uses  you  find  for  surface  tension  is  in  the 
transference  of  writing-ink  from  pot  to  paper — and  you  know  the 
effect  of  any  greasiness  of  paper  or  pen — and  the  next  commonest 
is  blotting-paper. 

§  343.  The  measurement  of  surface  tension  by  the  clinging  ring. 

Make  a  ring  of  thin  copper,  about  3  cm.  diameter  and  1  cm.  high ; 
fit  it  with  three  wires  so  that  it  can  hang  horizontally  from  a  balance- 
hook,  in  contact  with  liquid  in  a  watch-glass  on  a  bridge  over  the 
balance-pan.  Boil  your  ring  in  caustic  soda,  and  keep  it  in  clean 
water  ;   rinse  it  in  your  liquid  before  use. 

Weigh  it  as  it  hangs  pulling  at  the  clinging  liquid  surface  in  the 
watch-glass  ;  break  it  loose  and  weigh  again.  The  experiment  is 
easiest  to  do,  and  perhaps  to  understand,  with  soapy  water,  which 
gives  a  2  or  3  mm.  wide  curtain  film,  joining  ring  and  liquid,  all 
round. 

The  difference  measures  the  cling  of  the  surface  all  round  the 
outside  and  all  round  the  inside  of  the  edge  of  the  ring,  a  total  length 
of  2  X  TT  X  its  mean  diameter  ;  and,  when  you  have  converted  it 
into  dynes,  by  multiplying  by  g 

Diff.  of  wt.  in  gm.  X  g  =  27r  x  mean  diam.  ring  x  T 

where  The  Surface  Tension  T  is  the  pull  in  dynes  exerted  across  each 
centimetre  width  of  surface,  see  Fig.  110,  T. 

§  344.  Capillarity — from  capilla,  a  hair — is  the  name  given  to  the 
creeping  of  liquid  up  narrow  crevices  and  tubes  ;  of  water  through 
wood  or  brickwork,  of  oil  up  a  wick,  etc. 

Take  two  clean,  §  341,  glass  plates  wet  with  the  liquid  under 
investigation,  keep  them  apart  at  the  edges  by  pins  or  something, 
strap  an  elastic  band  round,  and  stand  them  upright  in  a  saucer  of 
the  liquid.  It  rises  between  them,  as  magnified  in  Fig.  110,  left; 
and  the  higher  the  closer  they  are,  as  shown  standing  in  the  dish, 
their  right  edges  touching. 

Surface  tension  and  capillary  rise.  The  strong  surfaces  of  the 
films  of  liquid  clinging  to  the  two  plates  A  and  B,  each  pulling  with 
force  T  dynes  per  cm.  width,  together  haul  up  the  previously  flat 
surface  of  the  liquid  to  a  height  h  cm.,  and  there  it  hangs  in  a  ham- 
mock-like curve,  which,  if  the  distance  apart  of  the  plates  be  b  cm., 
sustains,  per  1  cm.  width  of  plate,  a  volume  of  liquid  hbcc,  weighing 
hb  X  density  gm.  ;   or  a  downward  force  of  hbdg  dynes. 

/.  2T  =  hbdg 

There  is  a  little  more  difficult,  but  more  generally  useful,  way  of 
looking  at  this.  Provided  that  6  is  small,  a  millimetre  or  less — 
and  we  are  not  interested  in  anything  larger — the  curve  is  practically 
a  semicircle,  of  radius  r  =  J6,  of  Curvature  1/r. 

The  atmosphere  presses  on  the  broad  flat  surface  of  the  liquid, 
and,  therefore,  just  beneath  that,  the  hydrostatic  pressure  is  the 
atmospheric  pressure.     At  a  height  of  liquid  h  above  it,  the  hydro- 


344] 


THE   LIQUID   SURFACE 


261 


static  pressure  inside  the  (hollow)  surface  is  evidently  less,  by  that 
due  to  the  column,  i.e.  hyhdg,  §  103,  and  this  diminution  of  pressure 
has  been  caused,  over  an  area  6  X  1  sq.  cm.  =  2r  sq.  cm.,  by  a 
combined  pull  2T  dynes. 

So  we  can  write  2T  =  hbdg  as  2T  =  hdg  -\-  2r 

or  T  X     =  hdg 

That  is,  Inside  a  hollow  surface,  the  hydrostatic  pressure  in  a  liquid 
is  diminished  ;  by  the  product  of  the  surface  tension  and  the  curvature. 


Fig.  110. 


For  instance,  inside  a  capillary  tube  of  radius  r,  the  *  meniscus ' 
is  a  hemisphere,  of  curvature  1/r  both  ways;  taking  account  of 
both,  the  liquid  evidently  rises  twice  as  high 


TX  (^^  +  bj=h'dg     or  TJ^rh'dg 


which  tallies  with  the  argument  that  an  upward  pull  all  round  the 
priphery  27rr  X  T,  lifts  a  volume  irr^'  of  liquid,  each  c.c.  of  which 
is  pulled  earthward  by  dg  dynes. 


k 


262  MOLECULAR    PHYSICS  [§345 

§  345.  You  all  do  experiments  with  capillary  tubes  in  the 
laboratory  :  recollect  to  have  the  tubes  perfectly  clean ;  to  keep 
measurements  in  cm.,  radii,  and  dynes ;  to  use  the  mean  radius, 
rejecting  any  considerably  flattened  tube ;  and  to  record  the 
temperature. 

The  results  we  have  calculated  evidently  give  the  Bule  :  The 
height  to  which  water  creeps  in  a  crevice  or  tube  is  inversely  proportional 
to  its  width  or  diameter. 

Also  you  see  that  different  liquids  creep  up  the  same  tube  to 
distances  proportional  to  their  surface  tensions  divided  by  their 
densities. 

The  Surface  Tension,  T  dynes  per  cm.  width,  at  15°  C,  of  Mercury  is  547, 
of  Water,  and  dilute  sulphuric  acid,  72;  benzene,  CSg,  olive  oil,  about  32; 
paraffin  oil,  soapy  water,  alcohol  and  acetic  acid,  about  24;  ether  16;  and 
liquid  air  at  —  186°,  13. 

Mercury  in  a  glass  tube,  water  in  a  greasy  or  waxed  tube,  any 
liquid  in  a  tube  which  it  does  not  wet,  is  pushed  down  instead  of 
drawn  up.  The  bulged-up  meniscus  meets  the  surface  at  an  obtuse 
angle  of  contact,  and  has  a  radius  =  r-^cosine  of  this  angle.  A 
barometer  with  a  narrow  tube  reads  low  by  cos  140°  ^  I3r  cm. ; 
but  nobody  dreams  of  trusting  one,  because  a  trace  of  dirt  affects 
the  angle  greatly.  Do  not  say  that  '  mercury  does  not  wet  glass 
because  this  angle  exceeds  90°,'  for  that  is  only  a  truism  :   see  §  37. 

§  346.  This  reduction  of  hydrostatic  pressure  by  T/r  accounts  for 
the  tenacious  adhesion,  even  in  a  vacuum,  of  two  pieces  of  plate  glass 
between  which  a  drop  of  water  has  been  squeezed  (or  the  end  gauges, 
§  152).  The  pressure  all  over  the  area  of  the  flattened  patch  of 
liquid  with  its  strongly  concave  edges.  Fig.  110,  bottom,  is  reduced 
by  T  -f-  half  distance  apart,  so  the  closer  they  are  the  tighter  they 
cling.  Squashed  mercury  drops  force  plates  apart,  for  their  edges 
are  convex,     tby  these  things. 

The  experiment  of  partly  filling  a  tumbler  with  water,  placing  a 
card  on  it,  and  inverting  without  spilling,  Fig.  110,  right,  is  similarly 
explained.  It  has  nothing  whatever  to  do  with  '  pressure  of  the 
atmosphere,'  for  the  pressure  inside  as  the  air  becomes  saturated 
is  greater  than  atmospheric,  and  there  is  the  weight  of  water  to  be 
sustained  as  well,  much  or  little  making  no  difference  to  success. 
What  happens  is  that  the  water  between  the  outer  edge  of  the 
tumbler  and  the  card  shrinks  to  a  sharp  concavity,  and  the  reduced 
pressure  throughout,  due  to  this,  holds  up  card,  water,  and  all; 
in  fact,  the  more  the  weight  of  water  the  closer  the  card  pulls  up. 
Drop  ether  on,  so  as  to  weaken  the  edge  in  parts,  and  the  experiment 
fails. 

§  347.  Liquid  cylinders  and  drawn  fibres.  The  quiet  cylindrical 
stream  from  a  water-tap  is  in  unstable  equilibrium,  for  if  a  vibration 
cause  a  momentary  thinning  at  one  place — a  smaller  radius  of 


§348]  THE   LIQUID   SURFACE  263 

curvature — T/r  increases  there,  and  pushes  the  liquid  into  the  wider 
parts,  thus  corrugating  the  stream  and  speedily  nipping  it  into 
drops.  Such  jets  are  sometimes  very  sensitive,  and  will  magnify 
the  ticking  of  a  watch  pressed  against  the  tap  into  a  succession  of 
noisy  splashes.  The  newly-formed  drops  vibrate  from  egg-shaped 
to  turnip-shaped,  giving  the  jet  the  bulbous  appearance  you  have 
noticed.  Common  shot  are  the  drops  into  which  slender  streams  of 
melted  lead  break  up,  and  solidify,  as  they  fall  down  the  shot-tower. 

Viscosity  brings  vibrations  to  a  standstill ;  if  a  liquid  be  very 
A'iscous,  the  small  pressure-differences  will  not  succeed  in  breaking 
up  the  stream ;  notice  the  difference  between  the  noisy  broken 
trickle  of  hot  water  into  your  tea-cup,  and  the  quiet  unbroken 
stream  of  the  more  viscous  milk.  If  it  is  very  viscous,  it  remains  in 
long  strings,  as  treacle,  seccotine,  or  rubber  solution  drying  '  tacky  '  ; 
squirted  viscose  setting  in  a  hardening  solution  to  artificial  silk, 
cuprammonium  silk  drawn  to  ten  times  its  length  after  leaving  the 
jet ;  natural  silk  and  spider  threads  ;  glass  tubing,  '  quartz  fibres  ' 
drawn  by  bow  and  arrow  from  a  drop  of  melted  silica,  etc.,  etc. 

But  water  must  break  into  drops  on  a  wetted  fibre,  or  wire,  and  a 
similar  beading  of  sticky  drops,  easily  seen  with  your  pocket-lens, 
gives  the  roundabout  threads  of  the  garden-spider's  web  their  efficacy 
as  fly- catchers. 

§  348.  Capillary  action  in  the  Soil  feeds  us  all ;  moisture  creeping 
up  the  inter-granular  crevices  from  the  stores  of  water  below,  and 
the  higher  the  finer  are  the  chinks.  So  that  light  soil  consisting  of 
visible  grains  '  burns  up,'  while  a  heavy  soil,  in  which  microscopic 
particles  pack  closer,  is  still  watering  its  crop.  If  the  top  four  inches 
of  soil  be  kept  hoed  up,  and  therefore  loose,  and  all  its  crannies 
wide,  moisture  that  has  risen  to  the  solid  surface  beneath  cannot 
continue  its  upward  creep  :  it  can  vaporize,  of  course,  and  the  vapour 
can  diffuse  up  through  the  thick  badly  conducting  blanket,  but  that 
is  a  different  matter  from  the  sun  blazing  straight  on  the  water- 
bearing surface,  and  all  its  vapour  being  carried  away  directly  by 
the  wind.  We  have  grown  fine  onions  in  a  torrid  summer  with  hoe 
alone  instead  of  watering-can,  and  since  the  discovery  of  this 
'  dry  tilth,'  millions  of  acres  of  wheat  are  raised  on  an  annual 
rainfall  of  4  in. 

Fig.  Ill  illustrates  the  transition  from  Mud  to  Dust. 

In  A  the  water-logged  soil  contains  no  air,  the  grains  are  copiously 
lubricated  and  move  at  a  touch  ;   the  Mud  is  thin. 

B  you  get  on  the  beach  ;  whenever  your  foot  disturbs  wet  sand  it 
packs  less  tightly,  water  sinks  in  to  fill  the  larger  interstices,  and  the 
sand  goes  drier  and  harder,  like  C  :  relieved  of  the  pressure,  it 
repacks  spontaneously,  and  exudes  water  into  your  footprints. 
You  can  do  the  same  with  wheat-starch  and  water — it  will  stoutly 
resist  your  prodding  finger,  yet  run  quietly  as  cream. 

In  C  plenty  of  air  has  got  in,  the  granules  are  connected  by  thick 
necks  of  water ;   not  very  sharply  concave,  the  hydrostatic  fall  of 


264 


MOLECULAR   PHYSICS 


[§348 


pressure  T/r  in  them  is  not  great,  yet  the  area  of  cross-section  of  the 
necks  is  large,  and  they  tie  the  grains  together. 

In  D,  evaporation  has  thinned  the  necks  of  water,  and  the  curva- 
ture of  their  concavity  is  sharp,  T/r  is  large,  their  areas  are  still 
ample,  and  the  soil  is  firmly  bound  together.  Time  was  when  we 
always  contrived  to  bike  over  the  sandy  roads  of  the  East  Suffolk 


Fig.  111. 


heaths  the  day  after  rain,  for  the  next  day  came  E,  too  much 
desiccation ;  T/r  is  larger  than  ever,  but  the  necks  are  vanishingly 
small,  and  it  takes  little  force  to  break  them  and  reduce  the  soil  to 
Dust. 

And  when,  after  hiking  on  the  roads  all  day,  you  wash  a  handker- 
chief and  spread  it  on  the  looking-glass  to  dry  smooth  by  morning, 
vou  are  evoking  precisely  the  same  actions. 

§  349.  Drops  and  bubbles,  both,  are  bound  together  by  surface 
tension.  A  liquid  on  a  surface  it  has  not  wetted — ^water  on  dust  or 
wax,  mercury  on  the  bench  top — shows  in   its  smallest  drops  an 

almost  perfect  spherical  shape,  the  pressure  inside  T  x(-H — 

2T/r  throughout,  for  the  weight  of  liquid  is  still  small  compared  with 
this  ;  §  102.  Bigger  drops  must  increase  their  curvature  at  the 
bottom,  to  sustain  the  increased  hydrostatic  pressure  there  ;  the 
rain- drops  that  hang  on  the  window- sash  have  two  curvatures  on 
their  ends,  against  only  one  on  their  straight  sides. 

The  tiny  glistening  blobs  inside  the  canvas  of  your  tent  hold  back 
the  rain  with  2T/r ;  touch  them  and  break  their  sharp  curvature, 
and  through  it  comes.  Drops  of  petrol,  ether,  etc.,  are  much  smaller 
than  water  drops  ;  for  their  surface  tensions  are  so  small. 

A  capillary  tube  will  not  serve  to  pump  water  continuously  to  a 
level  just  below  the  natural  rise,  for  it  would  have  to  flow  out  of  a 
side  spout  in  drops,  and  these  would  bulge  outwards,  whereas  the 
rise  depends  wholly  on  the  inward  bulge. 

The  pressure  2T/r  acts  also  on  the  contents  of  a  Bubble  in  a  liquid 
— for  this  see  §  280 — while  inside  a  Soap-Bubble  is  4T/r,  for  the  film 
has  two  surfaces  :  removing  the  pipe  from  your  mouth  and  pointing 
it  at  a  candle,  the  flame  is  blown  aside,  and  the  more  violently  the 
smaller  r.  A  soap -film  open  to  the  air  on  both  sides  must  be  either 
flat  or  saddle-shaped,  its  curves  equal  and  opposite. 

Large  drops.  The  difficulty  of  the  weight  of  a  drop  distorting  it 
from  the  spherical  shape  can  be  met  by  floating  it  in  another  liquid 
of  the  same  density.     It  is  true  that  the  surface  tension  available 


§350] 


THE   LIQUID   SURFACE 


265 


to  hold  it  together  is  now  only  the  difference  of  those  of  the  two 
liquids,  but  if  one  be  water,  more  than  half  of  it  is  left.  By  pipetting 
aniline  slowly  into  warm  water  you  can  get  a  '  drop  '  as  big  as  a 
bilUard  ball,  pulsating  balloon-like  when  prodded ;  or,  very  easily 
indeed,  melt  naphthalene  in  hot  water,  and  watch  the  great  clear 
drops  being  towed  about  by  tiny  vapour-bubbles,     do  this. 

§  350.  Altered  Vapour  Pressure  over  curved  surfaces.  Since  in 
a  closed  vessel  containing  only  liquid  and  its  vapour.  Fig.  112,  left, 
the  Hquid  rises  a  little  distance  ^  in  a  capillary  tube,  the  Saturated 
Vapour  Pressure  must  be  less  at  the  concave  surface  in  the  tube 
than  at  the  flat  surface  outside,  by  the  weight  of  the  column  of 
vapour  above  which  the  liquid  has  crept ;  i.e.  there  is  a  diminution 
of  Vapour  Pressure  on  the  hollow  side  of  the  surface  of  ^  X  density 
of  vapour   x  g  dynes/cm. 2,  while  inside  the  liquid  is  the  already 


^7f 


B 


Fig.  112. 


discussed  diminution  of  Hydrostatic  Pressure,  h  x  density  of 
liquid  X  g  (which  we  saw  was  T  X  total  curvature  of  surface, 
§  344),  the  two  being  in  the  ratio  density  of  vapour /density  of  liquid. 

.',  Reduction  in  Vapour  Pressure  in  hollow  of  surface 

^density_^j^our   ^  j,  x  total  curvature  of  surface. 
,,       ,,    liquid 

Resorting  to  the  Kinetic  Theory,  suppose,  as  in  Fig.  112,  a  mole- 
cule, at  distance  D  below  the  surface,  and  the  next  jump  of  which 
has  a  probable  length  =  the  radius  shown,  and  is  equally  likely  to 
take  place  in  any  direction.  The  three  diagrams  show  that  its 
chance  of  jumping  through  a  flat  surface  A  is  greater  than  that  of 
getting  through  the  hollowed  surface  B,  and  less  than  through  the 
bulged-out  one  C. 

Hence,  from  a  concave  liquid  surface  a  molecule  has  less  chance  of 
escaping,  while  one  in  the  vapour  above  has  a  better  chance  of  falling 
in ;  both  effects  reduce  the  number  of  vapour  molecules  per  c.c, 
i.e.  the  vapour  pressure.  On  to  this  hollow  surface  vapour  may 
therefore  condense  from  an  Unsaturated  Atmosphere. 

Vegetable  and  animal  substances — cotton,  paper,  wood,  charcoal, 
wool,  hair,  catgut,  etc.— are  built  of  minute  cells,  with  intercellular 


266  MOLECULAR   PHYSICS  §  350 

crevices,  and  their  cell-walls  are  covered  with  pits,  folds,  and 
chinks.  Hence  their  hygroscopic  character  :  every  crevice  holds  a 
concave  water-molecule  trap.  All  get  too  damp  to  electrify  two 
minutes  after  taken  away  from  the  fire  :  your  clothes  will  always 
steam  and  lose  weight  by  ounces  in  front  of  it. 

Similarly,  the  Vapour  Pressure  inside  a  very  Small  Bubble  must 
be  far  below  the  normal ;  hence  the  difficulty  of  starting  boiling 
in  a  liquid  freed  from  gaseous  nuclei.  All  the  dodges  for  avoiding 
'  bumping  '  are  directed  to  giving  the  bubble  a  comparatively  flat 
surface  to  gather  on.     See  also  §  280. 

Conversely,  the  Vapour  Pressure  over  the  convex  surface  of  a  very 
Small  Drop  is  abnormally  high,  for  the  molecules  inside  get  a  better 
chance  of  jumping  out,  and  those  outside  a  less  chance  of  falling  in. 
Little  drops  will  therefore  evaporate  and  supersaturate  an  atmosphere 
with  vapour,  which  must  condense  on  the  flatter  larger  drops.  The 
big  drops  grow  by  the  self-sacrifice  of  the  little  ones,  a  process  which 
is  always  going  on  in  clouds.     See  also  §  354. 

§  351.  Films  and  froth.  Clean  liquids  do  not  form  persistent  films. 
Glistening  bubbles,  or  froth,  on  a  pond,  are  a  hint  not  to  drink,  nor 
even  smell.  On  such  a  surface,  spray-drops  from  your  paddle  would 
not  run  yards  before  breaking  in,  as  they  will  on  the  strong  clean 
surface  of  the  stream.  Any  hanging  film,  such  as  the  wall  of  a 
bubble,  must  be  a  little  stronger  at  the  top,  to  bear  its  own  weight, 
i.e.  there  must  be  a  mixture  of  substances,  and  a  little  automatic 
rearrangement  in  superficial  concentrations. 

That  means  an  alteration  in  the  relative  numbers  of  molecules  of 
the  constituents  of  the  mixture  which  form  the  surface  layer.  Is 
there  any  evidence  for  this  ? 

If  you  let  permanganate  solution  ooze  along  a  tube  packed  with 
inert  siUca  powder,  it  loses  its  colour  by  the  way.  Or  if  you  shake 
up  violet  ink  and  water  with  fuller's  earth,  or  brownish  solution  of 
crude  sugar  with  charcoal  powder,  and  let  settle,  or  filter,  the 
liquid  is  freed  from  the  colour,  which  is  left  sticking  to  the  surface 
of  the  grains — which  is  enormous,  1  c.c.  of  gas-mask  charcoal  is 
computed  to  have  a  total  surface  of  1000  sq.  metres. 

We  even  have  evidence  how  the  molecules  arrange  themselves  in 
the  surface : 

From  time  immemorial,  oil  has  been  used  in  storms  to  smooth  the 
rough  surface  of  the  water  :  fish  oil,  oozing  out  of  pricked  canvas 
bags  slung  over  the  bows  of  the  boat.  Cheaper  hydrocarbon  oil  has 
never  superseded  it,  for  it  does  not  spread  so  well  or  so  far.  Com- 
paring §  337,  you  suspect  at  once  that  it  is  the  gripping  fatty  acid 
again,  spreading  until  every  molecule  has  hold  of  the  water,  jaws 
down  and  smooth  tails  up  ;  while  the  gripless  paraffin  merely  floats 
about.  [The  calming  effect  lies  in  this,  that  the  wind  begins  by 
ruffling  the  surface  of  the  water  into  minute  capillary  ripples  (con- 
trolled by  surface  tension,  §  392),  and  thus  gets  a  hold  on  it,  just  as 
you  get  foothold  on  a  wooden  floor  until  some  house-proud  body 


352] 


THE   LIQUID   SURFACE 


267 


waxes  it,  §  338.  Now,  ripples  mean  increased  surface,  and  local 
tearing  apart  of  the  packed  oil  molecules,  leaving  bare  spaces  of 
water,  which  has  three  times  the  surface  tension,  and  pulls  them  to, 
strongly  ;  so  the  surface  won't  roughen,  and  the  wind  can't  tear  off 
spray  and  fling  it  into  the  boat  and  swamp  her — for  it  isn't  big  ships 
that  use  oil.] 

Again,  well-conducted  soap-bubbles,  before  they  die,  put  a 
'  black  spot '  of  mourning  on  their  heads.  This  is  black  and  almost 
invisible  because  its  thickness  is  only  a  minute  fraction  of  a  wave- 
length of  light,  whereas  the  shiny  parts  of  the  bubble  are  wave- 
lengths or  more.  The  Royal  Institution  used  to  have  soap  films 
that  lasted  six  weeks  ;  why  didn't  they  evaporate  in  a  few  seconds  ? 
Answer,  the  film  has  no  water  in  its  surface,  being  faced  completely 
with  non- volatile  soap  :  here  is  a  section  of  a  vertical  olive-oil  soap 
bubble, 


ditto 

0         H      H 

H      H 

8         H    H 

H     H 

0 

C-(C)^( 
ONa    H 

>-C-(C),    C— H 
H     H      H         ^ 

•S  H-C-(C),-< 
=         H    H      I 

^    ( 

MC). 

-C 
NaO 

0         H 

H     H      H         % 

c 

litto 

C— (C)^C— C    (C)-    C    H  ^ 
ONa  H      HI      H      H 

a  *  double-faced  velvet '  of  sodium  oleate,  the  business  ends  stitched 
together,  the  pile  woven  together  midway  into  a  tough  fabric,  the 
surfaces  of  the  bubble  the  smooth  inert  hydrogens  that  glide  harm- 
lessly in  lubrication,  or  gleam  on  your  shoes.  The  thick  parts  of 
the  film  are  padded  out  with  water  in  the  middle. 

§  352.  Surface  energy.  Another  way  of  regarding  the  liquid 
surface  is  sometimes  useful.  Let  the  tension  T  dynes  pull  back  its 
1  cm.  crossbar  1  cm.,  creating  1  sq.  cm.  of  surface  and  doing  T  ergs  of 
ivork ;  this  Energy  is  stored  in  the  Surface  and  given  back  as  it 
contracts. 

Any  system  free  to  move  does  so  in  the  direction  of  diminishing 
its  Potential  Energy — cashes  all  it  can,  so  to  speak,  into  mobile 
kinetic — water  always  runs  downhill,  etc.  Soap  bubbles  contract 
into  spheres  either  because  of  a  tension  which  keeps  1/r  +  l/r 
constant ;  or  because  a  sphere  has  minimum  surface,  and  therefore 
minimum  Surface  Energy — simply  two  ways  of  looking  at  the  same 
thing. 

The  soap,  or  the  oil,  clings  in  the  surface,  or  the  dye  over  the  vast 
surface  of  the  clay,  because  its  T,  its  surface  energy  per  sq.  cm.,  is 
less  than  that  of  water,  and  therefore  the  energy  of  the  system  is  a 
minimum. 


268  MOLECULAR   PHYSICS  [§  353 

§  353.  This  surface-occupying  action  is  called  Adsorption,  and 
instances  of  it,  most  noticeably  perhaps  among  colloids,  are  innumer- 
able. The  water  in  §  342  was  adsorbed  on  the  glass.  Wireless 
valves  are  flashed  over  with  magnesium  during  evacuation,  §  107, 
to  combine  with  and  immobilize  the  air  or  water  molecules  adsorbed 
on  the  glass,  as  thick  as  they  can  stick.  The  problem  of  when  any 
particular  molecule  will  leave  go,  and  why,  is  one  of  the  Oordian 
knots  of  modern  physics,  and  the  magnesium  cuts  it.  Charcoal 
adsorbs  gases  up  to  several  times  its  volume,  hence  its  use  in  gas- 
masks, etc.  Colloids  have  an  ultra-microscopic  porosity,  and 
their  internal  adsorbent  surface  may  be  immense  ;  silica  gel  looks 
like  glass,  but  is  also  invaluable  in  gas-masks.  Humus  in  the  soil 
adsorbs  food-materials  and  confers  fertility.  Dyes  are  removed 
from  solution  by  adsorption  on  the  colloid  fibres  of  fabrics  or  tissues 
— and  here  is  an  instance  of  a  further  action  :  this  supply  of  surface 
energy  is  able  to  bring  about  chemical  reactions  which  would  fail 
without  it,  often  the  dye  is  driven  into  actual  chemical  combination. 

Another,  precipitated  BaS04  forms  excessively  fine  particles ; 
kept  warm,  the  comparatively  large  surface  energy  of  the  smaller 
particles  drives  them  into  solution,  which  deposits  on  the  larger 
(cf.  §  349),  and  the  precipitate  no  longer  passes  the  filter. 

§  354.  Say  you,  in  §  293  we  are  bidden  to  regard  a  liquid  surface 
as  the  '  envelope  '  of  the  high-jumps  of  rocketting  molecules,  pulled 
home  again  by  the  immense  attraction  of  the  main  bulk ;  how  can 
that  airy  nothingness,  that  anti-aircraft  barrage  in  petto,  have  a 
horizontal  '  surface  tension  '  ? 

The  molecules  at  the  tops  of  their  paths  are  moving  very  slowly, 
and  there  is  time  for  mutual  attraction  among  them  to  take  effect — 
a  very  little  one  compared  with  the  inward  one,  from  the  bondage  of 
which  it  takes  all  the  energy  of  the  '  latent  heat '  to  escape. 

Here  is  another.     A  little  drop  evaporates  unduly  fast 

(a)  because  molecules  have  a  better  chance  of  getting  out  and 

less  of  getting  back,  §  350. 
(6)  because  surface  tension  tightening  all  round  him  helps  to 

squeeze  him  out  of  existence, 
(c)  because  his  surface  is  far  larger,  compared  with  his  mass, 

than  that  of  a  big  drop  ;    and  that  additional  surface  energy 

helps  to  volatilize  him,  §  352. 
{d)  because  as  his  mass  diminishes  so  does  its  total  attraction, 

and  molecules  fly  right  away  into  vapour  more  easily. 

(a)  and  {d),  in  the  Brownian  motion,  §  367,  you  glimpse  the 

molecular  dance. 
(6)  you  have  measured  surface  tension, 
(c)  you  know  all  about  latent  heat. 

Which  is  true  ?    ALL  ! 


§  354]  THE   LIQUID   SURFACE  269 

They  are  different  ways  of  looking  at  the  same  thing,  from  different 
routes  of  attack. 

As  knowledge  increases,  these  three  chapters  are  blending  more 
and  more  into  one — '  For  as  yet  we  know  in  part.' 

Pray  recollect  this  when  you  are  reading  widely  divergent  views 
on  problems  of  Physiology  :  try  to  make  out  if  they  are  not  really 
the  same  in  different  guise. 

And,  later,  if  you  would  meet  the  World  philosophically — 
remember  a  little  instance  in  Natural  Philosophy. 


EXAM   QUESTIONS,   CHAPTER  XXIII 

1.  Explain  the  movements  of  camphor  on  water,  of  oil  on  water,  and  of 
water  on  oil.     (  X  2) 

2.  Distinguish  between  body  and  surface  forces  in  liquids.  Define  Surface 
tension,  and  give  experiments  which  illustrate  it.  Describe  a  method  of 
measuring  its  value ;  in  what  units  is  it  measured  ?     (  X  5) 

3.  Explain  the  rise  in  a  capillary  tube  of  a  liquid  that  wets  it. 

How  high  would  water  rise  between  parallel  plates  of  glass  0-5  mm.  apart 
(given  surface  tension  75)  ?     (  X  3) 

4.  What  is  meant  by  the  statement  that  the  surface  tension  of  oil  is  26 
dynes  per  cm.  and  the  angle  of  contact  between  the  oil  and  glass  is  26°  ? 

To  what  height  will  oil,  of  sp.  gr.  0-85,  rise  in  a  tube  of  diam.  0-4  mm,  ? 

5.  Calculate  how  high  water,  T  =  75,  will  rise  in  a  tube  1  mm.  bore,  and 
in  a  piece  of  wood  with  vessels  0-0013  cm.  diam.  What  would  be  the  effect 
in  a  narrow  tube  which  had  first  been  dipped  in  melted  wax,  and  drained  out 
while  hot  ?     (  X  2) 

6.  How  would  you  show  the  effect  of  temperature  upon  surface  tension, 
and  what  happens  at  the  Critical  Temperature  ?  Water  rises  to  H  in  a  0-5- 
mm.  tube,  and  a  liquid  of  s.g.  0-8  to  H  in  a  0-2-mm.  tube;  compare  their 
surface  tensions.     (  X  2) 

7.  Define  surface  tension.  A  loop  of  silk  is  dropped  on  to  the  surface  of 
a  soap  film,  and  the  film  inside  the  loop  is  pierced ;  what  happens  (a)  if  the 
film  is  flat,  (6)  part  of  a  big  bubble  ? 

8.  Calculate  the  pressure  inside  a  bubble  4  cm.  diam.,  blown  from  soap- 
suds T  =  25.     (  X  3) 

9.  A  tube  3  cm.  diam.  and  0-5  mm.  thick,  of  glass  sp.  gr.  2-5,  is  lowered 
vertically  into  a  solution  of  sp.  gr.  1-05  and  surface  tension  30  dynes  per  cm. 
To  what  depth  must  it  be  immersed  to  appear  neither  heavier  nor  lighter 
than  in  air  ? 

10.  If  this  tube  be  15  cm.  long,  and  closed  at  the  top  with  a  flat  end  the 
same  thickness  as  the  walls,  at  what  depth  will  it  float  if  held  upright  ?  Will 
it  make  any  difference  if  the  closed  end  is  downwards  ? 

1 1 .  Calculate  the  height  of  rise  in  a  capillary  coliunn,  and  show  that,  as 
in  the  middle  of  Fig.  110,  it  depends  on  the  diameter  at  the  surface  only, 
provided  the  liquid  be  once  drawn  up  into  the  narrow  part. 

12.  Water  drips  from  a  pipette;  discuss  the  effect  of  surface  tension.  A 
thin  cover-glass,  a  cm.  wide,  hangs,  like  a  picture,  from  the  hook  of  a  balance. 
Its  weight  is  w,  but  it  takes  W  gm.  to  pull  it  clear  of  a  liquid  surface  into 
which  it  is  dipping ;   calculate  T. 


270  MOLECULAR   PHYSICS 

13.  Define  surface  tension  and  specific  gravity. 

A  clean  glass  disc,  4  cm.  diam.,  hanging  vertically  from  a  balance,  weighs 
1-50  gm.  Water  is  poured  round  it  until  it  is  precisely  half  immersed,  and 
equilibrium  is  regained  at  1'75  gm.  More  water  is  poured  in  until  it  is  com- 
pletely inmiersed,  and  appears  to  weigh  0-90  gm.  What  do  you  deduce  from 
these  figures  ? 

14.  Explain  how  surface  tension  may  introduce  errors  in  the  readings  of 
a  common  hydrometer. 

A  hydrometer  consists  of  a  bulb  with  two  cylindrical  ends  of  radii  1-25 
and  0-31  cm.  If  it  is  placed  in  a  liquid  of  sp.  gr.  0-95,  the  height  of  the  emergent 
cylinder  in  one  position  is  1-24  cm.,  and  in  the  inverted  position  21-8  cm. 
Find  the  surface  tension. 

15.  A  drop  of  water,  squeezed  between  flat  glasses,  spreads  out  to  A  sq. 
cm.  area  and  t  cm.  thickness.  Show  that  the  force  between  the  plates  is 
2  AT  It. 

16.  Describe  several  instances  of  surface  tension.  A  sphere  of  water 
radius  R  is  sprayed  into  1000  drops;    calculate  the  work  done. 


PRACTICAL  QUESTIONS 

Compare  the  diameters  of  narrow  tubes. 

Measure  the  diameter  of  a  tube  by  the  rise  of  water  in  it. 

Compare  the  surface  tensions  of  two  liquids. 

Measure  T  by  the  suspended  ring  method. 


CHAPTER   XXrV 


DIFFUSION 


§  361.  If  a  few  drops  of  bromine  be  poured  into  a  tall  glass  jar 
which  stands  in  a  place  free  from  all  draughts  and  differences  of 
temperature,  their  red  vapour  is  seen  to  spread  slowly  up  the  jar, 
and  its  odour  is  presently  perceptible  in  the  room.  To  spreading 
such  as  this,  which  has  taken  place  without  regard  to  gravity, 
for  bromine  vapour  is  six  times  as  dense  as  air,  and  without  any 
help  from  differences  of  pressure  and  temperature,  the  name  of 
Diffusion  is  given. 

If  the  jar  were  full  of  water,  the  orange  hue  of  dissolved  bromine 
would  creep  upward  in  the  same  way,  but  far  more  slowly  :  a  matter 
of  days  and  weeks  before  it  reached  the  top.  Or  a  darker  dye 
spreads  equally  slowly,  whether  up  through  water,  or  up  or  down 
through  a  jelly,  showing  that  currents  have  nothing  to  do  with  it — 
such  currents  of  denser  solution  as  stream  down  from  a  lump  of 
sugar,  held  in  a  spoon  high  up  the  side  of  the  teacup  (and  even 
then  you  must  stir). 

§  362.  Measurement  of  rate  of  diffusion.  The  rates  of  inter- 
diffusion  of  pairs  of  Gases  have  been  measured  by  enclosing  them  in 
the  halves  of  a  vertical  cylinder  with  a  diaphragm  in  the  middle, 
the  lighter  gas  being  in  the  upper  half.  The  diaphragm  is  cautiously 
sUpped  out  and,  after  a  definite  interval,  replaced  ;  the  contents  of 
each  half  are  analysed,  and  the  rate  is  calculated. 

It  is  found  that  lighter  gases  diffuse  faster,  hydrogen  and  marsh 
gas,  or  hydrogen  and  air,  interdiffuse  nearly  five  times  as  fast  as 
air  and  carbon  dioxide,  the  latter  and  nitrous  oxide  going  one- 
third  slower  still.  Hence  diffusion  meters  can  be  used  as  '  firedamp 
detectors.' 

Hydrogen  travels  through  air  about  half  as  fast  as  heat  through 
copper.  Two  inches  of  COg  at  the  bottom  of  a  2-ft.  tall  jar  spreads 
uniformly  through  it  in  2  hr.  Just  exactly  as  in  the  diffusion  of 
Heat  by  conduction,  §  241,  the  time  taken  is  proportional  to  the 
square  of  the  distance  to  be  travelled ;  the  COg  would  have  filled 
a  little  jar  nearly  uniformly  in  a  minute  or  two  ;  but  it  evidently 
requires  some  process  far  more  violent  than  Diffusion  to  save  us 
from  stifling  beneath  a  city  atmosphere. 

Stirring,  which  brings  together  portions  of  widely  different 
concentrations  in  very  thin  streaks,  hence  ensures  rapid  and 
complete  mixing  by  diffusion.  Instance  the  streakiness  on 
stirring  together  syrup  (or  whisky)  and  water,  and  its  quick 
disappearance. 

271 


272  MOLECULAR   PHYSICS  [§  363 

§  363.  Diffusion  through  porous  diaphragms.  Experiments  on 
diffusion  with  fluids  in  open  contact  are  so  liable  to  be  disturbed 
by  currents  that  sheets  of  porous  solids  are  commonly  put  between 
them,  to  stop  this  wholesale  mixing. 

The  spontaneous  diffusion  that  goes  on  through  these  plates 
is  to  be  distinguished  from  the  more  familiar  Filtration,  Trans- 
piration, or  Effusion,  in  which  any  particles  smaller  than  the  pores 
are  driven  through  pell-mell  by  a  one-sided  gas  pressure,  or  by  weight 
of  liquid.  This  has  to  be  avoided  by  keeping  the  pressures  on  both 
sides  equal  throughout. 

It  was  with  '  plates  '  made  from  fine  plaster,  meerschaum,  etc., 
that  Graham  experimented  in  1 850 .  They  show  typically  the  greater 
speed  of  diffusion  of  lighter  gases.  There  is  a  striking  lecture 
experiment  in  which  an  inverted  porous  battery-pot  is  sealed  on 
the  top  of  a  long  tube  dipping  in  water.  Over  the  pot  is  held  an 
inverted  bell- jar  of  hydrogen  ;  a  rapid  stream  of  bubbles  drives  out 
of  the  tube.  The  bell-jar  is  removed,  and  the  water  quickly  climbs 
the  tube,  as  the  hydrogen  that  has  entered  the.  pot  diffuses  out  of  it 
again,  faster  than  air  can  enter,  even  with  the  diminishing  pressure 
inside  in  its  favour.  Again,  sal-ammoniac  vapour  is  passed  through 
red-hot  churchwarden  pipe-stems  ;  the  pipes  smell  of  ammonia, 
while  the  gas  that  emerges  at  the  far  end  reddens  litmus  ;  the  salt 
has  split  into  ammonia  (vap.  density  8-5)  and  hydrochloric  acid 
(18-2),  and  the  lighter  gas  has  escaped  more  rapidly  through  the 
clay  walls. 

Graham's  experiments  led  him  to  the  Law  that,  other  things 
being  equal.  The  rate  of  diffusion  of  a  gas  is  inversely  proportional 
to  the  square  root  of  its  density. 

This  law  tallies  with  Kinetic  Theory.  The  molecules  of  gases  at 
the  same  temperature  possess  the  same  average  kinetic  energy 
Jw?;2,  hence  their  speed  vazlj^/m,  see  §201.  And  since  by 
Avogadro's  law  the  number  of  molecules  per  c.c.  of  any  gas  at  the 
same  temperature  and  pressure  is  the  same,  m,  the  mass  of  one  mole- 
cule, is  proportional  to  the  density  of  the  whole  gas  ;  hence  molecular 
speed  varies  inversely  as  square  root  of  density.  You  will  admit 
that  the  rate  of  diffusion  is  proportional  to  the  speed  of  the  diffusing 
molecules  :   hence  the  Law. 

Each  gas  present  in  a  mixture,  on  either  side,  behaves  inde- 
pendently of  the  others,  simply  on  account  of  the  characteristically 
different  speed  of  its  molecules.  Each  will  in  time  reach  the 
same  partial  pressure  {i.e.  molecular  population  density)  on  either 
side,  but  the  lightest  reaches  equilibrium  first. 

§364.  Selective  transmission  of  gases.  The  diaphragms  dis- 
cussed above  are  porous  in  the  ordinary  sense,  that  a  little  pressure 
will  drive  any  gas  through  them.  But  there  are  several  things, 
commonly  regarded  as  quite  '  air-tight,'  which  are  permeable  by 
particular  gases.  Red-hot  platinum  is  permeable  to  hydrogen  : 
a  tiny  blind-ended  platinum  tube  is  sealed  to  a  vacuous  X-ray  tube, 


§  366]  DIFFUSION  273 

§  912  ;  heated  in  a  spirit-lamp  flame  it  at  once  admits  traces  of 
pure  hydrogen.  Thin  india-rubber  balloons  blown  with  COg  soon 
collapse,  and  oxygen  passes  through  them  2^  times  as  fast  as  nitrogen. 
Perhaps  one  may  say  that  the  gas  selectively  dissolves  in  the  solid, 
diffuses  about  in  it,  and  evaporates  off  from  the  other  side  ;  as  when 
it  quickly  finds  it  way  through  a  soap-bubble. 

§365.  The  diffusion  of  Liquids  through  membranes.    Osmosis. 

The  passage  of  liquids  through  ordinary  porous  materials  is  usually 
a  mere  question  of  Filtration,  a  gross  mechanical  process,  forced 
by  pressure,  or  induced  by  the  capillary  drag  of  surface  tension. 
The  chemist  typically  uses  a  filter-paper  to  retain  undissolved, 
and  transmit  dissolved,  substances  ;  it  takes  a  specially  fine  one  to 
retain  BaS04  ;  the  porcelain  tubes  of  the  Pasteur  or  Berkefeld 
filters  sterilize  water  because  their  pores  are  too  small  to  admit 
bacteria  :  there  has  been  a  long  wrangle  over  '  filter-passing 
viruses,'  but  some  are  now  admitted  to  be  actual  organisms  smaller 
than  bacteria.  Finally,  there  are  toxins  and  salts  no  known  filter 
can  stop,  and  these  we  may  call  '  dissolved.' 

We  have  seen  that  Diffusion  in  Liquids  is  a  very  slow  proceeding 
kinetically,  on  account  of  the  dense  crowd  a  molecule  has  to  jostle 
through.  It  follows  that  '  porous  '  pots  and  papers  are  inefficient 
in  studying  it,  the  least  alteration  in  pressure  causing  an  infiltration 
that  quite  swamps  its  slow  effects.  Much  less  porous  partitions 
must  be  used,  '  water-tight '  things  like  bladder,  parchment  paper, 
etc.  Parchment  paper  is  made  by  dipping  soft  paper  into  strong 
sulphuric  acid,  or  caustic  soda,  for  a  moment,  and  washing  copiously 
— the  same  process  as  '  mercerising '  cotton.  The  flattened  fibres 
swell  to  a  permanent  turgidity,  closing  the  passages  between  them 
to  crevices  beyond  the  ken  of  the  ultra-microscope — smaller  than 
some  molecules,  but  still  bigger  than  others. 

Liquid  Diffusion  through  these  is  designated  Osmosis;  by  this 
the  Living  Cell  of  plant  or  animal  takes  up  its  nutriment  from,  or 
gives  out  its  elaborated  or  its  waste  products  to,  the  watery  fluids 
bathing  its  walls.  Accordingly,  the  most  interesting  part  of  the 
subject,  and  the  most  studied — that  to  which  we  shall  confine  our- 
selves here — is  that  dealing  with  the  diffusion  of  water  and  sub- 
stances dissolved  in  it. 

§  366.  Graham  observed  that  parchment  paper  permits  the 
passage  of  crystallizable  substances,  *  crystalloids,'  from  solution 
on  one  side  to  weaker  solution  on  the  other,  but  does  not  transmit 
gum,  albumen,  starch,  globulins,  etc.,  *  colloids '  {colla  =  glue). 
On  this  he  founded  the  process  of  Dialysis  :  a  little  drum,  the 
*  dialyser,'  containing  mixed  solutions,  is  floated  on  water  ;  only 
the  crystalloids  pass  through  the  parchmentized  paper  bottom,  and 
very  much  more  rapidly  when  hot.  This  process  is  useful  in  medico- 
legal work  for  separating  traces  of  mineral  poisons  and  alkaloids 
from  the  mass  of  colloids  which  mask  chemical  tests  ;   and  batteries 


274  MOLECULAH  PHYSICS  [§306 

of  hundreds  of  foot-long  saccate  tubes  of  collodion  are  kept  at  work  j 
in  the  preparation  of  anti-rabies  vaccine. 

The  ultimate  particles  of  colloids  in  solution  have  in  several 
instances  been  actually  detected  by  the  *  ultra  '-microscope  ;  and 
freezing-point  determinations,  §  377,  show  that  their  mass  is  scores 
or  hundreds  of  times  that  of  a  molecule  of  crystalloid.  The  natural 
explanation  therefore  is,  that  the  colloid  particles  are  too  big  to  get 
through.  I 

All  the  membranes  used  in  the  study  of  Osmosis — ^parchment  > 
paper,  copper  ferrocyanide,  etc. — are  colloid  in  character.  Jellies 
are  typical  colloids  :  some  ultra-microscopists  have  claimed  that 
they  have  a  fibrillar  sponge-like  structure,  in  which  vacuoles  of 
liquid  are  dispersed  :  it  seems  likely  that  the  shadow  has  been 
mistaken  for  the  substance,  and  that  they  are  more  like  a  mass  of 
boiled  sago,  solid  granules  with  inter-granular  crevices  through 
which  crystalloids  can  creep  almost  as  readily  as  through  open  water 
— instance  the  spreading  of  the  red  dye  when  '  raspberry  '  jelly 
lies  on  '  lemon  '  jelly — but  the  bigger  colloids  are  much  hindered. 

The  amoeboid  extravasation  of  leucocytes,  through  the  walls  of 
the  capillaries  in  the  vicinity  of  a  lesion,  is  a  proceeding  on  a  far  larger 
scale  than  that  contemplated  here. 

§367.  The  Brownian  Motion.  Robert  Brown,  the  botanist, 
watching  in  1827  the  bursting  of  some  asclepiad  pollen-grains 
under  his  microscope,  observed  a  perpetual  jiggling  motion  going 
on  in  the  granular  matter  exuded.  Naturally,  he  took  this  at  first 
for  a  sign  of  life,  but  went  on  to  find  exactly  the  same  thing  in  a 
wide  variety  of  other  minute  particles,  including 
burnt  ash,  and  all  sorts  of  inorganic  matter. 

Brown's    microscope    is    shown    complete    in 
Fig.   113  :    it  was   a    l/32nd-in.   focus   biconvex 
lens,    and   he   very  wisely  preferred  it  to   any 
other  available  at  that  time,   v.    §  629.      Your 
Fig.  113.  modern    sixth,    however,    backed    by    its    eye- 

piece, has  much  better  seeing -power ;  so  take 
the  slightest  trace  of  '  burnt  umber  '  in  water,  or  '  ivory  black,' 
or  '  carmine,'  or  even  fine  crushed  ash,  and  do  not,  from  sheer 
laziness,  go  without  a  sight  of  this  most  fundamental  movement 
in  Nature  :  every  particle  afloat  is  dancing  an  aimless  little  jig ; 
and  the  smaller  the  merrier,  as  you  can  see  best  with  an  '  ultra-micro- 
scope,' §642. 

For  eighty  years  the  mysterious  ceaseless  movement  was  ascribed  to 
whatever  force  happened  to  be  in  vogue — surface  tension,  electricity, 
etc. — ^until  Perrin  in  1908  settled  the  question  for  good,  and  showed 
it  to  be  the  motion  of  bulky  partners  in  that  eternal  dance  of  the 
molecules  the  very  existence  of  which  had  been  a  matter  of  pure 
faith  with  the  physicist.  In  that  dance,  the  essential  is,  that  the 
average  energy  of  motion,  ^mv^,  of  every  partner,  is  the  same, 
irrespective  of  its  mass. 


§  368]  DIFFUSION  275 

Perrin  obtained  particles  of  measured  mass  by  pouring  alcoholic 
solution  of  gamboge  into  water  ;  in  this,  of  course,  the  resin  is 
insoluble,  and  was  immediately  thrown  out  as  a  cloudy  suspension 
of  minute  spherules.  This  he  'fractionated'  by  systematically 
repeated  centrifuging,  until,  after  months,  he  obtained  fractions 
containing  spherules  of  very  uniform  size.  This  he  measured  by 
counting  in  long  rows  under  the  micrometer  microscope,  and  then 
deduced  their  mass  m  from  their  density,  which  was  that  of  a  solution 
in  which  they  refused  to  centrifuge. 

Your  observation  of  the  amazing  irregularity  and  complete 
aimlessness  of  the  motion  of  these  particles,  jostled  by  invisible 
molecules  on  all  sides,  will  show  you  that  any  measure  of  their 
actual  average  velocity  v  is  hopeless. 

But  suppose  you  tore  up  slices  of  bread  and  threw  the  fragments 
on  a  pool  full  of  small  fry.  All  would  soon  be  seen  performing 
aimless  movements,  the  crumbs  with  great  apparent  activity  as  two 
or  three  tiddlers  snatched  them  hither  and  thither,  larger  pieces  less 
easily  moved  and  more  frequently  pulled  at  by  more  numerous 
mouths,  and  a  whole  slice  hardly  visibly  moving ;  yet  the  average 
energy  of  motion  of  each  is  the  same,  and  equal  to  that  of  one  fish. 
Presently  you  would  notice  that  the  smaller  faster  pieces,  for  all 
their  aimlessness,  were  slowly  being  drifted  away  in  all  directions 
more  than  were  the  larger  sluggish  ones,  and  you  might  make 
out  a  relation  between  a  '  Rate  of  Diffusion '  and  *  Mass  of 
Particle.' 

Under  a  micrometer  ruled  in  squares,  Perrin  watched  many 
hundred  particles  for  2  min.  each  :  then  plotting  on  a  target, 
from  the  centre,  the  direction  and  distance  each  one  had  made  good, 
the  target  looked  as  if  fairly  struck  by  a  charge  of  shot ;  counting 
up  the  hits  in  successive  rings,  he  deduced  the  average  rate  at  which 
they  had  diffused  out  from  the  centre. 

Then,  by  Kinetic  Theory,  he  calculated  how  far  particles  of  that 
same  mass  ought  to  diffuse,  considered  as  big  molecules.  His 
smallest  ultra-microscopic  particles  were  1000  million  times  the 
mass  of  an  air  molecule,  his  largest  15,000  times  more  massive  still ; 
his  liquids  varied  in« viscosity  from  1  to  125. 

The  results  of  this  difficult  research  are  best  expressed  by  quoting 
Avogadro's  number  (of  molecules  in  1  gm.-mol.)  as  calculated  from 
them  :  70,  55,  72,  78  (big  grains),  64,  69  :  the  average  of  two  dozen 
other  methods  for  this  number  is  62  ( x  lO^^j, 

As  to  actual  molecules,  *  each  of  those  of  the  air  we  breathe  moves 
with  rifle-bullet  speed,  flies  in  a  straight  line  between  two  *  collisions  * 
1 /10,000th  mm.,  and  is  therefore  deviated  5000  million  times  per 
second.  Three  millions  in  line  measure  1  mm.,  and  20,000  million 
make  a  thousand  millionth  of  a  milligram.' 

§368.  Osmotic  Pressure.  When  one  succeeds  in  getting  a 
sufficiently  tight  membrane,  one  studies  Osmosis  in  another  way, 
originating  in  an  observation  made  by  the  Abb^  Nollet  in  1748. 


276  MOLECULAR   PHYSICS  [§  368  -~ 

He  found  that  a  bladder  full  of  sweet  wine  swelled  and  burst  in  water, 
while  one  of  water  collapsed  when  immersed  in  wine. 

The  diffusing  water  forces  its  way  into  the  sugary  solution  even 
in  spite  of  a  pressure  which  increases  until,  if  the  membrane  can 
sustain  it,  it  reaches  the  maximum  Osmotic  Pressure  characteristic 
of  the  solution  and  its  concentration.  , 

Domestic  cookery  affords  excellent  illustrations.  Mushrooms 
sprinkled  with  salt  slowly  exude  a  dark  juice  which,  boiled  with 
spices,  constitutes  ketchup.  The  salt  dissolves  in  their  superficial 
moisture  to  a  strong  brine,  the  watery  cell  sap  '  exosmoses  '  through 
the  cell  walls  to  dilute  it.  More  salt  dissolves,  and  the  process 
goes  on  until  the  cells  are  drained  almost  dry.  Again,  it  is  desired 
to  stew  some  hard  windfall  apples.  Cut  up,  covered  with  sugar, 
and  left  over-night,  there  results  a  sjrrup  on  which  float  shrivelled 
pieces,  tough  as  leather.  On  the  contrary,  cut  up  and  stewed  in 
plain  water,  the  apples  swell  and  their  cells  burst  to  a  pulp  which 
can  now  be  sugared  ad  libitum.  In  the  former  case  water  passed 
from  the  unripe  cell  sap  into  the  stronger  syrup,  in  the  latter  case 
water  '  endosmoses  '  into  the  acid  sap  until  the  cell- walls  give  way. 

The  process  can  be  followed  under  the  microscope,  using  any  cells  . 
you  can  find  with  strongly  coloured  contents,  such  as  those  of  the 
filamentous  algae,  or  of  the  beaded  hairs  on  the  stamens  of  the  garden 
spider-wort  {Tradescantia  virginica).  Examining  under  a  high 
power,  irrigate  with  strong  brine  or  syrup.  The  protoplasmic 
lining  of  the  cells,  the  '  primordial  utricle  ' — the  live  cell  itself — 
will  be  seen  to  leave  the  cell  walls,  and  contract,  as  the  water  of 
the  cell  sap  passes  through  it,  out  into  the  strong  solution.  The  cell 
is  '  plasmolysed.'  Irrigated  now  with  fresh  water,  it  expands 
again — in  fact  the  blue  cells  of  the  staminal  hairs  become  more 
turgid  and  threaten  to  burst,  like  the  apple  cells. 

The  crispness  of  a  fresh  green  leaf  is  due  to  the  turgidity  of  its 
cells,  and  this  is  maintained  by  the  '  endosmotic  '  diffusion  of  the 
watery  stem-sap  into  their  more  concentrated  contents,  up  to  an 
osmotic  pressure  of  20  atmos.  or  more.  It  takes  a  hard  pinch  to 
really  damage  the  '  soft '  tissue  stiffened  out  with  this  pressure. 
Contrast  the  flagging  leaf  from  which  water  vapour  has  transpired 
without  renewal. 

Loss  of  turgidity  paralyses  the  cell.  Hence  it  is  that  the  micro- 
fungi — moulds,  bacteria,  etc. — although  their  capability  of  pro- 
ducing high  osmotic  pressures  gives  them  enormous  activity,  can 
make  no  headway  in  well-boiled  jam,  for  this  represents  a  solution 
more  concentrated  than  ('  hypertonic  '  to)  their  cell-contents,  and 
plasmolyses  them. 

Early  experiments  on  Osmotic  Pressure  were  those  of  Pfeiffer 
and  de  Vries.  They  soaked  purple  epidermal  cells  of  the  leaves  of 
Tradescantia  discolor  {Zebrina ;  get  a  bit  from  any  greenhouse)  in 
1%  KNOg  (saltpetre)  solution  ;   the  cells  reached  a  healthy  equili- 


§  369]  DIFFUSION  277 

brium  condition  in  an  hour.  They  were  then  irrigated  with  various 
solutions,  and  would  show,  by  incipient  plasmolysis,  any  variation 
corresponding  to  0-1%  KNOg.  The  following  is  an  extract  from 
their  list : 


Equivalent  to,  or  '  isotonic  '  with,  1  % 
KNO3  solution  {decinormal)  are  : — 


5%  cane-sugar 

2-7%  glucose 

0-58%  common  salt  NaCl  . 

1*4%  glycerine 

2-0%  potassium  citrate     . 

1-8%  magnesium  sulphate  MgSOj 

41%  gmn  arabic 


Osmotic  pressure  of 
1%  solution. 

0-7    atmos. 

1-25  „ 
61 

2-55  „ 

1-75  „ 

1-95  „ 

0085  .. 


§369.  Solutions  'isotonic  with,'  i.e.  of  equal  osmotic  pressure 
with,  the  Blood,  are  of  the  very  greatest  importance,  and  too  much 
heed  cannot  be  paid  to  them.  '  Normal  saline  '  must  be  used 
in  the  micro-examination  of  tissues.  Lotions  to  be  applied  to 
inflamed  or  sensitive  surfaces  should  put  no  osmotic  strain  upon 
them  ;  a  nasal  douche  must  approximate  not  only  in  temperature, 
but  also  in  saline  content,  0-9%  NaCl,  to  the  mucosa  of  the  nose ; 
the  efficiency  of  a  well-known  disinfectant  largely  depends  on  its 
dilution,  as  directed,  whereby  it  becomes  a  salt  solution  of  this 
strength,  and  the  contained  hypochlorite  diffuses  into  the  tissues  on 
perfectly  level  terms,  without  the  slightest  osmotic  check.  The 
grievous  pain  of  some  bulky  hypodermic  injections  is  due  to  the 
operator's  neglect  of  this  essential  means  of  avoiding  shock  :  the 
sterile  salt  and  sugar  transfused  into  a  patient's  veins  to  stave  off 
collapse  from  loss  of  blood  must  have  this  equivalent  concentration. 
If  stronger,  the  corpuscles  and  living  cells  plasmolyse ;  if  weaker, 
they  may  burst. 

Thirst  is  relieved  by  weak  salt  water  which  up  to  isotonic  strength 
has  hardly  any  taste  :  used  to  restore  their  salt  wastage  in  excessive 
perspiration,  by  the  miners  in  the  hot  Staffordshire  mines,  it  has  done 
away  with  a  most  baffling  debilitating  disease.  Brine,  '  hypertonic 
saline,'  has  been  used  to  draw  infected  fluid  from  wounds  ;  its  violent 
plasmolysis  of  the  gastric  epithelium  gives  it  its  emetic  value  : 
the  hydragogue  action  of  full  doses  of  epsom  salts  is  due,  in  all 
probabiUty,  to  osmotic  drainage  from  the  walls  of  the  gut  into  the 
concentrated  solution. 

The  wine  and  oil  of  the  Good  Samaritan  made  no  bad  dressing 
for  a  wound  :  no  antiseptics  can  reach  deeply  enough  into  infected 
tissues,  infection  must  be  pushed  out  by  drainage  from  within,  and 
this  would  take  place  copiously  in  the  osmotic  effort  to  dilute  the 
sugary  and  spirituous  liquor ;  while  the  oil,  in  which  no  germ  can 
thrive,  prevents  local  drying  and  cracking,  and  shuts  out  any  further 
infection  from  without. 


278 


MOLECULAR   PHYSICS 


[§370 


§  370.  Measurement  of  osmotic  pressure.  An  Osmometer  which 
has  given  us  good  service  for  low  pressures  and  bulky  molecules  is 
shown  in  Fig.  114,  in  it  fish  swim-bladder  is  gripped  between  leaden 
grids,  rebated  into  the  mouths  of  two  bronze  capsules,  provided 
with  narrow  '  standpipes  '  which  act  as  pressure  gauges.  But  the 
classic  cell  used  for  physical  measurements  of  osmotic  pressure  is 
composed  of  a  membrane  of  the  colloid  brown  copper  ferrocyanide, 
precipitated  in  the  pores  of  a  small  porous  battery-pot  to  give  it^ 
the  needful  mechanical  strength,  and  shown  as  a  thin  layer  in  the 

middle  of  the  wall  of  the 
broken  piece  in  Fig.  1 15. 
The  jar  is  attached  to  a 
mercury  gauge,  and  is 
then  filled  with  the 
solution,  sealed  up,  and 
plunged  into  water. 
The  gauge  rises  hour  by 
hour  as  the  water  slowly 
crowds  in  up  to  the  high 
osmotic  pressure  of  the 
solution. 

This  makes   a   good 
semi-permeaWe     mem- 
brane,  for   it  will  not 
transmit      sugar      and 
many      other     organic 
substances  at  all,  while 
it  is  quite  permeable  to 
water.      A    well-made 
cell,   indeed,    transmits 
only  small  traces  of  the  alkaline  chlorides  and  nitrates,  but   one 
usually  has  to  be  content  with  less  perfection  than  this,  and  to 
shun  these  very  easily  diffusible  substances. 

NOTE.— The  rest  of  the  chapter  refers,  QUANTITATIVELY, 
only  to  DILUTE  SOLUTIONS,  very  little  has  been  made  out  about 
strong  solutions. 

§371.  Theory  of  osmotic  pressure.  We  have  stepped  from 
coarse  to  fine  and  finer  holes  through  our  dividing  wall,  and  from 
visible  to  ultra-microscopic  colloid  particles.  We  see  no  reason 
to  suppose  that  the  membranes  with  the  most  stuffed-up  pores  that 
we  can  utilize  differ  essentially  in  any  other  way  from  those  with 
larger  pores.  Nor  that  molecules  of  sugar,  etc.,  '  in  solution  '  differ 
in  any  physical  essential,  except  size,  from  those  of  gum  or  the  ultra- 
microscopic  particles  we  watch  in  Brownian  movement.  So,  if 
they  cannot  get  through  a  semi-permeable  membrane,  but  the  mole- 
cules of  solvent  water  can,  we  can  put  it  down  to  the  holes  in  the 
wall  being  too  small  for  the  big  chaps,  but  big  enough  for  the  water 
molecules.     Let  us  see  what  happens. 


Fig.  114. 


Fig.  115. 


§  371]  DIFFUSION  279 

On  one  side  we  have  '  Solution '  containing  particles  of  possibly- 
very  mixed  sizes,  from  broken  bits  of  molecules,  molecules,  couples 
of  molecules,  companies  of  molecules,  up  to  ultra-microscopic  or 
microscopic  grains  kept  too  much  alive  to  settle  :  every  one  of  these 
particles  has  the  same  average  energy  of  motion,  §  202  ;  and  none 
can  get  through  the  wall.  Some  may  be  positively  berried  round 
with  molecules  of  solvent ;  m  is  greater  and  \v^  correspondingly 
less.  Conceivably  some  may  be  clinging  to  solvent  to  quite  a  distance, 
as  the  stone  of  a  mango  clings  to  the  pulp  ;  again,  as  with  water 
clinging  to  a  moving  ship  or  air  to  a  swinging  pendulum,  the  only 
effect  in  calculation  is  that  they  are  unexpectedly  increased  in  mass 
and  diminished  in  speed.  All  these  we  class  as  molecules  of  *  Solute,' 
and  each  moving  m<iss  counts  one.  The  only  condition  is  that  ail 
shall  join  in  the  molecular  dance  :  if  Gravity  is  too  much  for  any 
one,  and  it  sinks  to  rest  on  the  floor,  like  big  starch-grains  or  other 
British  heavy-weights,  it  goes  out  of  count  at  once. 

Mixed  with  these  are  very  many  (the  solution  is  dilute)  little 
Solvent '  molecules  which  pass  the  frontier  in  perfectly  free  inter- 
course with  their  unmixed  fellows  on  the  other  side.  These  little 
molecules  may  not  all  be  the  same,  e.g.  we  might  have  whisky  and 
water  as  the  solvent,  but  they  can  all  get  through — ^if  not,  they  are 
Solute  molecules. 

If  molecules  get  stuck  and  block  the  gateways  to  any  extent, 
the  experiment  comes  to  a  standstill,  and  the  only  thing  to  do  is  to 
scrap  it  and  start  afresh  ;  we  can't  legislate  for  that  nonsense. 

Now,  by  §  102,  III,  the  pressure  in  a  fluid  at  rest,  the  weight  of 
which  can  be  neglected,  is  the  same  throughout,  i.e.,  so  far  as  the 
freely  inter-communicating  Solvent  is  concerned,  there  is  no  pressure 
to  take  into  account. 

Turn  to  the  Solute  molecules  ;  they  batter  on  their  side  of  the  wall, 
each  hitting  a  blow  independent  of  its  size,  the  standard  average 
blow  for  any  free  molecule  at  that  temperature  (cf.  §201).  We 
have  kept  them  well  apart — the  solution  is  quite  dilute,  for  any 
accuracy  in  practice  — as  far  apart  as  gas  molecules  commonly  are, 
probably  ten  or  more  solvent  molecules  lie  in  line  between  any 
two  of  them ;  anyway,  they  move  as  freely  among  their  fellows 
as  do  gas  molecules  :  their  aggregate  bombardment  of  the  wall  is 
just  the  same  as  that  struck  by  the  same  number  of  molecules  of 
any  Gas  at  the  same  temperature  and  molecular  population-density. 

Therefore  their  Osmotic  Pressure  is  not  only  equal  to,  but  is  the 
same  thing  as,  the  Gas  Pressure  produced  by  the  same  number  of 
molecules. 

Take  a  single  instance.  Pfeiffer  found  that  1%  sugar  solution, 
i.e.  I  gm.  in  100  c.c,  at  7°  C,  gave  an  osmotic  pressure  of  50-5  cm.  of 
mercury.  You  will  find  that  the  vapour -pressure  of  C12H22OH 
calculates  out  at  50-8  cm.,  reckoning  that  1  mol  wt.  (342)  in  1  litre 
should  give  22-32  atmos. 


280  MOLECULAR   PHYSICS  [§372 

§  372.  Osmosis  being  of  the  most  intense  importance  to  medicals, 
you  may  just  as  well  cultivate  clear  ideas  about  it.  There  are  those 
who  will  fall  foul  of  the  foregoing  argument  :  bear  with  them  if 
you  must,  but  beware  of  getting  mixed ;  as  in  all  physical  contro- 
versy, accept  nothing  that  asks  you  to  jump  ;  cling  tight,  hand  over 
hand,  from  beginning  to  end  : 

Myself  when  young  did  earnestly  frequent 
Doctor  and  scient  and  heard  great  argument 
About  it  and  about,  but  evermore,  came  out 
By  that  same  door  as  in  I  went — 
until  I  fixed  on  the  plain  discussion  I  have  set  before  you. 

Some  have  claimed  that  H2O,  (H20)2,  and  (H20)3,  §  265,  play  the 
great  part ;  but  at  least  half  the  osmotic  research  of  the  world  is 
done  on  water -free  solutions. 

It  is  said  that  surface  tension,  solution  in  the  membrane,  ad- 
sorption, chemical  action,  electrical  attraction,  control  the  action  of 
the  membrane — all  of  them  are  manifestations  of  one  dominant  inter- 
particular  force,  in  the  end  making  it  hard  or  easy  for  one  set  of 
particles  to  pass;  the  idea  most  easily  expressible  by  'gateway.' 
Take,  if  you  like,  one  instance.  Grease  a  patch  on  a  filter-paper  and 
then  filter  cod-liver-oil  emulsion  through  it ;  oil  oozes  through  the 
greasy  patch,  water  through  the  rest  of  the  paper  :  there,  now  go 
and  join  the  surface-energy  actionists. 

It  is  urged  that  if  the  molecular  bombardment  of  the  solute 
molecules  provides  a  great  pressure,  then  that  of  the  thousand 
times  more  numerous  solvent  molecules  would  burst  any  vessel 
containing  them.  But  in  an  open  vessel  the  pressure  is  the  same 
throughout,  §  102,  III,  and  is  only  that  of  the  atmosphere  upon 
the  open  top.  Think  of  the  cloud  of  rocketing  droplets  of  §  293  ; 
on  the  '  envelope  '  plane  just  exactly  touching  them  at  the  top  of 
their  leap  they  press  not  at  all ;  on  one  a  millimetre  lower  down 
they  would  rain  blows  by  thousands  :  the  first  is  a  free  liquid  surface, 
every  molecule  at  the  end  of  its  flight  from  the  attraction  of  the  mass, 
the  second  is  an  attempt  to  press  the  wall  closer  in.  Fill  a  bottle 
completely  with  water,  and  insert  a  straight  loose-fitting  cork ; 
hold  it  by  the  neck  and  smack  the  cork  with  a  board,  i.e.  try  to  drive 
a  few  more '  solvent '  molecules  into  the  body  of  the  bottle  ;  up  jumps 
the  pressure  and  out  drops  the  bottom. 

There  is  no  need  to  stop  short  at  '  gateways  '  as  meaning  just 
gaps  in  a  board  fence  ;  they  may  be  intricate  gateways  through  the 
hills,  long  tedious  passes  through  mountain  country.  This  gradual 
soaking-through  of  solute  into  their  plant-cells,  in  the  experiments 
of  Pfeiffer  and  De  Vries ;  how  is  that  to  be  explained  ?  The 
demonstration  of  the  streaming  of  the  protoplasm  in  the  staminal 
cells  of  Tradescantia,  the  cells  of  Elodea,  or  many  others,  is  a  common- 
place of  biology;  the  layer  is  thick,  and  it  is  fluid.  There  is  no 
solid  ground  into  which  to  drive  the  gateposts ;  fluid  gives  way 
continuously  to  any  persistent  force,  however  small,  §  335  ;  given 
time,  the  solute  will  struggle  through — ^unless  it  is  poisonous,  and 


§  373]  DIFFUSION  281 

the  protoplasm  solidifies  (coagulates),  and  then  the  specially  interest- 
ing application  to  live  matter  ceases. 

§  373.  In  the  light  of  this  theory  let  us  glance  at  one  or  two  of 
the  myriad  biological  instances  of  Osmosis. 

The  smallest  of  the  Protozoa — those  with  no  pseudopodia  nor 
pulsating  vacuole — have  to  depend  entirely  on  the  endosmosis 
of  nutrient  solution  and  the  exosmose  of  excretory  materials. 

In  that  same  way  lives  the  foetus  in  utero,  through  the  placenta. 

Later,  the  cells  lining  our  alimentary  canal  become  bathed  on  the 
one  side  by  solution  rich  in  sugars,  food-proteins,  and  their  '  break- 
downs,' and  on  the  other  side  by  the  body-lymph  ;  and  their  duty 
is  to  effect  a  transfer. 

They  cannot  do  this  by  osmosis  only. 

Do  not  get  the  idea  that  Osmosis  does  everything ;  it  does  half. 
Chemical  action  and  Osmosis  share  the  job  between  them  :  chemical 
action  is  the  bricklayer  who  builds  up,  who  makes  bricks  move  up- 
wards instead  of  '  naturally  '  downwards  ;  osmosis  is  the  labourer 
who  brings  along  the  supply  of  bricks  and  mortar,  as  the  builder 
combines  them  into  structures  which  no  longer  cumber  the  scaffold, 
and  are  largely  beyond  his  grasp. 

Food  materials  diffuse  into  the  cell  because  they  are  going  from 
stronger  to  weaker  solution  :  the  cell  is  not  so  impermeable  as  to 
pass  water  only,  and  the  digestive  ferments  break  things  down 
until  they  arrive  at  molecules  small  enough  to  pass  in.  Inside, 
the  cell  maintains  the  solution  weaker,  by  continuously  building 
the  incoming  molecules  into  combinations  too  '  large  '  to  pass  out 
that  way,  and  then  presenting  them  to  the  surfaces  bathed  by 
lymph,  which  is  short  of  them.  The  entering  molecules  cannot, 
as  a  rule,  '  gate-crash  '  the  thick  tough  protoplasm  ;  but  some — 
notably  the  sugars — do  pass  rapidly  into  the  blood,  unchanged. 

In  Respiration,  an  oyster  maintains  a  copious  flow  of  '  oxygen 
solution  '  over  the  thin  tissue  of  its  gills,  and  the  oxygen  diffuses  in, 
to  reach  the  same  '  partial  pressure  '  in  its  body. 

The  wriggly  '  blood-worm  '  larva,  common  in  old  water-butts 
and  woodland  pools,  of  Chironomus,  the  gnat-like,  but  harmless, 
Harlequin  Fly,  ventures  to  the  surface  only  at  night.  Its  red  blood 
is  a  solution  of  haemoglobin,  which  '  forms  a  loose  compound  witli 
oxygen,'  perhaps  so  tethering  several  oxygens  that  together  they 
count  only  as  one  against  free  oxygen,  which  therefore  continues 
to  diffuse  into  the  gills  to  a  many  times  greater  extent.  By  day, 
below  in  the  mud,  the  oxy- haemoglobin  lets  go  its  oxygens  into 
the  weakening  blood-stream.  Higher  animals  have  gone  a  step 
further,  and  packed  their  haemoglobin  itself  into  corpuscles,  them- 
selves membranous ;  and  living  in  air,  tbey  have  developed  a  vast 
lung  surface  for  gaseous  endosmose  and  exosmose. 

In  an  active  leaf -cell,  molecules  of  sugar  are  being  manufactured 
chemically  during  dajdight,  and  the  osmotic  pressure  rises  propor- 
tionally to  their  number,  and  we  have  seen  that  it  rises  high.  To 
keep  from  bursting,  the  number  of  molecules  must  be  kept  down 


282  MOLECULAR   PHYSICS  [§373 

somehow,  so  the  soluble  CgHjaOg  loses  an  HgO,  and  becomes 
CgHjoOg,  which  is  much  less  soluble,  and  builds  up,  layer  on  layer, 
the  sphere -crystals  we  presently  see  growing  under  the  microscope 
as  starch- grains.  Each  of  these  counts  only  one,  osmotically,  and 
the  situation  is  saved.  By  night,  in  abundance  of  water  and  no 
light,  they  hydrolyse  to  sugar  again,  dissolve  and  are  passed  out 
by  osmosis  into  the  weaker  solution  in  the  vessels,  for  the  use  of  a 
chemical  builder  elsewhere. 

§  374.  Let  us  therefore  calculate  the  Osmotic  Pressure  of  a 
solution  containing  n  mols.  of  solute  substance  mixed  with  100  mols. 
of  solvent,  i.e.  n  mol.  wt.  in  gm.  of  Solute  dissolved  in  100  mol. 
wt.  in  gm.  of  Solvent. 

Li  practice,  n  is  much  less  than  1 .  Neglecting  the  small  additional 
volume  this  causes,  the  mixture  has  a 

Volume  =  100  X  volume  of  the  mol.  wt.  in  gm.  of  solvent 
=  100  X  mol.  wt.  -f-  density,  of  solvent. 

1  gm.-mol.  gasified  at  normal  atmospheric  pressure  occupies  22,300 
(1  +  ^°/273)  c.c.  Hence,  applying  Boyle's  Law,  n  gm.-mols.  in  the 
given  volume  would  exert  a  Gas  Pressure 

n  X  22300  (1  +  ^7273) 
—  —  \      I      /       /  atmos. 


100  X  vol.  of  gm.-mol.  of  solvent 

n        vol.  of  gm.-mol.  of  a  gas 
100        ditto  of  liquid  solvent 


=  Osmotic  Pressure  of  n  per  100  molecular  solution. 
As  an  illustration,  taking  alcohol  as  solvent,  C2H5HO  has  mol. 
wt.  46  and  density  0-80,  so  that  the  volume  of  its  gm.-mol.  =  46  — 
0-80  =  57-5  c.c.  ;  hence  at  15° 

Osmotic  pressure  =  ^^  X  ^??50x^i+15/?Ii)  X  76  cm.   mercury 
=  311  cm.  mercury  for  a  1  per  100  molecular  solution 

a  pressure  that  no  osmotic  pot  stands  without  leaking.  Pfeiffer's 
1%  sugar  solution  was  a  1/342  mol.  in  100/46  mols.  =  0-13  mole- 
cular %. 

§375.  Lowered  vapour  pressures  of  solutions  of  non-volatile 
solutes.  Life  is  short,  and  the  preparation  of  satisfactory  '  osmotic 
pots  '  proves  often  exasperatingly  slow  and  tiresome,  consequently 
the  surface  of  an  evaporating  solution  of  a  non-volatile  substance 
is  welcomed  as  providing  an  excellent  sort  of  '  semi-permeable 
membrane,'  ready-made  and  free  from  leakage.  The  substance 
cannot  get  out  into  the  vapour  {e.g.  the  distillation  of  pure  water 
from  sea- water),  whereas  all  vapour  molecules  are  able  to  pass  back 
into  the  liquid  as  usual.     (For  a  volatile  solute,  such  as  alcohol, 


§375] 


DIFFUSION 


283 


it  is  of  course  useless,  being  all  leak.)  Hence  the  concentration  of 
molecules  in  the  vapour  over  the  solution  is  less  than  at  the  same 
temperature  over  pure  solvent,  wherefrom  all  molecules  are  able  to 
escape.  As  the  argument  leads  us  to  expect,  if  among  100  mols. 
of  solution  there  are  n  of  dissolved  substance,  the  vapour  pressure 
is  diminished  by  n%  of  its  normal  saturation  value.  (Practically, 
between  0-9671  and  1-ln,  according  to  solvent.) 

Thus  in  Fig.  116  (which  is  purely  diagrammatic  and  not  to  general 
scale)  the  upper  curve  represents  the  saturated- vapour-pressure 
temperature  curve  of  pure  solvent,  Fig.  82,  and  the  lower  curve, 
everywhere  16%  below  it,  the  vapour  curve  of  a  solution  of  w  =  16. 


100  104 


Fig.  116. 


On  the  top  left  is  suggested  the  surface,  up  through  which  the  O's 
cannot  pass,  while  their  '  opposite  numbers '  above  are  quite  free  to 
drop  down,  resulting  in  the  equilibrium  condition  shown  below,  with 
only  5/6ths  as  many  molecules  in  the  vapour  space  as  if  the  O's 
could  pass  into  it.  (Experimentally  this  strength  of  solution  is 
excessive.) 

This  result  can  be  verified  by  sending  a  slow  stream  of  air  first 
through  the  solution  and  then  through  pure  solvent.  The  loss  of 
weight  of  the  solvent  represents  the  addition  required  to  fully 
saturate  the  air,  and,  expressed  as  a  fraction  of  the  weight  of  all 
the  solvent  vapour  (collected  in  drying  tubes)  is  equal,  of  course,  to 
the  fractional  lowering  of  saturated  vapour  pressure  due  to  the 
dissolved  substance. 

As  an  instance,  the  atmosphere  of  the  saltings  is  less  humid,  on 
the  whole,  than  that  of  the  fresh-water  marshes. 


284  MOLECULAR   PHYSICS  [§376 

§376.  Raised  boiling  points  of  solutions  of  non-volatile  solutes. 

But  vapour-pressure  apparatus  is  none  too  easy  to  construct  and 
use,  and  a  further  step  can  be  taken  towards  increased  experimental 
convenience. 

To  reach  the  full  saturation  pressure  of  the  vapour,  the 
capacity  of  the  (100 — n)  mol.  to  vaporize  must  be  increased  until 
equal  to  that  of  the  100  mol.  of  vapour  to  liquefy.  This  is  done 
by  increasing  their  molecular  speed,  i.e.  by  raising  the  temperature 
of  the  solution.  The  saturation  pressure  convenient  in  practice 
is  1  atmo.,  for  that  is  the  pressure  of  the  vapour  when  a  liquid 
visibly  and  steadily  boils  in  an  open  vessel.  Thus  finding  the  rise 
in  boiling  point  of  the  solution,  above  that  of  the  pure  solvent, 
is  an  indirect  means  of  finding  the  solution's  deficiency  in  vapour- 
producing  power,  and  hence  of  finding  n. 

In  Fig.  115  the  16%  deficiency  of  pressure  VU  at  100°  is  made 
up  by  heating  the  solution  to  104°,  its  vapour  pressure  rising  to 
the  atmospheric  at  W,  where  it  boils. 

The  precise  relation  between  the  deficiency  of  vapour  pressure 
and  the  rise  of  temperature  necessary  to  make  it  up,  is  evidently 
settled  by  the  slope  of  the  curve  near  the  boiling  point.  This,  of 
course,  has  been  determined  by  the  experiments  of  §  281  on  the 
solvent  (the  curve  for  dilute  solutions,  from  which  alone  accurate 
results  are  obtainable,  is  closer  and  more  parallel  than  solution  curve 
in  figure).  As  explained  in  §  282,  the  curve  is  almost  identical 
for  all  liquids,  provided  only  that  the  temperature  scale  is  stretched 
or  compressed  as  a  whole,  so  as  to  bring  the  normal  boiling  point 
of  the  liquid  under  the  76-cm.  pressure  point  on  the  curve. 
Measurement  of  the  scale  diagram,  Fig.  82",  will  show  that  for 
2°  or  3°  above  the  boiling  point  the  vapour  pressure  increases 
2-7  cm.  Hg  for  1°  rise  of  temperature,  so  that  the  1%  molecular 
solution,  with  its  1%  deficiency  of  vapour  pressure,  =  760  -!-  100 
=  0-76  cm.  Hg,  must  be  raised  an  additional  0-76  4-  2-7  =  0-28°  C. 
to  boil  it.  (The  direct  experimental  figure  for  ether  as  solvent  is 
0-284°  and  for  carbon  disulphide  0-31°.) 

In  practice  a  very  delicate  thermometer  is  inserted  in  the  solvent, 
kept  boiling  as  steadily  as  possible,  by  the  aid  of  beads,  §  337,  or 
other  '  anti- bumpers.'  Then  the  substance  is  put  in  and  the  rise 
observed.  The  thermometer  must  dip  in  the  liquid;  the  pure 
vapour  leaves  the  solution  superheated  (unsaturated),  but  cools 
probably  in  the  first  centimetre  of  its  path  to  its  normal  temperature 
of  saturation,  i.e.  in  the  body  of  the  flask  it  is  no  hotter  than  before, 
cf.  §  195.  A  reflux  condenser  returns  the  boiled- away  solvent  to 
keep  the  strength  of  the  solution  constant. 

The  thermometers  mostly  used  are  '  Beckmann  '  instruments, 
with  bulbs  as  big  as  a  filbert,  and  stems  graduated  to  1/100°,  8°  long. 
This  is  enough,  as  only  differences  of  temperature  have  to  be  observed. 
To  fit  them  for  all  solvents,  both  in  this  section  and  the  next,  they 
have  a  little  reservoir  at  the  top  of  the  stem  where  spare  mercury  is 
retained,  or  brought  into  use  when  required  (most,  of  course,  for 
freezing  points). 


378] 


DIFFUSION 


285 


§377.  Lowered  freezing  points  of  solutions.  When  a  dilute 
solution  freezes,  pure  ice  separates  out  (in  fact,  freezing  is  the  easiest 
way  of  preparing  water  of  the  utmost  purity  from  ordinary  distilled 
water).  Now,  the  vapour  pressure  of  the  subliming  solid,  pure  ice, 
is  less  than  that  of  '  under-cooled  '  water  at  the  same  temperature 
(taking  ice  and  water  throughout  as  types  of  the  solid  and  liquid 
states  of  the  solvent,  whatever  it  may  be),  for  solid  bondage  hinders 
molecules  escaping  more  than  does  liquid  bondage,  and  the  ice 
vapour-pressure  curve  slopes  back  from  0°  more  steeply  than  the 
water,  Fig.  116.  Thus  it  presently  cuts  the  solution  curve  at  X, 
and  this  is  at  the  freezing-point  temperature  of  the  solution. 

For  at  temperatures  to  the  right  of  X  a  piece  of  ice  placed  on  the 
solution  has  a  vapour  pressure  higher  than  that  of  the  solution, 
and  will  evaporate,  and  vapour  will  condense  into  the  solution. 
The  total  rate  of  escape  of  the  100  mol.  from  the  ice  exceeds  that 
of  the  100 — n  from  the  solution.  Per  contra,  below  X,  vapour 
from  the  solution  would  deposit  on  the  ice,  which  would  therefore 
grow,  i.e.  the  liquid  continuously  freezes.  X  is  thus  the  only  point 
where  ice  and  liquid  exist  in  equilibrium  together,  as  many  molecules 
leaving  the  liquid  and  re-precipitating  on  the  ice,  as  leave  the  ice 
and  drop  back  into  the  liquid. 

The  same  equilibrium  holds  for  the  submerged  parts  of  the  ice, 
for  if  it  did  not,  we  might  have  ice  and  solution  at  a  perfectly 
uniform  temperature  throughout,  and  the  liquid  evaporating  and 
continuously  '  snowing  the  ice  under  '  from  above,  while  it  continu- 
ously dissolved  it  from  beneath,  a  Perpetual  Motion. 

The  slope  of  the  ice- vapour  line  having  been  found  by  experiment, 
the  result  is  that  it  cuts  the  vapour- pressure  curve  of  a  1  mol. 
per  100  mol.  solution  at  a  temperature  below  the  freezing  point  for 
various  solvents  as  follows  : 


Solvent. 

Mol.  Wt. 

Freezing 
Point. 

Depression  for 
1  mol./lOOmols. 

Water       

Benzene  

Acetic  acid         

Formic  acid        

18 
78 
60 
46 

0° 

5-4° 

16-5° 

8° 

1-05° 
0-65* 

0-60° 

§  378.  Summarizing,  for  a  solution  containing    n    gm.-mol.  of 
substance  dissolved  in  100  gm.-mol.  of  any  solvent  : 


Osmotic  Pressure 

at  any  temp. 


n        volume  of  gm.-mol.  of  a  gas 


100 


X 


ditto  of  liquid  solvent 


atmos. 


Lowering  of  Vapour  Pressure 
at  any  temp. 


100 


X  vap.  press,  of  solvent. 


I 


286  MOLECULAR   PHYSICS  [§  378 

Rise  of  Boiling  Point  at  atmospheric  pressure  =  n  x  0-28°  C. 

Lowering  of  Freezing  Point 

=  71  X  0-65°  C.  for  several  organic  solvents 
or      n  X  1-05°  C.  for  water. 

The  last  two,  being  experimentally  easier,  are  in  common  use  by 
the  chemist  to  find  molecular  weights.  Taking  100  mol.  wt.  of 
solvent  in  centigrammes,  say,  and  adding  w  eg.  of  substance, 
the  change  of  temperature  observed  is  divided  by  the  0-28°  or  0-65°, 
and  gives  n,  the  number  of  molecules  added. 

.*.  Mol.  Wt.  of  substance  =  w  -^  n. 

§  379.  For  reasons  not  yet  too  well  known,  the  above  relations 
have  still  to  be  applied  with  some  caution,  but  one  or  another 
usually  gives  satisfactorily   accurate  results  with  dilute  solutions. 

Two  major  discrepancies  are  observable.  The  first  is  illustrated 
by  Water  giving  an  altogether  too  big  depression  of  the  freezing 
point.  Many  very  varied  considerations  point  to  liquid  water  being 
mostly  composed  of  (H20)2,  v.  §  265,  a  typical  Associated  Solvent  : 
as  each  moving  particle  counts  only  1,  plainly  that  brings  a  1  in  100 
mol.  solution  in  water  up  towards  a  double  strength  of  1  in  50  ; 
hence  the  increased  effect.  On  the  other  hand,  in  some  solvents, 
solutes  fail  to  break  down  into  single  molecules,  and  the  effects 
remain  too  low  ;  these  are  Associating  Solvents. 

The  second  is  observed  with  solutions  which  can  conduct  electricity, 
solutions  of  electrolytes,  §  851 .  They  appear  to  contain,  when  dilute, 
double  the  expected  number  of  molecules ;  e.g.  sea-water  is  a  3% 
salt  solution  =  3/59  in  100/18  =  roughly  1  mol.  NaCl  per  100  mol. 
H2O,  and  should  freeze  at  —  1'05°  C.,  whereas  it  actually  freezes 
at  —  2°  C.  And  with  calcium  chloride,  CaClg,  with  3  atoms  in  the 
molecule,  nearly  3  times  the  expected  change  is  observable. 

This  supports  the  Ionic  Theory,  that  molecules  of  salts  in  dilute 
solution  spontaneously  split  up  into  bits  (sometimes  individual 
atoms),  electrically  charged,  called  Ions,  acting  kinetically  as  unit 
particles,  and  electrically  as  the  carriers  of  electricity  through  the 
solution.  See  §  853,  where  you  will  find  why  Water,  although  an 
associated  solvent  itself,  is  facile  princeps  at  bringing  this  Dissociation 
of  Solutes  about. 


DIFFUSION 


287 


EXAM  QUESTIONS,   CHAPTER  XXIV 


To  a  medical  student,  the  most  important  chapter  in  the  book.  Get  hold 
of  a  clear  idea  of  what  Osmosis  is,  and  does,  and  is  not  and  does  not  do,  before 
you  begin  to  use  it  familiarly  in  Physiology.  Keep  all  quantities  packed  in 
molecular  form,  and  so  avoid  muddles  comparable  only  with  loose  matches 
in  your  pocket. 


1.  Describe  experiments  to  test  whether  the  diffusion  of  gases  has  any 
connection  with  their  densities,  and  explain  how  they  do  test  the  question. 

On  what  does  the  rate  of  diffusion  depend  in  (a)  liquids,  (6)  gases  ? 

2.  Give  instances  of  Osmosis. 

What  is  meant  by  the  osmotic  press\u*e  of  a  substance  in  solution,  and 
what  is  its  relation  to  the  gas  pressure  of  the  same  substance  volatilized  ? 
(X2) 

3.  Give  a  short  account  of  Osmosis. 

Calculate  the  osmotic  pressure  at  0°  C.  of  a  solution  containing  10  gm. 
of  cane-sugar  per  litre  of  solution.  (Mol.  wt.  of  cane-sugar  =  342.  One 
gm.-mol.  of  hydrogen  at  0°  C.  and  760  occupies  22-3  litres.) 

4.  Explain  how  an  osmotic  pressure  is  produced,  and  measured. 
Calculate  that  of  a  15%  sugar  solution  (CiaHjaOn),  given  that  for  2  gm. 

of  hydrogen  per  litre  it  is  22-3  atmos.     (  X  2)        ' 

5.  Describe  some  instances  of  Osmosis,  and  show  how  this  process  can  set 
up  differences  of  fluid  pressure. 

Calculate  the  osmotic  pressure  of  a  deci-normal  solution  of  sodium  acetate 
at  30°  C.  In  what  circumstances  might  the  calculated  value  for  a  solution 
be  exceeded  ? 

6.  How  would  you  determine  the  freezing  point  and  boiling  point  of  an 
aqueous  solution  ?  How  are  these  related  to  the  corresponding  temperatures 
of  pure  water  ? 

7.  How  are  freezing  point  and  boiling  point  affected  by  (a)  pressure,  (6) 
dissolved  salts  ?     Mention  everyday  applications. 

8.  Calculate  the  osmotic  pressure,  against  alcohol,  for  a  solution  of  25  gm. 
of  glycerine,  mol.  wt.  92,  density  1-25,  in  880  gm.  of  alcohol,  mol.  wt.  46, 
density  0-80. 

9.  Draw  a  curve  indicating  generally  the  change  in  maximum  vapour 
pressiu^e  of  water  between  0°  and  100°  C.  Show  that  the  boiling  point  of  a 
solution  is  higher  than  that  of  pure  water. 

10.  A  liquid  is  heated  in  a  loosely  covered  vessel :  trace  the  changes  in 
the  air  space  above  it  as  it  rises  to  the  boil.  When  does  this  occur,  and  what 
is  the  effect  of  putting  in  10%  of  a  soluble  non-volatile  solid  of  mol.  wt.  double 
that  of  the  liquid  ? 

11.  A  current  of  dry  air  is  blown  through  fresh  water  and  then  through 
salt  water.  All  are  initially  at  the  same  temperature.  Explain  all  that  may 
be  observed,  and  state  what  would  happen  if  the  air  went  through  the  salt 
water  first. 

12.  4  litres  of  air  at  17°  and  76  cm.  and  dew-point  6-5*'  are  bubbled  through 
water  and  become  saturated.  How  much  water  is  taken  up  ?  Water  v.p. 
17°  =  1-44  cm.,  6-5°,  0-72  cm.,  22-3  litres  vapour  at  0°  and  76  cm.  weigh 
18  gm. 

13.  Would  the  amount  taken  up  from  salt  water  be  more  or  lees  ?  If  the 
salt  water  b.  pt.  were  102°,  how  much  would  be  taken  up  ? 


WAVES 

CHAPTER   XXV 
THE  PERIODIC  MOTION   OF  A  PARTICLE 


§381.  Simple  Harmonic  Motion.  A  'particle'  in  periodic 
motion  is  repeating  the  same  movement  in  equal  intervals  or  periods 
of  time. 

Take  a  '  particle  '  of  steel  on  the  tip  of  the  prong  of  a  tuning-fork  : 
struck  softly,  the  prong  swings  right  and  left,  and  the  fork  emits 
a  note  which  is  used  for  tuning  all  the  instruments  of  the  orchestra 
indifferently.  It  has  no  family  affinity  with  any  particular  one  of 
them  ;  in  its  one  pure  note  no  trace  of  any  other  can  be  detected 
by  any  known  means  ;  it  is  a  Simple  Tone  ;  the  motion  producing 
it  is  rightly  called  '  simple  harmonic  ' ;  the  prong 
is  moving  with  Simple  Harmonic  Motion.  As  the 
prong  vibrates  elastically  its  opposite  faces  are 
alternately  shortened  and  lengthened  :  to  a  first 
approximation  its  movement  resembles  Fig.  117, 
where  the  elastic  springs  connecting  the  rocking 
end  piece  represent  the  elastic  strength  of  the 
steel.  The  little  triangles  at  the  base  are  similar 
to  the  long  triangle,  so  that  the  stretching  or 
compression  of  the  springs  is  exactly  proportional 
to  the  width  of  swing  of  the  prong  (but  if  the 
motion  becomes  large,  as  dotted,  the  propor- 
FiG.  117.  tionality  fails,  and  one  hears  the  twang  of  a  hard- 

struck  fork). 
By  Hooke's  Law,  §  142,  the  forces  exerted  by  the  springs  are 
proportional  to  their  extensions  or  compressions  which  are  zero  in 
mid- swing,  straight  up  ;  so  that  we  arrive  at  this  :  A  particle  which 
moves  in  a  straight  line,  under  the  control  of  a  force  attracting  it  to, 
and  proportional  to  its  distance  from,  the  middle  point  of  the  line, 
moves  with  Simple  Harmonic  Motion. 

How  can  we  graph  this  ?  Turn  back  to  Fig.  17,  plan  :  here  the 
bob  moves  round  the  circle  because  there  is  always  the  radial  force 
BC  pulling  it  towards  the  centre.  Resolve  this  into  two  components 
BD  and  BE.  Now,  suppose  we  are  looking  along  EB,  i.e.  are  out 
in  front  of  the  pendulum,  on  a  level  with  it,  so  that  its  motion 
controlled  by  BE  is  in  the  line  of  sight  itself,  and  consequently 
invisible  ;   then  its  component  motion  right  and  left  appears  as  the 

288 


381 


PERIODIC   MOTION 


289 


lotioii  in  a  straight  line  of  a  particle  controlled  by  a  force,  BD, 
jwards  the  middle  point,  and  always  proportional  to  its  distance 
rem  it. 

This,  the  motion  to  and  fro,  along  a  diameter,  of  a  foot  of  the  perpen- 
'icuhir  dropped  on  it  from  a  point  moving  uniformly  round  the  circle 
herefore  tallies  with  our  definition  of  a  Simple  Harmonic  Motion,' 
lid  provides  the  sought-for  means  of  graphing  it. 
(It  coincides  with  the  motion  of  a  '  simple  pendulum,'  so  long  as 
he  curve  of  the  latter 's  path  is  unnotic cable.) 
Now,  don't  go  and  mix  up  the  tracing  point  moving  with  con- 
enient  uniformity  round  its  purely  constructional  circle,  with  the 
\ctual  Mass  plunging  to  and  fro,  with  always  varying  speed,  along 
ts  straight  line  path. 

In  an  old  type  of  steam  fire-pump,  Fig.  118,  the  vertical  piston- 
)ump-rod  bears  a  long  slotted  cross-head,  in  which  a  crank-pin 
vorks  to  drive  a  fly-wheel,  etc.  Assuming  the 
iy-wheel  speed  constant,  the  vertical  motion 
)f  the  pump  is  evidently  a  S.H.M.,  for  the  slot 
s  the  perpendicular  to  the  vertical  diameter. 
And  the  right  and  left  motion  of  the  pin  in  the 
slot  is  another  S.H.M.  (90°  different  in  phase). 

In  the  circular  diagram  marked  like  a  clock- 
face  in  Fig.  119,  the  dots  on  the  vertical 
liameter  show  the  positions  of  the  point  moving 
in  a  vertical  S.H.M. ,  at  equal  intervals  of 
time.  But  it  is  more  graphic  to  put  each  on 
a  diameter  of  its  own,  spaced  horizontally  at 
•'qual  times  apart  as  shown,  and  so  produce 
the  Sine  Curve  (or  Cosine  Curve).  This  curve 
would  be  obtained  by  carrying  a  card  horizon- 
tally past  a  pencil  on  the  pump-rod  ;  or  by  sliding  a  plate  at 
uniform  speed  below,  and  at  right  angles  to,  a  pendulum  the  bob 
of  which  is  a  can  of  sand  with  a  hole  in  the  bottom.  It  is  roughly 
Itained  by  the  small  boy  as  he  ambles  beside  a  wall,  chalk  in  hand. 
The  following  particulars  of  a  S.H.M.  must  be  defined  : 
The  Amplitude,  a,  is  the  maximum  distance  from  the  centre.  It 
i^  the  radius  of  the  circle  ;  half  the  length  of  a  pendulum  swing ; 
the  height  or  depth  of  the  curve  from  the  centre  line. 

The  Phase  of  the  particle  expresses  its  position  at  any  moment. 
It  is  usually  defined  by  the  angle  the  corresponding  point  in  the 
I'ircle  would  have  moved  from  its  starting-point,  e.g.  the  point 
<'  (Fig.  119)  is  either  in  phase  60°  (II  o'clock)  when  moving  down- 
wards, or  in  phase  300°  (X  o'clock)  when  moving  upwards. 

The  particle  passes  through  all  phases  once  in  each  completed 
motion. 

The  Period,  Periodic  tiw^,  or  Time  of  Vibration  or  Oscillation,  T,  is 
the  time  taken  to  complete  one  whole  motion,  i.e.  the  interval 
of  time  between  two  successive  passages  of  the  particle  through 
the  same  phase. 

L 


Fio.   118. 


290 


WAVES 


[§38 


The  complete  vibration  or  oscillation  is  the  whole  motion  the  li 
and  bach,  e.g.  'the  time  of  oscillation'  of  a  'seconds'  pendulu 
is  2  sec,  each  single  '  swing  '  or  '  stroke  '  occupying  1  sec. 


Fig.  119. 

Never  call  a  single  swing  a  vibration  or  oscillation,  for  this  hi 
caused  much  confusion. 

The  Frequency,  n,  is  the  number  of  vibrations  per  second ;  it 
the  reciprocal  of  the  periodic  time  T  in  seconds, 

1 

§  382.  The  particle  reaches  full  speed  in  mid-swing,  when  tl 
accelerating  force  is  exhausted  and  the  decelerating  force  has  n( 
begun — the  pendulum  has  run  down  the  hill  and  is  about  to  sta 
the  upward  climb.  Inspection  of  either  of  the  Figs.  17,  118,  ( 
119,  shows  that  this  speed  =  that,  parallel  to  it,  of  the  point  movh: 
in  the  circle,  of  radius  a,  circumference  27ra,  in  time  T,i.e.v  =  27ra/'' 

At  this  moment  the  particle  m  is  free  from  force,  all  its  energy 
kinetic,  and  =  \mv'^  =  Jm  X  ^i-kVIT"^.  Presently  the  particle 
at  the  end  of  the  swing,  and  all  this  energy  is  stored  as  potenti 
energy  of  displacement  against  elastic  forces,  gravity,  electric 
force,  etc.,  as  the  case  may  be.  Generally,  its  energy  is  part  kineti 
part  potential,  and  they  always  add  up  to  this  same  Energy  of 
Vibrating  Particle,  which,  since  1/T  =  n,  can  also  be  written  as 

\  mass  X  4:7r^amplitude)^  X  (frequency)^ 


§  383.  The  S.H.M.  is  of  all  vibrations  the  most  easily  and  natural 
produced,  it  is  by  far  the  simplest  to  study  scientifically,  ai 
fortunately  any  persistent  periodic  motion  whatever  can  be  analys 
into,  or  built  up  as  the  resultant  of,  a  series  of  S.H.M.'s.  For  instanc 
the  violent  motion  of  a  shuttle,  or  of  a  ball  bounced  on  the  pav 
ment,  such  as  drawn  out  on  a  moving  time-sheet,  would  gi 
curves  like  Fig.  120  (H,  K) ;  or  indeed  any  sort  of  wriggles,  zigzag, 
saw-teeth,  or  battlements,  provided  they  do  not  overhang  an 
require  time  to  go  back  on  itself. 

The  analysing  process  is  too  difficult  for  us  here,  but  the  buildin 


■'] 


PERIODIC   MOTION 


291 


I  to  Compound  Harmonic  Curves  is  easy  enough  :  just  add  the 

,  iltaneous  displacements. 

The  curve  C  in  Fig.  120  is  got  by  adding  the  heights  of  the  two 
ii\  es  A  and  B  (amplitudes  2:1,  periods  3  :  2)  above  the  centre 
K  .  depths  below  being  reckoned  minus.     Three  or  any  number 

S.H.M.'s  may  be  compounded  in  this  same  way. 


HtiFbFt     PAPA  K 

Fig.  120. 

Compound  harmonic  curves  fall  into  two  general  types  : 

{n)  If  one  component  is  much  the  strongest  and  slowest,  the 
Ms  that  this  persists,  merely  battered  by  the  others  from 
>imple  sine  shape  into  some  sort  of  regular  zigzag.  Such 
[5  the  air  motion  produced  by  a  musical  instrument.  In  Fig.  120 
r  and  G  show  resultant  motions  obtained  by  adding,  to  a 
ifundamental '  vibration,  another  of  half  the  period  and  half  the 
mplitude.  In  G  there  is  a  phase  shift  of  90°  from  the  F  condition, 
ero  points  being  simultaneous,  instead  of  maxima. 

(6)  If  there  are  two  components  not  very  unequal  in  frequency, 
he  motion  waxes  and  wanes.  An  instance  of  this  on  the  grand 
cale  occurs  in  the  Tides,  springs  and  neaps.  As  everyone  knows, 
hese  are  due  to  a  solar  pull  of  period  12  hr.,  and  a  lunar  pull,  pro- 
lucing  2 J  times  the  amplitude,  and  of  period  about  30/29  of  12  hr. 
^'ig.  D  represents  the  resultant  Tidal  rise  and  fall  for  just  over  the 
ortnight.  In  E  there  is  the  same  relative  frequency,  but  the 
implitudes  are  equal,  the  vibration  dies  down  to  nothing  and  rises 
o  double.     Both  illustrate  Beats,  in  Sound,  §  431. 


292 


WAVES 


[§  384 


§  384.  Composition  of  S.H.M.'s  at  right  angles.  In  that  delightful 
instrument,  the  Harmonograph,  described  in  all  works  on  popular 
science,  a  pencil  is  pushed  north  and  south  by  a  light  rod  connectin|; 
it  to  one  heavy  pendulum,  and  east  and  west  by  another  pendulunt 
Typical  curves  that  it  draws  are  shown  in  Fig.  121.  A  shows  the 
combination  of  two  equal  S.H.M.'s  at  right  angles  with  the  stated 
difference  of  phase.  Notice  among  the  curves  the  straight  line» 
for  0°  or  180°  and  the  circle  for  90°.  It  is  this  circle  that  was  rel- 
solved  into  the  two  equal  S.H.M.'s  at  right  angles  in  the  fu*e-pump, 
one  just  starting  as  the  other  is  in  mid- swing. 


Fig.  121. 


B  shows  a  S.H.M.  combined  with  one  of  twice  its  period  andi 
initially  60°  phase  difference.  There  are  many  other  curves 
depending  on  the  ratio  of  the  two  periods.  With  ratio  2:1,  a^ 
drawn,  the  tracing  point  completes  two  cross -journeys  during 
each  vertical  journey  ;  with  3  :  2  it  would  make  three  during 
each  two  verticals,  and  so  on.  If  the  periods  are  not  in  exact 
ratio,  the  figures  go  to  and  fro,  through  all  their  changes,  every 
time  that  one  motion  gains  a  whole  vibration  over  the  other.  It: 
is  this  inexactness,  together  with  the  gradual  dying  down  of  the 
motion,  that  gives  their  interwoven  beauty  to  the  harmonograph 
curves.     And  see  §  432.  ^ 


^386]  PERIODIC  MOTION  293 

§  385.  Forced  oscillations.  So  far  nothing  has  been  said  as  to 
how  the  oscillations  were  originated,  and  they  have  gone  on  freely 
under  their  own  natural  controlling  forces  in  their  own  natural 
period. 

Experience  assures  us  that  it  was  some  outside  force  that 
started  the  motion.  It  further  assures  us  that  using  forces  great 
enough  we  can  make  any  body  move  how  we  like.  A  load  on 
the  rope  of  a  crane  is  a  pendulum,  but  the  skilful  driver  slews  it 
round  and  deposits  it  where  required,  without  much  bother  from 
oscillation,  and  without  undue  delay. 

Now,  what  of  the  condition  of  affairs  intermediate  between  this 
close  artificial  control  and  free  natural  oscillation  ? 

It  is  a  state  of  oscillation  more  or  less  modified  by  external  forces, 
a  state  of  forced  oscillation. 


Fig.  122. 

Push  a  child  in  a  swing.  Holding  it,  you  can  walk  slowly 
backwards  and  forwards.  Increase  the  speed,  and  you  become 
aware  that  the  thing  has  a  tendency  to  swing  of  itself ;  sometimes 
it  moves  easily,  sometimes  pulls  you  along,  at  other  times  it 
resists  with  unexpected  force.  With  hard  labour  you  have  it 
swinging  with  the  frequency  you  choose,  but  it  will  probably  knock 
you  down  before  reaching  the  amplitude  the  youngster  demands. 

But  be  guided  by  the  swing  itself,  give  it  push  after  push  always 
at  the  right  time,  as  a  clock  does  to  its  pendulum,  and  with  little 
effort  you  get  an  ample  oscillation  ^mc^icaZ/?/  in  its  natural  periodicity. 

Try  again  to  drive  it  with  a  higher  frequency,  and  your  utmost 
exertions  hardly  shake  it  a  foot. 

Another  experiment  is  this  :  hold  up  a  simple  pendulum, 
oscillate  your  hand  horizontally  with  different  frequencies,  and 
observe  to  what  comparative  extents  the  bob  swings  for  each  ; 
see  Fig.  122 ;  notice  the  large  motions  of  the  hand  when  too  slow 
(left  fig.)  or  too  fast  (right  fig.),     do  this. 

§  386.  These  exemplify  a  perfectly  Greneral  Principle  : 

A  body  can  be  forced  to  oscillate  in  any  period,  but  the  forces 

required  become  less  and  less  the  nearer  that  period  is  to  its  natural 

one. 

Or,  conversely,  when  a  definite  force  is  applied  periodically,  the 

body  will  oscillate  in  that  period,  but  the  forced  oscillations  become 

large  only  when  near  the  natural  period  of  free  oscillation. 


294  WAVES  [§386 

There,  they  often  increase  enormously,  and  there  is  said  to  be 
Resonance  between  the  vibration  and  the  exciting  forces.  The 
term  is  borrowed  from  Sound  :  acoustic  instances  of  this  mechanical 
action  are  given  in  Chapter  XXIX. 

A  familiar  annoyance  arising  from  mechanical  Resonance  is  the 
exaggerated  jumping  vibration  of  the  railway  carriage  at  one 
particular  speed,  that  which  happens  to  bring  the  rail-end  jolts 
'  in  step  with  '  the  natural  frequency  of  bouncing  of  the  carriage 
on  its  springs.  Another  is  the  objectionable  jarring  of  car  or  steam- 
ship at  some  one  particular  engine  speed ;  and  yet  another  the 
excessive  response  of  a  ship  to  a  cross-sea  which  strikes  her  too  nearly 
in  her  natural  period  of  swing.  All  day  long  we  had  been  dipping 
our  forefoot  well  down  among  the  dolphins  off  the  Spanish  coast, 
while  astern  of  us  a  heavier  Castle  liner  rode  steady  as  Dover  pier, 
save  for  an  occasional  yaw  off  her  course.  Rounding  Finisterre, 
and  setting  our  course  30°  E.  for  Ushant,  increased  the  interval 
between  successive  seas  reaching  us  by  a  full  second,  from  a  little 
below  our  natural  period  of  pitch  to  nearly  hers  :  we  quieted  down 
at  once,  but  two  days  later  a  new  student,  who  had  travelled  aboard 
her  from  the  Cape,  spoke  with  horror  of  his  night  in  the  Bay. 

You  are  familiar,  too,  with  the  oft-quoted  military  regulation 
that  companies  of  soldiers  crossing  bridges  should  break  step ; 
lest  the  bridge,  with  a  period  too  near  the  regular  marching  time, 
and  therefore  in  resonance  with  the  impressed  force,  gradually  work 
up  to  a  destructive  vibration.  This  has  happened,  with  calamitous 
results,  and  in  what  type  of  bridge  can  be  deduced  from  the  tests 
of  the  Sydney  harbour  bridge  :  its  steel  arch  subsided  J  in.  under 
the  test  load,  a  cantilever  bridge  (Forth  Bridge  type)  would  have 
been  expected  to  yield  4  in.,  and  an  ordinary  suspension  bridge 
of  the  same  size  (Clifton  type)  perhaps  4  ft. 

Those  who  have  stumbled  the  length  of  a  road  suspension-bridge, 
swaying  and  plunging  in  a  northerly  gale  of  fair  Provence,  with  that 
in  its  teeth  which  deluded  us  both  into  mistaking  an  ice-crystal  in 
the  limestone  bank  for  calc-spar,  can  appreciate  all  the  more 
Mr.  Punch's  skit  on  the  whole  business,  wherein  he  depicts  a  company 
of  two  recruits  breaking  step  as  they  emerge  through  the  machico- 
lated  gateway  on  to  the  stern  stone  arches  of  the  mighty  mediaeval 
fortress-bridge  of  Cahors. 

§  387.  A  very  simple  little  home-made  apparatus,  Fig.  123,  can  assist 
us  to  study  both  the  Combination  of  S.H.M.'s,  and  Resonance.  Wind  thin 
(s.w.g.  26)  steel  wire  on  a  J-in.  mandrel,  and  make  two  helical  springs  of  30 
turns  each;  clamp  their  upper  ends  and  hang  from  their  lower  ends  2-oz. 
weights  made  of  strip  solder,  and  snipped  to  make  their  times  of  vertical 
boimcing  equal,  and  about  a  second. 

To  observe  the  composition  o/  S.H.M.'s,  loop  a  thread  from  weight  to  weight, 
and  watch  the  vertical  movement  of  the  bottom  of  the  bight : 

Set  either  boimcing,  the  bight  rises  and  falls  half  as  far. 

Set  both  bouncing  side  by  side,  it  moves  full  distance;  i.e.  two  S.H.M.'s, 
of  the  same  period,  amplitude,  and  phase,  add  to  a  S.H.M.  of  the  same  period 
and  doubled  amplitude.  Or,  if  one  moves  farther,  the  amplitudes  are  simply 
added. 


187]  PERIODIC   MOTION  295 

Set  both  bouncing  equally,  but  left-right,  in  opposite  phases,  and  the  bight 
does  not  rise  or  fall ;   if  unequally,  it  moves  their  difference. 

Bounce  them  in  different,  but  not  opposite,  phases,  and  they  add  up  to 
the  same  period  and  an  intermediate  amplitude. 

Hang  an  extra  2  or  3  in.  of  strip  on  one,  so  as  to  slow  its  period,  set  them 
going,  and  now  you  observe  D  or  E,  Fig.  120,  according  as  the  amplitudes 
differ  or  are  equal,  the  faster  going  through  all  its  phases,  with  reference  to 
the  slower,  once  between  minima  of  movement. 

Hang  a  lot  of  weight  on,  so  that  the  periods  differ  consider- 
ably, and  the  loop  dances  more  like  C,  Fig.  120,  in  what  looks  at 
first  a  very  irregular  motion  indeed. 

To  study  Resonance,  unload  again  to  equal  frequency,  and 
instead  of  thread  use  chain,  or  thick  soft  woolly  cord  of  appreci- 
able weight. 

Set  one  bouncing,  the  other  gradually  begins,  and  presently 
has  absorbed  the  whole  of  the  motion  and  reduced  the  first  to 
rest ;  then  the  first  retaliates  and  gets  it  all  back,  and  so  on. 

With  a  heavier  cord,  '  closer  coupling,'  the  interchange  is 
faster,  for  the  individual  impulses,  lifting  and  lowering,  are 
stronger. 

Load  them  out  of  tune,  and  now  the  second  takes  up  some 
of  the  first's  motion,  and  then  gives  it  back,  and  so  on.  The 
worse  out  of  time,  the  less  is  picked  up ;  for  the  sooner  the 
fijrst  starts  pulling  directly  opposite  to  how  it  started,  having 
got  more  and  more  out  of  phase  at  each  swing  imtil  now  it  is 
180°  away. 

Badly  out  of  time,  very  little  transference  of  motion  occurs ; 
but  with  a  '  heavy  coupling  '  some  can  always  be  forced. 

To  get  some  idea  of  Damping,  let  the  bight  of  the  rope  rub  ^'^o*  123. 
against  a  vertical  rod  with  less  or  more  friction. 

Two  further  instances  of  Resonance  can  be  contrived,  each  with  a  single 
spring  : 

Load  one  lightly  so  that  its  period  of  swing  as  a  '  simple  pendulum  '  is  the 
same  as  its  period  of  boimce.  Start  it  doing  either,  and  watch  how  the  motion 
keeps  changing  completely  from  one  to  the  other. 

Fasten  a  W-shaped  load  on  the  other,  and  open  its  wings  until  the  p>eriod 
of  spin  about  a  vertical  axis  is  equal  to  that  of  bounce  :  start  bouncing,  and 
again  see  the  regularly  repeated  interchanges  of  energy  between  the  two 
motions. 

[A  passenger  ship  is  built  to  have  a  period  of  roll  very  different 
from  that  of  pitching,  usually  twice.  For  if  the  two  periods  are 
anywhere  near  equal,  she  is  perpetually  changing  her  mind  from  the 
one  movement  to  the  other ;  and  the  resultant  uneasy  motion  has 
been  known  to  unsettle  sensitive  people.] 

Calling  these  two  Oscillator  and  Resonator,  and  supposing  the 
Resonator  is  1  %  slower,  what  happens  is  this  :  Resonator  gets 
a  push  from  Oscillator,  and  starts  swinging  in  its  own  natural  period, 
then  along  comes  another  push  from  Oscillator,  just,  but  only  just, 
1%  too  soon  for  it.  On  the  next  swing  it  is  2%  too  soon,  and 
so  on.  The  Resonator  adds  these  impulses  together  as  a  succession 
of  S.H.M.'s  3-6°  apart  in  phase,  and  increases  its  swing  up  to  the 
fiftieth.  But  the  fifty-first  is  half  a  period  too  soon,  it  pulls  in 
direct  opposition  to  the  first,  and  so  on,  the  succeeding  impulses 
np  to  the  hundredth  wiping  out  the  effect  of  the  first  fifty.  The 
resonator  therefore  keeps  on  getting  up  a  small  swing  and  dying 


296  WAVES  [§  387 

down  again,  and  this  imperfect  resonance  can  never  cause  a  strong 

movement. 

This  adding  up  of  successive  equal  impulses  with  a  constant 

phase  difference  is  easily  effected  graphically,  as  in  Fig.  124.     Little 

equal  vectors,  representing  the  impulses,  are  joined  tail  to  head,  each 
succeeding  one  turned  through  a  small  angle 
=  common  phase  difference,  and  the  straight 
closing  side  of  the  polygon  thus  formed  gives  the 
magnitude  and  phase  angle  of  the  resultant. 
With  100  small  impulses  and  3-6°  phase-lag,  the 
polygon  becomes  a  practically  continuous  circle, 
with  a  maximum  resultant  0-50,  min.  0-100, 
max.  0-150,  and  so  on.  If  the  impulses  gradually 
become  weaker,  the  polygon  curls  gradually  closer 
into  a  spiral. 

§  388.  Effect   of   '  Damping '    on     Resonance. 

An  oscillatory  motion  which  gradually  dies  away 
owing  to  its  energy  being  either  spent  in  over- 
coming friction,  or  '  radiated  '  out  as  vibration  of 
the  supports,  sound,  electromagnetic  waves,  etc., 
is  described  as  '  damped.' 

Without  air  friction  a  clock  pendulum  would 
get    up    an    indefinitely    great    ampHtude    as   it 
continually    added    up    the    effects    of    impulses 
Fig.  124.  always  in  phase  with  its  natural  swing,  and  never 

lost  anything.  1  part  in  1000  parts  away  from 
this,  only  500  impulses  would  be  accumulated  before  they  had 
drifted  round  into  opposition  and  begun  to  destroy  the  motion. 
The  difference  due  to  this  imperfect  resonance  is  the  difference 
between  an  indefinitely  great  number  and  500,  that  is  : 

With  but  slight  damping,  resonance  is  strong  and  its  position  very 
sharply  marked. 

But  if,  after  a  half-dozen  pushes  or  so,  a  swing  had  been  worked 
up  which  takes  nearly  all  the  applied  force  to  keep  up  its  vigour, 
constantly  sapped  by  friction,  etc.,  the  difference  between  the 
two  previous  cases  quite  disappears.  In  fact,  a  '  mis-tuning ' 
of  1  in  20  would  still  supply  enough  impulses  to  work  up  the  full 
resonance  possible,  although  this  would  be  only  a  small  fraction  of 
the  maximum  obtainable  with  good  tuning  and  little  friction. 

With  heavy  damping,  resonance  is  weak  and  its  position  poorly 
marked. 


CHAPTER  XXVI 


WAVE  MOTION 


§391.  Wave  Motion.  Suppose  a  long  row  of  particles  con- 
nected by  some  means  which  can  transmit  a  force  from  one  to 
the  next — a  long  line  of  angler's  split  shot,  for  instance,  strung  an 
inch  apart  on  a  thread  of  the  thinnest  elastic,  with  an  inch  left 
at  the  beginning.  Pull  this  in  any  way  you  like,  and  so  displace 
the  first  shot.  As  it  moves  it  gradually  stretches  the  next  inch 
of  elastic,  which  begins  to  pull  on  the  second  shot,  i.e.  to  impart 
momentum  to  it.  It  moves,  and,  stretching  the  next  inch  of 
thread,  begins  to  hand  on  momentum  to  the  third,  and  so  on, 
and  soon  every  particle  in  turn  is  performing  the  same  motion  as 
its  neighbour  before  it,  but  a  little  later.  An  alternating  pull 
on  the  end  sets  up  a  typical  Running  Wave  Motion,  caused  by  every 
particle  in  a  series  'performing  exactly  the  same  periodic  oscillation ; 


Fig.  125. 


hut  each  later,  or  lagging  a  little  in  'phase  behind  its  neighbour  on 
the  side  whence  the  motion  arrives  ;  while  it  equally  leads  the  oscillation 
of  its  farther  neighbour. 

The  stronger  the  elastic  links  the  less  they  stretch  to  transmit 
a  given  force,  and  the  quicker  and  with  the  less  phase  difference 
the  successive  particles  have  to  respond ;  but  the  heavier  the 
particles  the  slower  they  get  into  motion,  and  the  greater  their 
phase  lag.  In  fact,  the  speed  of  travel  of  every  sort  of  wave 
depends  upon  (is  the  square  root  of)  the  quotient  of  a  quantity 
analogous  to  elastic  force,  by  a  quantity  analogous  to  mass. 

To  the  definitions  given  concerning  the  motion  of  a  single  particle 
must  now  be  added  the  following  : 

The  velocity  of  travel  V  of  the  wave  is  the  speed  with  which 
any  one  selected  wave  form  travels  forward. 

The  wave-length  is  the  distance  between  two  successive  particles 
in  the  same  phase  of  their  motion,  e.g.  between  two  crests  (0°) 
or  between  two  points  such  as  PQ  (half-past  ten  phase),  Fig.  126. 

297 


298  WAVES  [§  391| 

In  order  that  a  succession  of  waves  of  length  L  may  continu 
to  spread  from  a  source  vibrating  n  times  per  sec.  (period  T  =  IjntY^ 
sec),  the  first  wave  must  travel  away  a  distance  nL  in  the  second  J 
to  leave  room  for  the  rest  that  are  produced.  | 

.-.  V  =  ?iL 

Sjieed  of  travel  of  waves  =  frequency  X  wave-length 
This  is  the  fundamental  equation  of  all  wave  motion.     Get  tol 
know  it  through  the  headline  in  the  Radio  Times,  where  n  kilocycles 
X  L  metres  wave-length  always  =  300,000,  the  speed  of  travel! 
of  the  waves  in  kilometres  per  second.  i 

A  wave  *  front '  is  a  theoretical  surface  drawn  through  all  I 
adjacent  particles  which  are  in  the  same  phase.  i 

I 

§392.  Water  Waves.  Most  familiar  of  all  wave  motions  isi 
the  deep-water  wave.  It  is  a  commonplace  that  floating  weed  only 
sways  about  while  the  wave  form  rolls  on,  but  the  oarsman  or  the 
swimmer  has  a  much  more  definite  impression.  Swimming  tol 
meet  the  sea,  a  wave  rushes  towards  him.  Unwillingly  he  is' 
drawn  forward  to  meet  it,  but  just  as  its  half -yard  or  so  of  dread 
altitude  looms  before  him,  blotting  out  the  view  and  walling  him 
in  a  vale  of  utter  loneliness,  he  is  lifted  right  up,  apparently  only 
condemned  to  receive  a  mouthful  off  the  crest.  But  no ;  time  is 
gained  to  lift  him  that  last  few  inches  by  an  unexpected  retreat 
before  the  crest,  he  is  already  caught  and  borne  back  in  the  wave, 
and  suffers  a  loss  of  headway  until  half-way  down  its  back  starts 
a  swift  swing  through  the  shallow  trough  to  meet  the  next  comer. 

All  this  is  consistent  with  the  water  particles  revolving  in 
vertical  circles  with  fixed  centres,  and  moving  at  the  top  in  the 
direction  of  travel  of  the  wave.  At  A  in  Fig.  126  selected  equi- 
distant drops  are  represented  by  the  dots  moving  in  successive 
circles  with  a  phase  lag  of  45°.  B  shows  the  effect  of  an  increased 
ampUtude  without  corresponding  increase  of  wave-length,  the 
crest  becomes  more  peaked  until  ultimately  it  is  bound  to  break 
into  white  horses.  Diminished  amplitude  give  low  round  tops 
such  as  characterize  the  Swell  into  which  storm-waves  die  down 
as  their  violence  abates.  Wide  but  slow  revolutions  produce  a 
heavy  swell,  adjacent  circles  would  differ  less  in  phase  than  in  A 
(or  the  45°  circles  would  be  spaced  wider  apart,  as  C)  :  the  amplitude 
is  considerable  and  the  wave-length  and  speed  are  great. 

Heights,  however,  have  had  to  be  exaggerated  to  make  a  diagram  : 
instead  of  the  height  of  Deep-water  waves,  from  trough  to  crest, 
being  l/8th  their  length  from  crest  to  crest,  as  in  A,  it  is  observed 
to  be  only  one-thirteenth.  B,  of  course,  is  nothing  but  artist's 
licence  :  the  white  horses  that  cap  the  waves  when  a  fresh  wind 
blows  are  due  to  its  quite  local  action  in  raising  smaller  ripples  on 
their  backs,  and  then  blowing  these  to  bits  on  the  exposed  tops. 

The  height  of  waves  in  feet  is  1-5  X  V  'fetch'  in  sea  miles, 
i.e.  the  sea  room  the  wind  has  had  to  work  them  up  in.     Hence 


392] 


WAVE   MOTION 


299 


the  great  size  of  the  waves,  600 — 1000  ft.  long,  perpetually  rolling 
eastward  round  the  world  in  the  unbroken  Southern  Ocean,  where 
gales  blow  for  great  distances  on  end. 

It  is  easy  to  connect  the  height  of  a  single  wave  with  its  speed 
of  travel.  For  the  greatest  energy  a  c.c.  of  water,  m,  can  acquire 
would  be  that  due  to  faUing  the  whole  height  of  the  wave,  mgh,  and 
this  would  give  it  a  maximum  speed,  when  its  energy  was  wholly 
kinetic,  obtained  by  putting  ^mv^  =  mgh,  or  v  =^J  Igh,  as  in  §  121, 
and  this  is  the  maximum  speed  about  the  wave,  the  speed  of  the 


€LiZlOi2)i^«riU(^ 


e©9O90GGe©9O 

I     I    I    I     I      I       I  ^    I      I     I    I   I     I 


Fio.  126. 


wave  itself ;  no  actual  particle  moving  at  more  than  a  quarter  as 
much,  in  its  circle. 

(Similarly,  you  will  find,  squaring  the  formula  of  §  396,  \  DV*  = 
E,  the  kinetic  energy  and  the  pressure  energy,  respectively,  of  1  c.c. 
of  the  medium.) 

As  you  are  aware,  however,  water  waves  travel  in  company — 
in  groups — and  the  group  travels  only  half  as  fast  as  the  waves 
themselves,  so  that  they  are  continually  being  born  at  the  rear  of 
the  group,  growing  in  size  as  they  reach  its  middle,  and  djing  away 
as  they  reach  the  front — you  can  see  this  by  throwing  stones  into 
a  pond — and  this  complicates  matters,  making  the  foregoing 
calculation  approximate  only.  The  following  three  formulae  are 
as  near  as  can  be  given  ;  they  are  really  only  variants  of  the  same 
statement,  and  one  or  other  of  them  may  interest  you  when  at  sea, 
or  on  a  steep-to  coast : 


300  WAVES  [§392 

Speed  of  deep-sea  wave,  in  knots  =  3  X  period  in  seconds. 

,,  =  5  X  V  height  in  feet. 

^[  ''  „  =  4/3  X  V  length  in  feet. 

or       3-3  X  \/  length  in  fathoms. 

Atlantic  waves  are  commonly  from  160  to  320  ft.  long  (so  that  for 
them,  very  roughly,  speed  in  knots  is  '  rather  more  than '  height  in 
feet) ;  the  greatest  length  ever  recorded  was  half  a  mile  in  the  S. 
Pacific. 

The  amplitude  of  wave  disturbance  becomes  so  very  small  at 
a  depth  of  one  wave-length  that  delicate  '  gravity  '  determinations 
have  been  made  in  submerged  submarines. 

In  shallow  water  the  circles  flatten  into  ellipses,  as  the  up-and- 
down  supply  of  water  is  limited,  and  the  speed  decreases  to 

Shallow  speed  in  knots  =  3-3  y'  depth  in  feet. 

Presently  the  backward  movement  at  the  bottom  of  the  ellipse 
is  so  much  hindered  by  friction  on  the  bottom,  that  the  front  of 
the  wave  is  starved  for  water,  and  the  crest  topples  over  the  familiar 
hollow  face,  which  shows  almost  the  path  of  the  particles.  The 
'last  wave  shows  the  flattened  elliptic  motion,  in  the  heaving  surge 
up  the  beach  and  the  subsiding  backward  scour. 

The  tide  rolls  slowly  in  on  very  shallow  sands,  as  one  long  straight 
wave.  This  makes  deeper  water,  and  the  wave  following  travels 
faster,  and  catches  up  and  rides  on  the  top  of  the  first ;  so  with  the 
next,  and  even  more  if  the  sands  are  wide,  and  they  add  up  into  a 
roaring  '  wall  of  water  '  which  advances  at  speed  you  can  estimate 
by  the  formula  :  such  are  the  tides  at  Mont  St.  Michel,  and  the 
Bore  gathering  strength  over  the  shallows  of  the  Severn. 

We  all  know  how  the  third  wave  is  the  biggest ;  or  else  it  is  the 
seventh,  or  the  ninth,  or  the  eleventh.  Well,  did  you  ever  watch 
and  count  ?  It  appears  that  waves  do  usually  arrive  as  a  compound 
harmonic  series,  in  which  more  than  one  periodicity  can  be  detected, 
something  between  conditions  C  and  D,  Fig.  120  ;  in  addition, 
inshore  there  is  the  varying  backrush  from  the  beach  adding  itself  in. 
Aboard  ship,  one  watches  for  her  forefoot  going  down  with  a  good 
splash,  again  a  harmonic  combination  of  her  own  natural  period  of 
pitch  with  that  of  arrival  of  the  oncoming  waves. 

Tiny  ripples  are  called  Capillary  Waves,  and  are  controlled  almost 
entirely  by  the  surface  tension  of  the  water,  Chapter  XXIII.  The 
surface  vibrates  something  like  a  stretched  membrane  or  string,  and 
the  ripples  approximate  to  the  type  to  be  described  next,  none  of 
them  anything  like  so  complicated  as  gravity  water  waves. 

§  393.  Waves  of  transverse  motion.  The  next  type  of  wave  is 
seen  in  a  jerked  rope  fast  at  the  end,  or  a  vibrating  string.  Here 
there  is  very  little  lengthwise  motion  possible,  and  all  particles 
move  simply  across  the  direction  of  travel  of  the  wave,  up  and  down 
along  the  diameters  as  the  vertical  components  only  of  the  construe- 


§  395]  WAVE   MOTION  301 

tional  small  circles  in  Fig.  126,  T.  They  need  not  be  actually  con- 
fined to  these  lines,  but  seen  from  the  side  must  appear  to  be.  They 
include  not  only  the  up-and-down  waves  of  a  shaken  rope  or  table- 
cloth, or  the  straight-line  vibrations  of  plane  polarized  light,  but 
also  circular  '  skipping-rope  '  motion,  or  the  irregular  vibrations  of 
ordinary  light,  which  are  merely  confined  to  planes  transverse  to 
the  waves'  travel.  The  typical  wave  form  is  now  a  Sine  Curve  like 
Fig.  119,  but  recollect  that  that  was  a  diagram  on  a  Time  base, 
whereas  now  both  co-ordinates  represent  lengths,  and  the  whole 
might  be  obtained  as  an  instantaneous  photograph. 

§  394.  The  speed  of  travel  of  waves  along  a  stretched  string  is 

found  thus  : 

Suppose  a  complete  circular  ring.  Fig.  127,  such  as  one  can 
easily  throw  along  a  rope  on  the  ground,  and  now  suppose  that 
the  string  is  being  hauled  back  just 
as  fast  as  the  ring  runs  forward. 
Then  we  have  a  ring  of  rope  which 
maintains  its   position,   in   space, 

but  the    circumference   of  which  

is    travelling    round  at  speed  v.  ^iq   127 

By  §  86  there  is  tension  in  it,  just 

as  in  the  rim  of  a  fly-wheel,   of  mv^.     This  must  =  T,  the  pull 

along  the  string  in  dynes,  or  else  one  would  overcome  the  other 

and  upset  the  equilibrium. 

/T        q      j  '  Ipull  on  string  in  dynes 


T  I' 

A/ —  or  Speed  in  cm.  per  sec.  =  \r 


mass  of  I  cm.  of  string 

Now,  there  is  no  need  for  the  ring  to  be  complete,  for  the  tension 
is  the  same  in  every  bit  of  it,  and  nothing  has  been  said  about  the 
radius  of  the  ring,  which  may  therefore  be  anything  and  vary 
anyhow ;  i.e.  a  distortion  of  any  shape  whatever  travels  on  the 
string  at  the  speed  we  have  found.     See  §  437. 

§  395.  Waves  of  *  longitudinal '  motion.  In  the  thh-d  type  of 
wave  the  only  motion  of  the  particles  is  to  and  fro  along  the  line  of 
travel  of  the  wave  itself. 

In  Fig.  126  L,  the  particles  perform  their  little  harmonic 
movements  along  the  horizontal  diameters  of  the  little  circles,  and 
become  crowded  together  and  scattered  alternately,  and  pass 
on  waves  of  compression  and  rarefaction  at  a  speed  far  greater 
than  their  own  motions.  Such  waves  can  be  seen  running  up  and 
down  a  long  wire  helix  when  its  end  is  pulled  straight  down  and  let 
go ;  the  spires  close  together  and  open  apart  periodically.  They 
run  on  a  piece  of  stretched  rubber  tubing  slipping  jerkily  back  through 
wet  fingers  ;  they  produce  a  shrill  sound  when  a  glass  rod  is  rubbed 
lengthwise  with  a  wet  leather  ;  they  travel  in  air  or  any  other 
substance  as  the  longitudinal  waves  of  compression  and  rarefaction 
convejdng  sound. 


302 


WAVES 


[§395 


It  is  a  slow  solitary  wave  of  this  type  that  clatters  along  a  checked 
goods-train ;  and  occasionally  a  few  to-and-fro  impulses  of  it  can 
be  felt  by  anyone  standing  in  a  long  passenger  train  as  it  starts. 

'  Transverse  waves   have  been  aptly  called  '  Shake-waves,'  and 
longitudinal  waves  '  Push- waves  ' ;    and  you  see  their  difference 
is  just  exactly  that  between  shaking  hands  with  a  man  and  punching  ^ 
his  head. 

§  396.  The  speed  of  longitudinal  waves  is  calculated  thus  : 

A  and  B  are  two  planes,  1  sq.  cm.  in  area,  moving  at  speed  V, 
and  maintaining  fixed  positions  in  the  wave  (just  as  the  fore-and- 
aft  edges  of  a  ship's  rudder  do  in  her 
stern  wave  :  this  by  way  of  a  rough 
illustration,  but  the  wave  now  under 
consideration  is  of  a  very  different 
kind).  For  this,  the  same  mass  m 
enters  the  space  AB  per  second  at 
A  as  leaves  it  at  B,  or  else  AB's  con- 
tents would  vary  in  quantity,  i.e.  it 
would  be  moving  about  in  the  wave. 
Let  u  and  w  be  the  actual  very 
small  forward  speeds  of  particles  at 
A  and  B,  due  to  the  compression 
somewhere  behind  B.  A  therefore 
catches  up  sparser  particles  at 
greater  speed  V  —  u  and  B  closer 
ones  at  Y  —  w.  Divide  the  speed,  which  =  the  volume  caught 
per  second  by  the  square-centimetre  plane,  by  the  volume  that 
contains  1  gm.  [=  a  at  A  and  6  at  B],  and  we  get  the  mass  caught 
m  =  (V  —  u)  -^  a  and  lost  m  =  (V  —  w)  ~  b. 


f 


^ 


^V. 


Fig.  128. 


.\  U 


am. 


w  =  y  —  hm. 


There  is  another  condition  of  permanence  :  the  resultant  force 
constantly  acting  on  AB  is  equal  to  the  increase  of  momentum 
that  takes  place  inside  it  per  second. 

The  force  is  the  forward  difference  of  pressures  Q  —  P  on  the 
square-centimetre  planes,  and  during  each  second  a  total  mass  m 
which  at  A  always  moved  at  speed  u  has  been  increased  in  forward 
speed  to  w 


Q  —  P  =  m{w  —  u) 

—  w(V  —  hm 


m'' 


Q 


V  +  am)  =  —  m^{b  —  a) 


increase  m  pressure 


b  —  a      increase  in  volume  of  1  gm. 


But  under  Elasticity,   §  143,  the  modulus  of  elasticity  E  was 
defined  to  be  the  ratio  of  the  increase  in  pressure  to  the  decrease 


§397]  WAVE   MOTION  303 

in  volume  it  causes  per  cubic  centimetre,  i.e.  per  volume  of  D 
grammes  [D  =  density]. 

.-.  —  m2  =  —  E  X  D. 

Now  m  =  c.c.  caught  per  sec.  x  mass  of  each  =  V  x  D  (slight 
increases  in  density  compensating  the  —  u  and  —  w). 

,\  V2D2  =  ED  /.  V  =  J? 

or  the  Speed  of  travel  of  a  Longitudinal  Wave  is  the  square  root 
of  the  quotient  of  the  Elasticity  of  the  medium  by  its  Density. 

This  applies  to  anything  :  rarefied  hydrogen,  water,  a  goods 
train,  the  '  Push- wave  of  an  earthquake,'  etc. 

This  latter  travels  under  the  drive  of  Young's  modulus,  §  142, 
at  speeds  from  5  to  8  miles/sec.  The  transversal  Shake- wave 
travels  about  half  as  fast,  under  the  control  of  the  much  smaller 
modulus  of  Rigidity,  §  143  ;  consequently  the  seconds  difference  in 
their  times  of  arrival  at  the  Seismograph,  x  4,  gives  an  idea  of  the 
distance  of  the  earthquake  focus.  Fluids  have  no  rigidity  :  it  is 
observed  that  the  shake-waves  always  come  via  the  crust  of  the 
earth,  and  never  through  its  core,  which  is  hence  deduced  to  be  a 
liquid  sphere  4000  m.  diameter. 

Happily,  there  is  a  homelier  instance  of  the  operation  of  this 
law,  to  be  found  by  anticipating  Sound  a  little,  and  it  is  second  to 
none.  When  you  are  stirring  up  sherbet  and  water  (or  effervescent 
salts,  if  you  have  outgrown  sherbet),  the  sound  of  the  spoon  falls 
softer  and  lower  as  the  liquid  swells  visibly  with  its  myriad  bubbles. 
True,  its  density  D  is  diminished  a  trifle,  but  a  gas  is  20,000  times 
as  compressible  as  water,  Chap.  IX,  Table,  so  that  E  of  the 
squashy  mixture  goes  down  to  a  small  fraction,  the  shock  of  the 
spoon  striking  the  glass  is  carried  only  idly  from  side  to  side,  and 
the  frequency  of  its  reflections,  i.e.  the  pitch  of  the  note  heard, 
goes  down  and  down. 

§  397.  Energy  carried  by  waves.  A  wave-train  carries  energy. 
One  can  do  work  at  the  far  end  of  a  rope,  or  throw  up  water  at  a 
distance,  by  setting  up  a  wave  motion.  Elastic  air  waves  carry 
sound,  or  sometimes  the  sudden  energy  of  explosions.  We  saw,  §  382, 
that  the  energy  of  a  vibrating  particle  =  ^mv^  =  ^m  x  ^iz^a^n^,  and 
now  in  wave  motion  the  mass  of  a  single  particle  has  to  be  increased 
to  the  whole  mass  of  all  the  particles  set  into  equal  motion  per 
second,  giving 

Power  =  energy  conveyed  by  wave  train  per  sec. 

=  J  mass  newly  disturbed  per  sec.  x  ^n^a^n^. 

Or,  the  energy  received  by  a  surface  per  second,  from  the  waves 
of  a  train  or  column  V  in  length,  A  in  area  of  cross-section,  in  a 
medium  of  density  D,  which  fall  upon  it  and  are  reduced  to  rest 

=  J  VAD  X  47r2a27i2 


304 


WAVES 


[§398 


§  398.  Doppler's  principle  deals  with  motions  of  observer  and  of 
source  of  waves. 

It  explains  the  change  in  musical  pitch  of  railway  whistles,  etc., 
as  heard  when  engine  or  observer  is  moving,  Chap.  XXVII,  Questions 
15 — 18  ;  by  the  change  in  vibration  frequency  of  light  of  definite 
character  (well-known  spectrum  lines)  from  moving  stars  and  far- 
distant  nebulae,  it  enables  us  to  calculate  their  speeds  in  the  line  of 
sight,  §  560  ;  coming  back  to  earth,  it  accounts  for  cars  meeting  the 
pedestrian  more  frequently  than  overtaking  him  ;  or  going  to  sea  : 

A.  Moving  observer. 

Sailing  out  against  the  waves,  they  pass  the  boat  more  fre- 
quently than  when  at  anchor,  and  sailing  with  them,  they  pass 
more  slowly.  If  their  speed  is  V  and  the  boat's  u,  the  speeds  of 
passing  in  the  three  cases  are  the  combined  speed  at  which  waves 
and  boat  rush  to  meet  each  other  Y  -\-  u  ;  V,  and  Y  —  u  the  speed 
at  which  the  waves  overtake  the  boat.     As  the  length  of  a  wave 


Fig.  129. 

remains  quite  unaffected  by  the  boat's  motion,  the  numbers  met 
in  a  given  time  are  also  in  the  same  ratios,  or  the  '  apparent  fre- 
quency '  is 

— ^— ,  I,  and  — :^ —  times  the  normal. 


B.  Moving  source  (not  applicable  with  precision  to  water  waves). 

The  source  of  the  waves  may  be  moving  at  speed  w  through  the 
medium  which  carries  them,  while  the  observer  is  at  rest.  From  a 
source  at  rest,  waves  spread  in  concentric  circles  ;  but  if  it  moves, 
the  successive  ripples  start  from  centres  farther  and  farther  from 
the  first,  and  Fig.  129  represents  their  distribution.  Each,  once 
started,  goes  on  spreading  from  its  own  centre  at  its  natural  speed  V. 

[We  cannot  deal  here  with  the  last  figure,  which  corresponds  to 
a  source  moving  faster  than  the  ripples,  e.g.  a  stick  drawn  through 
water,  or  a  rifle-bullet  in  air.]  But  when  w  is  less  than  V,  waves 
which  normally  occupy  a  space  V  get  squeezed  into  Y  —  w  ahead 


§  399]  WAVE   MOTION  305 

of  the  source,  and  spread  over  \  -\-  w  astern,  as  in  the  middle  figure. 
Their  lengths  alter  in  the  same  ratio,  and  as  all  are  travelling  at 
the  natural  speed  V,  the  number  that  pass  an  observer  ahead  is 

V  V 

increased  in  the  ratio  ^y and  astern  is  decreased  as  ~ 

\  —  w  V  +  1^ 

since  speed  -f-  wave-length  =  frequency.  Hardly  any  change  is 
noticed  by  an  observer  '  on  the  beam.' 

C.  If  u  and  w  are  small  compared  with  V  (as  they  usually  are), 
you  will  see  that  it  makes  no  appreciable  difference  whether  the  source 
or  the  observer  moves.  If  they  are  approaching  each  other,  the 
frequency  rises  in  the  ratio  (V  -|-  net  speed  of  approach)  :  V,  and 
if  receding,  it  falls  in  the  ratio  (V  —  net  speed  of  recession)  :  V 
(which  is  the  same  as  the  former  if  we  call  recession  a  minus  approach). 
You  need  no  formula. 


INTERFERENCE  OF  WAVES 

§  399.  We  saw  in  §  383  that  a  particle  disturbed  by  two  harmonic 
forces  will  vibrate  very  differently  at  different  times,  its  actual 
amplitude  gradually  alternating  between  the  sum  and  difference  of 
those  due  to  the  two  forces  independently. 

So  two  wave-systems  spreading  simultaneously  will  produce  very 
different  ampHtudes  at  different  places.  If  you  watch  the  waves 
coming  squarely  up  to  a  high  sea-wall,  you  can  see  and  follow 
reflected  waves  threading  their  way  back  through  the  oncoming 
wave-system.  As  you  do  so,  you  will  gradually  notice  that  the 
collisions  between  incoming  and  outgoing  waves  have  a  way  of 
occurring  in  certain  fairly  fixed  positions,  of  which  you  may  be  able 
to  locate  three  or  four,  farther  and  farther  out  from  the  wall :  here 
there  are  big  splashes,  quickly  subsiding  as  the  water  recoils  both 
ways  ;  in  between,  there  is  never  any  commotion  worth  looking  at. 

This  is  an  instance  of  the  Interference  of  two  running  wave 
systems,  producing  a  '  stationary  wave  system  ' ;  it  is  dealt  with 
fully  in  §  403.  Another  is  the  choppy  water  in  the  corner  of  a  dock, 
where  cross-reflections  from  the  walls  produce  a  local  bobbing  up 
and  down,  a  chequering  which  can  be  imitated  by  jarring  an 
oblong  dish  of  water. 

In  Fig.  130  let  P  and  Q  be  two  sources,  e.g.  two  prongs  of  a  tuning- 
fork,  vibrating  in  the  same  phase,  and  emitting  equal  wave-systems, 
of  which  the  solid  rings  represent  the  crests  and  the  pecked  circles 
the  troughs.  An}^  point  on  the  bisecting  axis  CC  is  equidistant  from 
both,  therefore  on  this  line  crest  arrives  with  crest,  and  trough  with 
trough,  amplitudes  are  doubled,  and  energy  quadrupled.  But 
along  \  J,  which  is  (a  hyperbola)  such  that  any  point  on  it  is  half  a 
wave-length  farther  from  P  than  from  Q,  P's  waves  ever^nvhere 
arrive  half  a  wave-length  behind  Q's  ;  crests  into  troughs,  the  motion 
is  destroyed,  and  no  energy  travels  there.     Along  the  next  hyper- 


^00 


WAVES 


t§399 


bola  11  the  difference  of  distance  is  a  whole  wave-length,  and  again 
crest  coincides  with  crest ;  along  the  next  there  is  1 J  difference, 
and  no  appreciable  resultant  motion.  Hence  there  is  a  steady 
pattern  of  quiet  rays  and  streams  of  short  ripples,  as  shown  on 


Fig.  130. 


the  right  of  the  diagram,  occupying  the  dotted  and  solid  hyper- 
bolas, worked  out  on  the  left,  from  the  intersections  of  the  circular 
ripples  that  a  snap  spark  would  enable  you  to  see  or  photograph. 

§  400.  Why  a  straight  wave  travels  straight  forward.  Now  let 
PQQ'  .  .  .  (Fig.  131)  be  a  straight  or  plane  wave-front,  i.e.  a  plane 
passing  through  many  particles  PQQ'  .  .  . 
vibrating  in  the  same  phase,  §381.  Each 
endeavours  to  send  out  its  own  circular 
ripples  in  all  directions.  P  and  Q  together 
would  produce  the  pattern  abeady  considered, 
but  now  Q'  Q"  .  .  .  join  in  with  their  rip- 
ples, '  interfere,'  and  cause  a  general  blur, 
and  the  only  parts  remaining  definite  and 
free  from  overlapping  are  the  little  arcs  p,  q, 
g'  ...  of  the  outermost  ripples,  which,  of 
course,  have  all  travelled  equal  distances 
from  their  sources. 

Together  these  coalesce  into  a  new  '  plane 
wave-front,'  and  we  see  that  a  plane  wave 
travels  forward  in  a  direction  perpendicular  to 
itself  without  alteration  of  shape.  (A  circular 
wave  will  spread  radially  into  a  larger  circle.) 
Backward  it  cannot  travel,  for  the  particles 
there  are  already  in  motion  ;  the  most  it  can 
do  is  to  reduce  them  to  rest,  and  that,  in 


Fig.  131. 


§401]  WAVE   MOTION  307 

the  absence  of  freshly  arriving  disturbance,  it  does.     Recollect  how 
smooth  a  surface  the  ripples  from  a  stone  leave  behind  them. 

You  see  these  long  straight  waves  steadily  rolling  in  with  the 
tide  on  a  very  shallow  beach  ;  they  are  the  essence  of  '  beam- 
radio  '  :  they  form  the  straight  ray  of  sunshine  that  streams  in 
through  a  crevice. 

§401.  Diffraction.  The  resultant  disturbance  goes  straight 
forward,  except  at  the  edges.  The  constituent  ripples  behave  like 
trees  growing  in  a  close  plantation.  These  lose  their  natural 
spreading  shape,  and  grow  straight  upward  only,  since  that  is  the  only 
direction  in  which  they  do  not  interfere  with  and  hinder  one  another's 
growth.  But  at  the  margin  of  the  wood  they  bear  spreading 
branches,  clad  with  foliage  almost  to  the  ground.  So  here,  we 
find  that  at  the  edges  mutual  interference  fails  to  prevent  the 
ripples  spreading  out  sideways  to  some  extent.  Fig.  131,  bottom. 

This  bending  round  the  corner,  into  the  *  shadow  '  of  the  obstacle 
which  has  limited  the  breadth  of  the  wave,  is  a  very  important 
characteristic  of  wave  motion,  and  is  known  as  Diffraction. 

It  is  easily  seen  behind  a  breakwater  ;  the  waves  gradually 
spread  into  the  calm  water  behind,  and  only  a  triangular  space  is 
completely  protected.  And  hiding  behind  a  corner  is  not  a  com- 
plete protection  from  the  waves  of  sound. 

It  appears  otherwise  with  Light,  and  the  sharp  shadows  thrown 
by  opaque  objects  were  long  a  difficulty  in  developing  the  wave 
theory  of  light.  But  closer  examination  shows  that  Ught  does 
spread  into  the  shadow  to  a  very  small  extent.  If  light  coming 
from  a  pinhole  in  a  card  with  a  bright  lamp  behind  it,  is  passed 
through  another  pinhole  a  foot  away,  and  then  received  on  a  third 
card  a  foot  beyond,  the  bright  circular  patch  is  much  larger  than 
the  hole,  and  the  smaller  the  holes  the  worse  is  the  discrepancy. 

But  this  is  not  a  fair  comparison,  nor  is  it  easy  to  make  one. 
Standing  on  the  breakwater,  we  see  the  first  half-dozen  or  more 
waves  gradually  curling  round  into  the  sheltered  water ;  but  the 
waves  of  ordinary  light  are  only  about  a  fifty-thousandth  of  an  inch 
long.  That  means  we  ought  to  be  inspecting,  with  a  microscope, 
the  space  within  a  ten-thousandth  of  an  inch  of  the  edge  of  the  pin- 
hole, instead  of  a  foot  away  from  it.  A  fiftieth-inch  pinhole  is  a 
thousand  wave-lengths  broad,  broader  than  the  North  Channel 
with  regard  to  the  Atlantic  swell,  and  that  does  not  diffract  round 
into  the  Irish  Sea  to  any  extent. 

Again,  sound-waves  are  a  few  feet  long  :  they  spread  well  over 
the  room  from  an  open  doorway  ;  but  a  train  plunging  into  a  deep 
cutting  goes  practically  out  of  hearing,  and  hills  or  large  buildings 
shut  off  the  sound  of  distant  bells  almost  as  soon  as  the  sight  of 
the  church  tower. 


Sog  WAVES  [§  401 

That  is,  when  the  observing  spaces  become  large  compared  with 
wave-lengths,  Diffraction  becomes  much  less  noticeable,  more 
definite  shadows  are  cast,  until  in  Light  it  requires  special  care  to 
observe  diffraction  at  all  (and  there  again  only  half  the  spreading 
occurs  with  violet  light  as  with  the  longer  ripples  of  red). 

The  theory  of  all  this,  developed  from  the  Principle  of  Inter- 
ference, is  too  long  to  put  in  here.  Return,  however,  to  a  sharp- 
ended  breakwater  for  an  illustration.  The  waves  that  escape 
past  it  ought  to  have  cut-off  vertical  '  gable-ends,'  Fig.  132.  The 
'  gable '  collapses  as  the  water  heaped  up  in  it  immediately  flows 
out  endways  into  the  calm  '  shadow.'  The  wave  travels  on  with 
a  sloping  end,  down  which  water  continues  to  flow  farther  and 
farther  out  into  the  '  shadow.'     This  keeps  on  flattening  the  slope. 


Fig.  132. 

so  that  the  flow  down  it,  i.e.  the  endways  extension  of  the  waves, 
presently  becomes  very  slow  compared  with  its  rate  at  first : 
diffraction  several  thousand  waves  beyond  the  obstacle  is  nothing 
like  as  noticeable  as  it  was  for  the  first  few  waves.  There  would 
be  a  return  flow  from  the  smooth  water  into  the  troughs,  which 
has  been  omitted  from  Fig.  132  for  clearness'  sake.  On  the  whole, 
no  water  flows  into  the  shadow,  only  the  wave  motion. 

§  402.  The  Diffraction  Grating.  Let  a  single  straight  wave-front 
strike  the  row  of  narrow  equidistant  obstacles  in  Fig.  133  (palings 
in  a  pond,  for  instance).  A  moment  after,  the  state  of  affairs  is  as 
represented.  Each  gap  has  let  through,  or  transmitted,  and  each 
obstacle  has  reflected  back,  a  separate  little  wave,  and,  the  spaces 
being  narrow,  these  spread  in  semi- circular  ripples.  In  any  direction 
PL,  not  one  ripple  is  sent,  but  a  succession  of  distant  ones,  their 
actual  distance  apart  depending  on  the  width  of  the  grating  spaces 
and  on  the  direction  of  PL. 


§403] 


WAVE   MOTION 


309 


This  can  be  heard  in  the  musical  sound  which  a  paled  fence  echoes 
to  a  sharp  footstep,  or  the  qu-u-urk  with  which  the  many-tiered 
seats  of  a  stadium  respond  to  a  rifle-shot,  the  rapid  string  of  Uttle 
echoes  blending  into  a  note,  a  Musical  Echo. 

It  is  vastly  important  in  Optics,  where  a  grating  with  perhaps 
15,000  spaces  to  the  inch  will  fling  off  light  of  different  wave-lengths 
(colours)  in  directions  PL,  PL',  etc.,  and  so  break  up  white  light 
into  colours.  If  instead  of  one  wave,  a  train  of  definite  wave- 
length falls  on  the  grating,  only  waves  of  that  length  can  exist 
anywhere,  all  others  getting  trampled  out  by 
interference,  and  these  can  pass  off  only  in 
certain  directions,  e.g.  in  the  figure  only  a  train 
of  6  mm.  length  can  pass  off  along  PL  :  in  a 
less-inclined  direction  only  shorter  waves  can 
pass  out. 

There  is  a  reflected  system,  such  as 
RM,  precisely  similar  to  the  transmitted 
system. 

If  several  definite  periodicities  can  be 
analysed  from  the  incident  disturbance,  several 
trains  of  diffracted  waves  will  spread  in 
definite  directions,  the  longer  waves  being 
thrown  off  at  greater  angles  :  red  is  more 
diffracted  than  blue  light.  The  grating  has 
analysed  a  disturbance  into  its  component 
S.H.M.'s,  §  381,  and  has  spread  them  out  to 
view  as  a  '  Spectrum.'  We  shall  return  to 
this  under  Light.  Fio.  133. 


REFLECTION  AND   REFRACTION  OF   WAVES 


§  403.     Waves 
back  or  reflected. 


beating  on  an  unyielding  surface  are  thrown 
If  circular  ripples  from  O  fall  on  the  flat  surface 
ACB,  a  point  on  the  ripple  which  would 
naturally  have  arrived  at  C,  has  had 
its  motion  reversed,  and  has  then 
travelled  without  change  of  speed  to  D. 
ADB  is  an  arc  exactly  equal  to  the 
original  one  AGB,  and  the  reflected 
ripples  spread  as  if  they  came  from  a 
point  I — a  '  virtual  image  ' — which  is 
perpendicularly  below  O,  and  as  far 
behind  the  reflector  as  0  is  in  front  of  it. 
Arc  ADB  centre  I  =  arc  AGB  centre  O. 

Your  eye  is  as  far  behind  the  looking-glass  as  in  front :    Echo 
dwells  deep  in  the  distant  grove. 


Fig.  134. 


310 


WAVES 


[§404 


§404.  Stationary  wave  motion.  The  choppiness  of  water  near 
reflecting  walls  and  *  interference  patterns  '  have  been  mentioned 
above,  §  399.  Let  us  see  how  this  so-called  '  stationary  wave 
motion,' — almost  as  contrarily  named  as  the  '  permanent  wave,' 
with  which  this  chapter  has  nothing  to  do — results  from  the  inter- 
ference of  running  waves.  Take  the  one  simple  and  important 
case  of  waves  meeting  perpendicularly  a  rigid  obstacle  ;  and  take, 
as  less  complex  in  construction  than  those  of  water,  waves  on  a 
rope  or  string  arriving  at  the  fixed  end,  Fig.  135. 

They  are  reflected  just  as  if  they  came  back  with  equal  wave- 
length, amplitude,  and  speed  from  an  '  image  '  source  beyond  the 
obstacle,    §  403.     The    direct    and   reflected   trains   interfere,   to 


y  V  /  \   y 


Fig.  135. 


produce  a  resultant  shape  obtained,  of  course,  by  adding  both  dis- 
placements together. 

At  the  fixed  obstacle  there  can  be  no  resultant  motion  ever,  there- 
fore the  train  travelling  to  the  left  must  produce  displacements 
there  always  equal  and  opposite  to  those  of  the  direct  train  moving 
to  the  right. 

In  Fig.  135  MR  =  ML,  and  both  are  increasing ;  with  this 
clue  the  reflected  wave-train  is  drawn  in  the  diagram,:  the 
arrow-marked  crest  above  R  will  reach  the  obstacle  at  the  same  in- 
stant as  the  bottom  of  the  trough,  travelling  out  at  L,  comes  into 
view ;  as  the  trains  pass  M  opposite  ways  MR  will  always  =  ML, 
and  M  remains  at  rest. 


§404]  WAVE   MOTION  311 

Adding  the  displacements  all  along  the  line,  one  finds  a  succes- 
sion of  points  N  N  at  which  the  two  displacements  are  always 
equal  and  opposite,  i.e.  no  motion  ever  occurs  at  these  points.  They 
are  Nodes,  and  remain  fixed  at  successive  half  wave-lengths  from  M. 

Half-way  between  them,  equal  and  similar  displacements  always 
come  to  be  added  together,  and  the  particles  at  these  Antlnodes 
vibrate  with  twice  the  amplitude  they  would  have  in  the  incident 
wave  alone'.     These  were  the  big  splashes  in  §  399. 

Whereas  in  Running  Waves  each  particle  performs  a  motion 
equal  in  amplitude  to  its  neighbours',  but  progressively  differing 
in  phase,  here  is  now  a  new  sort  of  undulation  in  which  each  particle 
performs  its  own  motion,  different  in  amplitude  from  its  neighbours'^ 
but  simultaneous.  Running  Waves  are  imitated  by  a  rotating 
corkscrew  seen  from  the  side,  these  *  Stationary  Waves  '  by  a 
rotating  zigzag. 

The  lower  figure  of  Fig.  135  shows  the  running  waves  and  the 
resultant  (thick)  '  stationary  wave  '  0-175  of  the  period  after  the 
upper  figure,  and  just  past  its  straight-line  mid-position,  when  one 
displacement  exactly  wiped  out  the  other  everjrwhere.  The  dotted 
lines  are  the  extreme  positions  of  the  '  stationary  waves,'  double 
the  amplitude  of  the  running  waves. 

The  argument  holds  for  longitudinal  waves  also,  e.g.  for  sound 
waves  at  a  wall ;  for  the  particles  next  the  surface  have  to  stop  there, 
at  rest.     See  later,  §  443,  etc. 

We  see  these  nodes  and  vibrating  segments  on  a  long  vibrating 
string.  Fig.  156 ;  quiet  nodal  lines  and  perturbed  antinodal  lines 
make  up  interference  patterns  on  water.  Fig.  130.  We  see  them  in 
the  longitudinal  motions  of  a  long  wire  helix  made  fast  at  the 
end — ^near  nodes  the  coils  are  alternately  squeezed  up  and  ex- 
panded, but  the  middle  one  does  not  move ;  near  antinodes  the 
coils  are  rushing  to  and  fro — ^we  can  detect  alternate  quiet  nodes 
and  windy  antinodes  in  '  pipes  '  resounding  to  a  high  harmonic. 

'  Reflection  from  a/ree  end  '  is  also  competent  to  set  up  stationary 
wave  motion,  but  there  is  a  difference  : 

Hang  up  two  pendulums  with  their  bobs  touching,  one  of  cork, 
the  other  of  lead.  Lift  and  drop  the  cork  bob  ;  it  hits  the  lead  and 
is  reflected  back  instantly ;  that  is  like  the  reflection  from  a  fixed 
end  considered  above.  But  lift  and  drop  the  lead  bob ;  the  cork 
flies  off  and  comes  back  to  return  the  blow  half  a  period  later. 

Again,  a  shunting  engine  bumps  into  a  train,  sending  a  wave 
of  compression  clattering  along  the  buffers.  The  last  truck  jerks 
out,  immediately  sending  a  wave  of  extension  back  along  the 
couplings,  and  then,  under  the  pull  of  its  stretched  coupling,  crashes 
back  and  starts  a  compressive  wave  half  a  period  later. 

This  '  reflection  from  a  free  end '  can  be  studied  in  the  wire 
helix,  and  it  occurs  at  the  open  ends  of  sounding  pipes.  The 
reflecting  place  is  one  where  the  motion  is  most  free,  i.e.  an  antinode 
(left-hand  end  of  Fig.  135  serves  to  show  it).  Reflection  of  light 
from  the  inside  of  the  surface  of  water-air  is  similar. 


312 


WAVES 
Fig.  136. 


[§404 


Fig.  137. 


405]  WAVE   MOTION  313 

Figs.  136  and  137  are  two  Charts  to  illustrate  Running  and 
Stationary  Wave  Motions. 

In  Fig.  136  you  see  that  the  satne  simple  harmonic  curve,  wave- 
length 1-5  in.  and  amplitude  0-1  in.,  makes  sixteen  steps  downward 
and  sixteen  steps  to  the  left  to  complete  one  period,  of  which  the 
chart  contains  two  in  depth.  As  time  travels  down  the  page,  the 
transverse  waves  run  out  to  the  left. 

Now  convert  this  into  a  moving  picture  of  Longitudinal  Waves  : 
DO  THIS.  Cut  a  slit  in  a  postcard,  3  in.  long  and  not  more  than 
I /loth  in.  wide,  lay  it,  upright,  on  Fig.  136,  and  move  it  left  and 
right  across  the  page.  Far  clearer  than  any  description  you  will 
see  how  the  air  particles  pulsating  to  and  fro  hand  on  the  waves 
of  compression  and  rarefaction  in  the  same  line — running  Waves 
of  Sound.     This  way,  the  charts  contain  two  and  a  half  periods. 

Now  look  at  Fig.  137.  Here  are  waves  which  don't  run  off  either 
way,  but  are  perpetually  changing  their  shape.  As  you  travel 
down  the  page,  in  time,  at  every  half-period  the  wave  curve  becomes 
a  straight  line,  a  Nodal  line  of  no  displacement.  Then  in  four  steps, 
a  quarter  period,  it  assumes  the  full  curve  of  Fig.  136,  from  which 
four  reducing  steps  bring  it  to  a  straight  line  ;  and  now  it  goes  through 
the  same  movement  the  other  way,  down  instead  of  up. 

Convert  it  into  Stationary  Longitudinal  Wave  Motion  by  your 
moving  slit,  as  before,  and  you  see  how  the  particles  alternately 
crowd  up  from  both  sides  to  a  Node,  and  then  rebound  and  leave  it 
both  sides,  to  crowd  to  the  next  node.  The  particle  half-way 
between  nodes  moves  most  freely ;  having  always  the  softest 
cushion  to  squeeze  against,  it  is  swinging  about  the  Antinodal 
position ;  you  can  thicken  in  the  nodal  lines  to  act  as  guides,  and 
rule  dotted  antinodal  lines  half-way  between  them. 

You  will  return  to  this  section  and  these  charts  again  and  again 
in  studying  Sound,  where  you  will  find  that : 

Strings  extend  from  node  to  node, 
Stopped  pipes  from  node  to  antinode, 
Open  pipes  from  antinode  to  antinode. 

§  405.  If  the  surface  AB,  Figs.  134,  138,  is  not  altogether  impene- 
trable, but  permits  the  wave-motion  to  pass,  in  part,  beyond  it — 
say  AB  is  the  edge  of  a  flat  submerged  rock,  or  a  shallower  part  of 
an  experimental  dish  of  water,  or  the  surface  of  a  wall  through  which 
sound  is  partly  audible,  or  of  glass  transparent  to  light  waves — then 
the  reflected  ripples  carry  back  only  part  of  the  energy,  and  more  or 
less  enfeebled  direct  ripples  continue  the  original  motion  over  the 
border,  but  always  with  an  alteration  of  speed. 


314 


WAVES 


[§405 


This  causes  Refraction.  We  have  stated  that  in  shallow  water 
waves  travel  slower,  and  have  found  the  same  in  media  of  greater 
density  (§  396).  CE,  where  AEB  is  the  new  wave-front,  is  less  than 
CG,  the  ripples  are  flattened  as  if  they  came  from  a  centre  at  a 
greater  distance  (but  are  now  not  quite  circular).  Conversely, 
if  the  medium  beyond  AB  transmitted  waves  faster,  AGB  would 
become  AFB,  and  the  ripples  spread  as  if  from  a  closer  centre. 

In  Fig.  138,  let  the  speed  of  wave-travel  in  the  medium  {i.e.  the 
substance  which  carries  the  waves)  above  AB  be  V,  and  in  the  lower 
medium,  v.  The  disturbance  that  would  have  spread  from  C  to  G 
with  speed  V,  now  travels  only  to  E,  with  speed  v,  .'.  CG/CE  =  Y/v. 
In  the  Spherometer,  §  152,  these  little  bulges  were  called  h,  and  it  was 

shown  that  the  radius  R  of  the  circle 
'  P  to  which  they  belonged  was  AC^/2h, 

~  i.e.  the  distances  to  the  centres  of 

these  ripple  systems  are  inversely  as 
the  bulges  CE,  CG,  etc.  In  point  of 
fact,  AEB  is  not  a  true  circular  arc, 
see  §  584,  but  if  short,  the  discrepancy 
is  trifling,  and  its  centre  P  is  V/v 
times  as  far  above  AB  as  the  actual 
centre  of  disturbance  0.  For  in- 
stance, for  Light  travelling  in  air  and 
water  V/v  =  4/3 ;  and  a  trout  sees 
the  fly  one-third  higher  above  the  surface  than  it  really  is. 

Per  contra,  turning  the  diagram  upside-down,  Light-wave  AGB 
spreading  at  speed  v  from  the  fish  in  water  becomes  AFB  at  V  in  air, 
where  CF/CG  =  Y/v,  to  the  angler  the  fish  appears  only  3/4  as 
deep  down  as  it  really  is. 

If  you  bring  0  so  near  the  surface  that  AFB  has  to  be  a  semi- 
circle, centre  C,  you  have  arrived  at  the  critical  condition,  with 
limiting  *  rays '  OA,  OB  :  total  reflection  ensuing  immediately  after. 

§406.    Reflexion  of  a  plane  wave  at  a  plane  wall.    Let  AC  be 

a  plane  wave-front,  §  400,  travelling  forward  at  speed  V,  and  incident 
upon  the  wall  AB  at  an 
angle  *  of  incidence  '  i. 

At  A  reflection  is  tak- 
ing place.  Presently,  by 
the  time  C  reaches  B,  the 
reflected  disturbance  from 
A  will  have  spread  to  D, 
where  AD  =  CB,  and 
DB  will  be  the  reflected 
wave-front,  built  up  as  in 
§  400  ;  which  evidently  leaves  at  the  same  angle  '  of  reflection  ' 
as  AC  arrived.     Hence 

In  Reflection,  the  angles  of  incidence  and  reflection  are  equal. 


Fig.  139. 


§408] 


WAVE   MOTION 


315 


§407.    Refraction  of  a  plane  wave  at  a  plane  boundary.    The 

disturbance  at  A  also  spreads  down  into  the  lower  medium,  but 
at  speed  v  (slower,  as  drawn.  Fig.  140)  and  arrives  at  E  by  the  time 
C  reaches  B,  and  the  refracted  wave  front  is  EB,  inclined  at  the 
angle  of  refraction  r  to  the  surface. 

Since  CB  and  AE  were  traversed  in  the  same  time,  they  must 
be  proportional  to  the  speeds  V  and  v  in  their  respective  media, 
.-.  CB/AE  =  V/v,  and  this  Ratio  of  Speeds  of  course  is  constant; 
it  is  called  the  Refractive  Index  of  the  second  medium  with  respect 
to  the  first,  and  is  usually  written  ^i  (Greek  m  ;   mu). 

Since  in  any  right-angled  triangle  the  length  of  a  side  divided  by 
the  length  of  the  hypotenuse  is  the  sine  of  the  angle  opposite  that  side 

In  triangle  BCA,  BC/BA  =  sine  BAG  ==  sine  i. 
„      „        BEA,  AE/BA=  sine  ABE  =  sine  r. 


"RG 

Divide,  BA  cancels  out,  -p^  = 


AE 


sme  I         ,      V 
=  also  — 


sme  r 


Hence,  in  Refraction,  the  ratio  of  the  sine  of  the  angle  of  incidence 
to  the  sine  of  the  angle  of  refraction  into  the  second  medium  is  constant, 
is  called  the  Refractive  Index  of  the  second  medium  {relative  to  the 
first),  and  is  actually  the  ratio  of  the  velocities,  in  first  and  in  second. 

It  is  easy  to  show  this  with  water  ripples  in  a  large  plate-glass 
tray,  co^itaining  barely  half-an-inch  depth  of  water  ;  AB  is  the  edge 
of  a  1/4-in.  glass  plate,  in  the  shallow  above  which  the  wavelets 
travel  slower. 


Fig.  140. 


Fio.  141. 


§408.  Total  Reflection.  When  the  incident  waves  are  nearly 
perpendicular  to  the  surface.  Fig.  140  becoming  Fig.  141,  and  sweep 
along  it,  CB  nearly  coincides  with  AB,  and  the  refracted  wave-front 
is  BE.  This  is  much  longer,  and  therefore  weaker,  than  AC,  from 
which  it  derives  its  energy. 

Conversely,  BE  emerging  into  the  faster  medium  would  become 
CA,  which  cannot  contain  all  the  power  of  an  energetic  BE,  and  much 
of  this  is  therefore  reflected  back.     It  is  very  remarkable  to  see  how 


316 


WAVES 


[§408 


determined  are  the  streams  of  wavelets  in  the  shallow  side  of  the 
trough  to  turn  down  again  over  their  own  slow  difficult  flats,  only 
mere  shadowy  continuations  of  them  venturing  out  into  the  deep 
water. 

When  the  waves  become  strictly  perpendicular,  AC  =  O,  and 
there  is  no  energy  to  produce  BE  :  the  last  light  of  the  setting  sun 
does  nothing  to  illumine  even  a  shallow  sea.  Conversely,  BE 
cannot  get  out  at  all,  but  is  totally  reflected,  and  so  are  all  waves 
beyond  it,  like  BE',  according  to  the  ordinary  law  of  reflection.  This 
totality  remains  incomplete  in  the  water-trough,  but  in  Light  it  is 
practically  complete,  and  unsuspected  scratches  on  a  totally- 
reflecting  glass  surface  glitter  in  the  midst  of  darkness  with  an 
unwanted  brilliance  that  seems  all  their  own. 

AE/CB  =  v/V,  and  now  putting  CB  in  coincidence  with  AB, 
AE/AB  =  sine  r  =  v/Y  =  l/[i.  Hence,  when  waves,  travelling 
at  slow  speed  v,  make  with  the  boundary  surface  of  a  faster  medium 
V  an  angle  greater  than  that  Critical  Angle  the  sine  of  which  is 
v/Y,  or  l/\i  of  their  own  medium,  they  suffer  Total  Reflection  hack 
into  their  slower  medium.     For  instances  of  this  see  §  491  in  Light. 

§409.    The  deviation  of  waves  passing  through  a  *thin  prism.' 

Suppose  plane  waves  fall  flat  on  AB,  Fig.  142,  one  face  of  a  narrow- 
angled  prismatic  space  ABC  in  which  they  must  travel  more  slowly 

{e.g.  the  tail  of  a  sandbank).  The  point 
B  of  the  wave  does  not  reach  C  until 
the  free  part  at  A  has  reached  E,  where 
AE/BC  =  speed  of  travel  outside  prism/ 
/speed  inside  =  Y/v  =  ^,  the  refractive 
index  of  the  prism  with  respect  to  the 
outer  space.  EC  is  therefore  the  position 
of  the  waves  as  they  leave.  Draw  CF 
parallel  to  BA,  small  angle  FCE  is  the 
change  of  direction  of  wave  front  and 
therefore  of  travel — the  Deviation — since 
the  waves  travel  perpendicularly  to  their 
own  fronts. 

Angle  A  of  prism  =  angle  ACF  =  arc 
AF  -^  radius  CF,  since  it  is  supposed  so 
small  that  the  difference  between  AF  and  the  arc  of  a  circle  is 
negligible. 

Similarly,  angle  D  of  deviation  =  angle  FCE  =  FE  -^  CF. 


Now 


AE 


AE 


.-.  AE 


AF 


_^  V       

BC  ~  V  ~  AF  V 

FE  =  AE  -  AF  =  (Y/v  -  1)AF 
D  _  FE/CF       V  _ 

A  ~"  AF/CF  -  ^  ~  ^  ~  ^  ~  ^• 
D  =  ((x-1)A. 


§409]  WAVE   MOTION  317 

You  can  easily  prove  for  yourself  that  for  any  waves  not  very 
far  from  parallel  to  AB  the  same  relation  holds  true. 

That  is,  provided  all  angles  are  small,  the  Deviation  produced 
by  a  Thin  Prism  is  obtained  by  multiplying  its  Angle  by  {the  ratio 
of  the  speeds  outside  and  inside  it,  less  1),  and  does  not  depend  on 
the  particular  angle  at  which  the  waves  strike  the  prism. 

After  using  a  pointed  sheet  of  plate  glass  to  show  this,  in  the  ripple- 
trough,  one  can  take  it  out  and  slant  the  whole  trough  to  imitate 
a  shelving  beach.  Then,  starting  straight  waves  in  any  direction  in 
the  deep  water,  one  sees  how  it  is  that,  whatever  way  the  wind  may 
be,  waves  roll  nearly  straight  in  up  the  beach,  as  you  have  so  often 
noticed.  For  the  inshore  end  of  a  passing  wave  must  travel  slower 
in  the  shallow  water ;  it  drags,  and  the  sea  end  over-runs  it,  and  swings 
the  whole  wave  round  more  and  more,  until  finally  it  makes  an 
almost  frontal  assault  on  the  shore. 


EXAM   QUESTIONS,   CHAPTER   XXV,   XXVI 

Questions  follow  later  in  Sound,  Light,  Alternating  Current,  High  Frequency, 
and  Radio,  for  these  two  chapters  are  simply  the  mechanics  of  periodic  motion. 

Throughout  Light  you  will  find  its  wave  character  insisted  on,  consequently 
the  wave  diagrams  134 — 142  should  be  puzzled  out.  Unfortunately,  wave 
diagrams  really  ought  to  be  moving  pictures,  and  attempts  such  as  139, 
140,  and  141  to  suggest  sequence  of  events  result  in  complication.  Try  to 
see  how  they  work,  however,  and  then  you  will  find,  in  Light,  very  different - 
looking  diagrams  which  are  really  only  these  with  the  wave-trains  reduced 
down  to  the  thickness  of  a  line — ^which  is  justifiable,  seeing  that  the  waves 
of  light  are  about  a  millionth  the  size.  Probably  it  will  be  these  simpler 
figures  you  will  learn  to  reproduce,  but  let  this  chapter  have  taught  you  their 
real  meaning.  Fig.  138,  however,  really  does  condense  a  lot  of  information 
in  a  very  small  space. 

I  believe  you  will  find  Figs.  136  and  137  thoroughly  useful  :  it  was  curious 
how  everyone  hated  the  sight  of  136  on  a  rolling  ship. 


SOUND 

CHAPTER   XXVII 
SOUND  TRAVEL 


§  411.  Production  and  Propagation.  An  exploding  cracker 
produces  a  sudden  outrush  of  air  straight  away  from  it,  and  a 
collapsing  vacuum-lamp  bulb  induces  a  sudden  inrush  from  all  sides 
straight  towards  it,  and  pops  most  satisfyingly. 

The  sudden  compression  of  air  in  the  outer  ears  when  diving  into 
water  gives  the  sensation  of  an  explosion. 

A  big  explosion  produces  a  pulse  in  the  atmosphere  that  can  be 
felt,  as  well  as  heard,  for  miles.  A  compression  travels  wave-like 
through  the  air,  breaking  windows  on  its  way  ;  its  shadow  has  been 
slow- motion-filmed  speeding  over  a  sunlit  plain,  and  the  shock 
felt  and  roar  heard  as  it  passed. 

* ...  In  college  fanes, 
Deep  organ-thunder,  rolling,  shakes 
The  prophet  blazoned  on  the  panes  ' 

and  the  lighthouse -syren  of  Alderney  rattles  the  crockery  on  our 
cottage  shelves. 

One  concludes  that,  physically  speaking,  the  ear  is  only  a  part 
of  the  body-surface  specially  sensitive  to  the  shock  of  impinging 
air,  and  that  sounds  are  heard  when  quick  compressions  or  expan- 
sions reach  it.  So  sensitive,  that  we  expect  to  hear  a  noise  from 
every  moving  thing  ;  if  we  don't,  we  think  its  motions  cat-like  and 
creepy. 

The  medium  of  transmission  need  not  be  air  ;  with  the  ears  under 
water  in  a  bath,  drops  falling,  or  noises  in  the  pipes,  are  heard  very 
distinctly  ;  miners  tap  on  the  wall ;  the  tracker  puts  his  ear  to  the 
ground ;  the  old-fashioned  physician  put  his  to  a  wooden  stetho- 
scope ;  solid  teeth  and  skull  are  commonly  employed  to  carry  sound 
to  the  '  inner  ear  '  in  deafness  due  to  outer  defect. 

That  an  elastic  material  medium  of  some  sort  is  necessary  is  proved 
by  the  Experiment  of  standing  a  cheap  clock  (in  the  tin  case  that 
lends  ferocity  to  its  tick),  on  some  tow,  inside  an  air-pump  receiver, 
and  exhausting  the  air.  The  tick  is  no  longer  heard,  nor  hardly 
the  ringing  of  the  alarm ;  the  tow  is  an  incoherent  sohd,  and  there 
is  little  air. 

318 


§413]  SOUND   TRAVEL  319 

Noise  and  note.  Air  pulses  travel  quickly  ;  a  single  one  requires 
very  special  means  of  study:  most  commonly  there  is  a  jumble  of 
irregular  ones  which  we  stigmatize  as  noise.  Fortunately,  it  is 
easy  to  produce  a  long  series  of  similar  impulses  by  the  use  of  some 
vibrating  body — card  pressed  on  a  cog-wheel,  fork,  string,  gong, 
whistle,  etc. — and  this  steady  succession  produces  a  '  musical 
note  '  which  can  be  studied  with  more  leisure  and  pleasure.  As  to 
the  boundary  between  noise  and  music,  of  course  no  two  people 
ever  agree. 

§  412.  Sound  waves.  We  know  that  these  things  set  up  a  running 
wave  motion  in  the  air,  spreading  spherically  through  it.  The 
waves  are  alternate  Compressions  and  Rarefactions.  Fig.  126  L 
bottom,  shows,  by  the  horizontal  spacing  of  the  lines,  two  compres- 
sions and  the  intervening  rarefaction ;  in  the  little  rings  above 
you  see  how  individual  particles  are  displaced,  right  or  left,  to 
produce  this  result.  They  blow  to  and  fro,  like  a  wee  changeable 
wind,  in  the  direction  of  travel  of  the  waves  they  hand  on. 

Fig.  143  is  drawn  from  an  actual  photo- 
graph, enlarged  2-5  times,  of  moving  float-      ~Z^2  -— - 

ing  specks  of  cork  dust  inside  a  horizontal  

organ  pipe,  sounding  loudly,  the  lower  of       ^TYTrr  

double  the  swing  (nearly  4  mm.)  and  there- 

fore  four  times  louder.     The  actual  wave-  

length,  on  the  same  scale,  was  1000  times     ^ 

as   much — 4  m.     Plainly  enough,   this   is  Fio.  143. 

*  longitudinal  vibration.' 

Turn  again  to  your  moving  picture,  Fig.  136,  and  watch  the 
pulsations  in  the  narrow  tube.  Thus  they  travel  in  the  narrow 
rubber  tubes  of  the  Stethoscope,  starting  from  the  little  pulsating 
patch  of  patient's  chest  delimited  by  the  '  mouthpiece,'  and  ending 
upon  your  ear-drums  :  a  stethoscope  merely  conserves  the  motion, 
and  the  ear-knobs  block  out  extraneous  noises ;  it  has  no  ampli- 
fying power. 

Such  are  Sound  Waves.  Transverse  motion  they  have  none,  for 
fluids  have  no  Rigidity,  §  143 ;  there  is  no  force  available  to  carry 
a  sidewise  motion  forward  to  particles  ahead.  Sound  cannot  be 
polarized,  as  can  Light,  §  651,  or  Radio  waves,  §  838. 

Above  open  ground  Sound  travels  out  in  hemispheres,  every 
radius  of  which  at  any  given  instant  is  identical  with  some  one 
particular  distribution  you  see  in  your  card  slit. 

§  413.  Speed  of  travel  of  Sound.  Direct  Methods.  That  sound 
travels  in  air  at  a  speed  which,  though  high,  is  far  from  instantaneous, 
is  familiar  to  everyone,  in  the  delay  between  the  fall  of  a  distant 
hammer  and  the  sound  of  its  blow,  between  the  visible  start  of  the 
100  yds.  and  the  snap  of  the  pistol,  between  the  puff  of  a  far-away 
steam  whistle  and  its  note,  between  noise  and  echo,  between  light- 
ning and  thunder,  seldom  half  a  minute  apart.    All  one  weary 


320  SOUND  [§  413 

summer  the  air  muttered  uneasily  in  quiet  Essex  gardens,  bringing 
the  message  with  which  the  guns  in  Flanders  had  laden  it  ten 
minutes  before. 

About  1708  the  earliest  extensive  experiments  on  the  speed 
of  sound  took  place  between  a  cannon  on  Blackheath  and  Upminster 
church,  12J  miles  away  across  the  Thames.  The  time  the  sight  of 
the  discharge  takes  to  travel  that  distance  is  inappreciable,  for  light 
travels  nearly  a  million  times  faster  than  sound.  The  report 
took  from  55J  to  63  sec,  according  to  the  wind.  For  the  air 
moving  as  a  whole  of  course  carries  all  contained  sound-waves 
with  it,  and  so  modifies  their  velocity  relative  to  the  earth.  The 
mean  of  many  results  with  winds  of  various  strengths  from  all 
points  was  1142  ft.  per  sec. 

In  1738  and  1822  various  French  and  Dutch  observers,  working 
in  fairly  calm  weather,  eliminated  wind  effect  by  firing  almost 
simultaneously  at  both  ends  of  11 -mile  distances.  The  wind 
accelerated  one  sound  as  much  as  it  retarded  the  other. 

The  experiment  was  repeated  in  this  way  in  1844,  in  Switzerland, 
with  the  guns  at  1800  ft.  and  8800  ft.  The  speeds  up  and  down  hill 
were  identical,  and  were  the  same  as  below  sea-level  in  Holland, 
showing  that  the  velocity  does  not  depend  on  the  pressure  of  the  air. 

In  1822  and  1890  Arctic  observations  gave  (1050  +  1  X  temp.  F.) 
ft.,  and  (333  +  0-6  temp.  C.)  metres,  per  sec,  between  —  40°  and  the 
freezing  point,  showing  how  the  velocity  increases  with  temperature. 

In  1905,  in  a  tunnel  2  miles  long,  it  was  proved  that  difference  of 
pitch  has  no  effect  whatever  on  the  velocity.  Were  it  otherwise,  indeed, 
a  tune  played  by  a  distant  band  might  become  confused,  and  the 
characteristic  quality  of  their  instruments  unrecognizable . 

The  speed  in  a  1-in.  pipe  is  about  1%  less. 

A  violent  explosion  wave  travels  faster  (in  accordance  with  theory) 
and  that  is  the  weak  point  of  the  gun  method  ;  for  100  yd.  or  more 
the  sound  is  certainly  '  violent.' 

But  you  can  get  a  very  fair  result  for  yourself  by  a  simple  echo 
method.  All  you  want  is  a  good  echoing  wall  in  a  quiet  place,  a 
foot-rule,  a  bob  on  a  bit  of  thread,  and  a  hook  or  a  friend  to  hang 
it  on.  Step  off  40  or  50  yd.  from  the  wall,  stand  and  clap  your  hands 
sharply.  The  echo  comes  back  at  an  interval  too  short  to  estimate  : 
multiply  the  interval.  You  cannot  clap  again  at  the  instant  the 
echo  returns,  because  that  would  drown  it ;  therefore  wait  an  equal 
time,  and  then  clap,  and  so  on.  This  is  not  so  difficult  as  it  sounds, 
because  clap  and  echo  will  alternate  like  the  tick-tock  of  a  clock, 
and  you  know  how  these  two  sounds  couple  themselves,  either 
one  way  or  the  other,  when  alternate  intervals  are  not  equal,  and 
the  clock  is  struggHng  on  with  lop-sided  ticks.  When,  after  a  little 
practice,  you  have  succeeded  in  this,  shorten  your  simple  pendulum 
until  it  beats  exactly  with  your  clapping,  one  single  swing  each 
time,  and  refer  to  Fig.  144. 

Thus  did  Newton,  in  the  cloister  of  Trinity,  when  the  old  knocker 
was  new,  that  they  tried  to  kid  me,  as  a  boy,  lifted  when  you  stamped 


§414]  SOUND   TRAVEL  321 

hard  on  a  particular  flagstone.  Or  else  you  count  up  strokes  per 
minute  by  aid  of  a  watch  ;  then  between  any  two,  sound  has  travelled 
to  the  wall  and  back,  and  might  have  done  it  again. 

Speed  =  4  X  distance  of  wall  x  claps  per  unit  time. 

On  the  right  of  Fig.  144  is  suggested  a  tuning-fork  and  a  resonance 
tube  ;  for  this  method  see  §  442. 


Fig.  144. 

The  accepted  value  for  the  Speed  of  Sound  in  dry  air  is  331  -S-f-O-B/®  C. 
m.  per  sec. 

This  is  1087  ft.  per  sec.  +  2  ft.  per  °C.  above  zero, 
or  371  yd.  per  sec,  plus  0-37  yd.  per  °F.  above  '  Temperate  * 
or  740  m.p.h.  at  the  freezing-point,  rising  to  760  m.p.h.  at  '  Tem- 
perate.' 

In  Water  the  speed  was  first  measured  one  night  in  1826  in  the 
Lake  of  Geneva.  The  hammer  of  a  submerged  bell  was  let  fall  by 
a  cord  which  simultaneously  dropped  a  lighted  match  into  powder. 
The  flash  was  seen  9  miles  away,  and  the  sound  listened  for  with  a 
large  ear-trumpet  having  a  membrane  stretched  across  its  mouth, 
under  water.     Speed,  1435  m./sec.  at  8°  C. 

Using  heavy  charges  of  explosive  the  shock  travels  faster,  e.g. 
2000  m/s  for  150  m.  from  41b.  of  gimcotton. 

§  414.  Theoretical.  In  §  395  it  was  proved  that  the  speed 
at  which  longitudinal  wave-motion  advances  through  an  elastic 
medium  is  the  square  root  of  its  elasticity  (d3mes/cm.2)  by  its  density 
(gm./cm.3),  provided  the  particles  themselves  move  but  little. 
This  is  a  wave  of  sound,  not  too  loud. 

For  sea-water  modulus  of  compression  is  2-33  x  10^®,  D  =  1-028 

.-.  S  =  VE/D  =  150,000  cm.  =  4900  ft./sec.  =  0-8  sea-mile. 

Submarine  Bell  signals  are  reliable  for  at  least  2  miles  under 
weather  conditions  which  would  render  aerial  signals  useless,  and 
up  to  15  miles  in  calm ;  and  many  lighthouses  are  equipped  with 
them.  The  ship  has  a  well-submerged  microphone  on  each  bow, 
and  each  is  switched  in  to  the  bridge  telephone  in  turn. 

[If  the  current  that  drops  the  bell-hammer  also  starts  a  wireless 

M 


322  SOUND  [§  414 

signal,  the  interval  between  them  on  the  telephone  in  seconds  =  IJ 
times  distance  in  sea-miles.] 

Several  makes  of  acoustic  depth -sounding  machine  are  on  the 
market.  In  one  *  Fathometer  '  a  hammer  strikes  the  hull  of  the 
ship  every  J  min.,  a  microphone  switches  in  and  converts  the  echo 
from  the  sea -bottom  into  a  flash  of  light  which  appears  opposite 
the  depth  on  a  graduated  dial.  In  another  pattern  the  hammer 
current  starts  a  pen  from  '  sea  level,'  and  it  marks  vertically 
down  until  the  echo  lifts  it  off  the  paper  band,  which  creeps 
slowly  along  :  the  lower  ends  of  the  close  parallel  lines  form 
a  complete  contour  of  the  sea-bottom  over  which  the  ship  is  steaming. 

For  ice  Young's  modulus  is  given  as  2-8  x  lO^o,  D  =  0-917 

/.  S  =  VE/D  =  1750  m./sec.  =  5800  ft./sec. 

and  an  echo  experiment,  of  similar  character,  has  measured  9000  ft. 
depth  of  ice  over  mid- Greenland. 

For  rock  Young's  modulus  averages  about  1-5  x  10^^,  D  =  2-5 

/,  S  =  a/E/D  =  nearly  8  km.  or  5  miles/sec. 

the  speed  of  sound,  or  an  earthquake  '  push-wave,'  in  the  earth's 
crust ;  a  score  times  that  in  air.  Using  a  ricketty  old  microscope 
60  yd.  away  from  a  pile-driver,  planting  the  foundations  for  a  new 
medical  school,  the  field  of  view  beginning  to  quiver  always  gave 
warning  of  the  coming  '  thump.' 

For  air  Newton  employed  in  this  formula  (which  he  discovered) 
the  result  of  his  friend  Boyle,  that  the  elasticity  =  the  pressure. 

For  if  PV  is  constant,  1  %  increase  in  P  will  cause  1  %  diminution 
in  V,  since  101  x  99  =  100  X  100  very  nearly. 

.  y  _  increase  in  pressure  _  0-01  P  _  p 
~  contraction  per  c.c.  ~    0-01    ~ 

Taking  atmospheric   P  =  1,013,000  dynes/cm.2  and  D  =  0-00129 

gm./c.c,  V  =  28,000  cm. /sec,  which  is  too  low. 

It  was  not  until  1822  that  Laplace  pointed  out  that  the  com- 
pression in  a  sound-wave  is  very  quick, 
whereas  that  in  a  Boyle  tube  is  slow. 

To  obtain  a  correction  for  this,  a  large  flask, 
Fig.  145,  containing  air  at  a  pressure  B,  a  little 
less  than  the  atmosphere  A,  is  suddenly  opened 
and  closed  by  a  sliding  plate.  Air  rushes  in 
to  raise  the  pressure  to  A,  but  the  sudden  com- 
pression heats  the  air  inside  (§291)  and  after  a 

V^oVw/     ^"^ ^  few  minutes'  cooling  to  its  original  temperature 

Fig.  145.  ^^^  pressure  has  fallen  somewhat,  to  C.  {i.e.  the 

oil  in  the  gauge  is  still  drawn  up  to  C,  by  the 

reduced  pressure).      That  is,  it  took  a  sudden  increase  A — B  to 

do  what  might  have  been  coolly  and  quietly  done  by  only  C — B, 


§  415]  SOUND  TRAVEL  323 

viz.  to  drive  a  little  air  into  the  flask  and  slightly  compress  its  con- 
tents ;  or  the  sudden  '  adiabatic  '  (§291)  elasticity  is 

,^^  times  the  slow  '  isothermal '  Boyle  elasticity  =  x  P. 

The  experiment  gives  this  ratio  1-40  for  air  and 


Q         /E  sudden  _    /I -40  x  P        /l-40  x  1,( 


,013.000 
•00129 
33,200  cm./sec. 


We  all  count  5  sec.  to  the  mile  to  find  out  *  where  that  one  went ' 
in  thunderstorms,  but  a  more  elaborate  system  of  range-finding  is 
valuable  in  locating  a  distant  gun.  Microphones  in  three  observa- 
tion-posts, not  in  line,  record  their  reception  of  its  sound  on  the 
moving  film  of  a  distant  string-galvanometer,  §  762,  alongside  the 
ticks  of  a  clock.  The  time-interval  between  first  and  second  gives 
the  difference  of  their  distances  from  the  gun,  which  therefore  lies 
somewhere  on  a  hyperbola  drawn  on  the  map,  with  these  posts  as 
foci,  and  this  common  difference.  The  difference  between  second  and 
third  gives  a  second  hyperbola,  which  cuts  the  first  at  the  gun. 


Speed  =  J- 


§  415.  From  this  relation 

^ratio  of  elasticities  x  pressure 
density 

you  see  that  in  gases  : 

A.  Change  of  gas  pressure  does  not  change  the  speed. 
For  doubling  the  pressure  would  halve  the  volume,  and 

therefore  double  the  density  also. 

B.  Speed  is  proportional  to  square  root  of  absolute  temperature. 

For  if  D  is  constant  P  will  increase  (or  if  P  is  constant 
V  will  increase  and  .*.  D  will  decrease),  proportionally 
to  the  absolute  temperature. 


Thus  speed  at  t°  =  V(273-f  ^)/273  speed  at  0° 

=  (^  +  i  •  2^)  X^^^  ^^^  g^^ 
which  for  Air  =  (l  +  ^)  330  =  330  -f  0-6<  m./sec., 

an  increase  of  speed  of  2  ft.  (in  1085)  or  60  cm.  per  sec.  per 
1°  C.  rise  of  temperature.  Notice  that  this  approximation 
holds  good  only  for  air,  and  at  ordinary  temperatures. 


324 


SOUND 


[§415 


C.  In  different  gases  the  speeds  are  inversely  as  the  square  roots 
of  the  densities. 

For  instance,  4  times  faster  in  hydrogen  than  in  oxygen  ;     ^ 
and 


Relative  speed, 
experimental. 

l/Vrelative 
density. 

Air          .          .          . 

coa,  N26    ;    ; 

NH3        .          .          . 

Argon     . 

1 

3-8 
0-80 

1-23,  slow 
0-92,  fast 

1/1 

1/3-8 

1/-81 

1/1-30 

1/-85 

The  ammonia,  NHg,  illustrates  a  discrepancy  that  must  occur  in 
this  law  when  the  ratio  of  elasticities  changes.  This  ratio,  7/5  for 
diatomic  gases,  becomes  about  5/4  for  steam,  or  ammonia,  with 
more  complex  molecules,  but  was  calculated  theoretically  to  rise 
to  5/3  for  monatomic  gases.  Upon  the  discovery  of  Argon,  its  den- 
sity was  measured  as  20  times  hydrogen,  but  its  atomic  weight 
defied  chemical  determination  :  the  precious  gas  was  admitted 
into  a  dust-tube,  §  443,  and  the  waves  of  sound  in  it  proved  to  be 
l/8th  longer  than  in  a  common  diatomic -molecule  gas  of  that  den- 
sity (tallying  in  that  respect  with  hot  mercury  vapour),  hence  it 
could  only  be  monatomic. 

We  shall  see  that  the  frequency  of  the  note  emitted  from  a  wind 
instrument  is  proportional  to  the  speed  of  sound  in  the  gas  which 
fills  it.  A  whistle  blown  with  hydrogen  therefore  becomes  very 
shrill  (and  feeble),  but  a  very  familiar  instance  is  the  sharpening 
of  the  hiss  of  an  unlit  gas-jet,  the  signal  we  all  listen  for  that  the 
air  has  been  blown  out  of  the  pipe  and  the  lighter  gas  has  arrived. 
Nitrous  oxide  is  half  as  dense  again  as  air,  and  accordingly  the 
'  laughter  '  induced  by  this  ansesthetic,  when  clumsily  administered, 
is  in  a  pitch  'y/(l/l-5)  =  1/1-22,  or  two  tones  lower  than  the  natural 
voice,  and  is  not  pleasant  to  hear. 

§  416.  Reflection  of  Sound.  We  notice  this  most  when  suddenly 
coming  from  the  open  country  into  space  confined  by  woods  and 
banks  and  buildings,  the  rattle  of  a  paled  fence,  or  the  sudden  sound 
of  a  cottage,  being  simply  reflections  of  the  unsuspected  noise  the 
car  itself  is  making.  The  bark  of  the  road-bridges,  and  the  roar  of 
cuttings  and  stations  and  tunnels,  are  just  the  returned  noise  of 
our  own  train,  and  taken  all  together  these  make  up  the  greater  part 
of  the  song  of  the  road  to  which  the  driver  comes  to  trust  to  tell 
him  where  he  is. 

Reflections  among  clouds  are  largely  accountable  for  the  roll  of 
thunder,  but  the  great  length  of  the  lightning  flash  compHcates 
matters,  and  the  loud  noise  of  a  solitary  aeroplane  dodging  between 
and  under  clouds  gives  more  definite  evidence. 


§  417]  SOUND   TRAVEL  326 

A  reflection  that  is  clear  and  definite  we  call  an  Echo,  and  we  get 
it  from  cliff  or  wall,  or  even  from  the  dense  summer  foliage  of  a  wood, 
for  the  waves  of  sound  are  several  feet  long,  and  such  surfaces  are 
less  rough  to  them  than  this  paper  is  to  the  minute  waves  of  light, 
i  ]cho  dwells  as  far  behind  the  reflecting  wall  as  the  source  is  in  front ; 
she  is  I  in  Fig.  134,  the  '  virtual  image  '  of  the  *  object '  O. 

With  curved  reflectors,  the  actions  we  shall  study  in  some  detail 
under  Light  are  obeyed,  of  course,  by  the  waves  of  Sound.  For 
instance,  the  obstructive  proscenium  arch  is  done  away  with  in  a 
modern  theatre,  and  the  player  stands  at  F,  Fig.  229  C,  near  the 
focus  of  a  semi-paraboloid  surface,  like  the  top  half  of  a  car  headlight, 
and  the  acoustic  improvement  throughout  the  auditorium  is  a 
revelation. 

In  the  U.S.  Capitol  is  an  old  assembly  hall  now  rightly  devoted 
to  statuary,  for  its  acoustic  properties  must  have  been  trying.  It 
has  a  quarter-ellipsoid  ceiling,  and  the  guide  collects  the  party  at 
I,  Fig.  229  B,  and  then  goes  across  and  talks  to  the  floor  at  O,  and 
is  heard  speaking  in  the  midst  of  them,  just  as  before. 

In  Whispering  Galleries  the  sound  does  not  leap  across  as  in  B, 
but  laps  round  the  wall,  suffering  repeated  reflections,  in  a  polygonal 
track,  in  which  it  can  be  heard  by  an  ear  close  to  the  wall. 

Sound  reflections,  of  course,  imply  Sound  Shadows,  behind  the 
reflectors.  The  travelling  of  street  noises  round  the  corner,  or 
of  noises  round  about  the  house,  is  largely  due  to  the  presence  of 
other  reflectors,  but  Diffraction  accounts  for  a  great  deal  of  bending 
of  sound  into  the  shadow,  for  the  waves  are  often  nearly  as  big  as  the 
obstacles,  and  you  get  Fig.  132  instead  of  the  broad  straightforward 
wave-front  of  which  Fig.  131  is  but  one  end.  Sound  waves  are  feet 
long,  and  light  waves  fifty-thousandths  of  an  inch,  and  that  is  why 
sound  shadows,  on  any  domes- 
tic scale,  are  so  vague  and 
uncertain  compared  with  light 
shadows. 

So,  when  it  comes  to  ex- 
periments of  laboratory  di- 
mensions, one  must  be  careful 
to  employ  sound-waves  of 
very  small  length,  such  as 
those  from  the  tick  of  a  wrist - 
watch.  Fig.  146. 

Various  experiments  can  be 
contrived,  but  that  of  Fig.  146  A  is  as  good  as  any  :  the  watch 
hangs,  in  a  good  light,  2  ft.  or  so  in  front  of  a  30-in.  paraboloid 
searchHght  murror  ;  standing  back,  and  looking  about,  you  find  the 
magnified  aerial  image  5  or  6  ft.  away,  and  placing  your  ear  to 
it  you  hear  it  ticking  ;  the  optical  and  acoustic  images  are  identical. 

§417.  Refraction  of  Sound.  We  have  seen  in  §405  that  Re- 
fraction occurs  when  waves  pass  into  a  medium  in  which  they  travel 


326  SOUND  [§417 

with  different  speed,  and  in  §  415  two  causes  of  change  of  speed 
were  pointed  out.  The  first — variation  of  temperature — occurs 
abundantly  in  the  atmosphere,  v.  Chap.  XXI,  and  the  result  is 
that,  over  any  considerable  distance,  sound  sometimes  gets  so  broken 
up  by  irregular  refractions  from  invisible  masses,  often  of  brilliantly 
clear  air,  that  sound  signals  become  very  unreliable.  Efforts  to 
counteract  this  by  the  employment  of  enormous  power  are  only 
very  partially  successful,  cf .  §  434  :  fog,  when  continuous,  favours 
the  carrying  of  sound,  but  not  when  patchy ;  submarine  signals 
have  far  less  trouble,  for  variations  in  water  density  are  trifling. 

§  415  C  provides  a  laboratory  experiment  on  sound  refraction  ;  the 
speed  of  sound  inCOgOrNgO  is  only  1/1-22  of  that  in  air  (its  refractive 
index  is  1-22) ;  accordingly,  blow  up  a  big  balloon  with  one  or  other 
of  these  gases,  and  arrange,  as  in  Fig.  146  B,  the  watch  at  about 
2  diameters  on  one  side,  and  your  ear  as  far  the  other  side  :  the  balloon 
acts  as  a  spherical  lens  and  concentrates  the  tick.  Don't  expect 
results  with  lesser  distances  ;  see  Fig.  200,  II,  etc. 

'  Musical '  Echo  has  been  mentioned  in  §  402.  If  the  echoing 
wall  consist  of  a  series  of  steps,  at  regularly  increasing  distances  from 
the  observer,  such  as  the  backs  of  the  seats  in  a  grand-stand,  a 
succession  of  little  echoes  of  a  sharp  crack  will  come  back,  separated 
from  one  another  by  twice  the  interval  of  distance  between  the 
reflectors,  for  the  sound  has  to  both  go  and  come  this  interval. 
Thus  if  the  seat-backs  were  3  ft.  apart,  a  *  train  of  waves,'  each  6  ft. 
long,  would  arrive.  This,  as  we  shall  see  very  shortly,  means  a 
musical  note ;  with  this  distance  a  short  guttural  F  or  G,  rising  to 
a  higher  note  as  the  common  interval  diminishes,  to  that  of  a  paled 
fence  by  the  roadside,  for  instance  ;  but  always  reminiscent  rather 
of  the  music  of  the  poultry-yard. 

§  418.  Wind.  Sound  '  carries  '  down-wind  mainly  because  the  air 
moves  much  faster  higher  up,  where  not  impeded  by  friction  with 
the  ground,  and  makes  the  spreading  waves  overhang  and  beat 
downwards.  Up- wind,  the  sound  lifts  off  the  ground  and  goes  up  : 
from  a  70-ft.  roof,  in  a  strong  westerly  breeze,  I  have  heard  conversa- 
tion on  the  ground-level  100  yd.  east. 


Fig.  147. 

To  get  some  idea  of  the  distortion  of  sound-waves  caused  by 
wind,  blow  on  a  hemispherical  soap-bubble.  In  Fig.  147  the 
speeds  of  the  wind  low  down  and  higher  up  are  marked  as  frac- 
tions of  the  speed  of  sound.  The  little  arrows  are  the  directions 
of  travel  of  the  wave-fronts  to  which  they  are  perpendicular. 

The  good  '  carrying  '  of  sounds  over  water  is  mainly  a  question 


§420]  SOUND   TRAVEL  327 

of  the  absence  of  obstacles  and  the  wind  eddies  they  cause.  The 
exaggeration  of  sounds  in  the  night,  when  they  may  be  audible 
10  or  even  20  times  as  far,  is  ascribable  to  the  profound  silence,  in 
the  absence  of  noises  due  to  wind  and  traffic.  The  cessation  of  the 
latter,  on  Nov.  11th,  makes  a  remarkable  difference,  in  the  very 
middle  of  the  Park. 

§  419.  Silent  areas.  Get  your  map  of  England  ;  draw  a  triangu- 
lar loop  Northampton-Canterbury-Ascot.  Inside  this  area  was 
heard  (and  felt)  a  terrific  explosion  at  a  chemical  works,  N.  of  Green- 
wich, Jan.  19th,  1917.  Its  irregular  shape  is  the  product  of  all  the 
causes  we  have  been  discussing.  Draw  another  line  from  Orfordness 
in  Suffolk,  to  Nottingham,  Stow  (Lines.),  and  thence  to  Cromer. 
Inside  this  area  also  the  explosion  was  heard,  though  a  good  many 
seconds  later  than  if  the  sound  had  travelled  direct. 

In  the  intervening  belt,  40  miles  wide,  nothing  was  heard,  and 
this  is  commonly  the  case  with  big  explosions,  experimental  and 
otherwise. 

For  explanation,  the  upper  Fig.  184  will  serve,  turned  upside 
down.  Seven  miles  above  us  is  the  Stratosphere,  kept  warm  by 
sunshine  above  all  clouds.  It  acts  towards  sound-waves  slanting 
up  from  below  as  does  the  hot  air  over  the  sea-  or  land -surface 
towards  Light-waves,  in  Mirage.  It  is  a  medium  of  greater  velocity, 
into  which  they  cannot  enter  very  obhquely,  but  are  totally  re- 
flected, and  reach  the  ground  again  farther  on,  having  crawled  a 
great  many  extra  miles  through  air  in  the  neighbourhood  of  —  60°  C. 

§420.  Loudness  of  Sound.  In  making  measurements  of  the 
loudness  of  a  sound,  an  arbitrary  standard  has  to  be  adopted, 
neither  small  nor  great ;  just  as  a  metre,  or  a  candle-power,  is  not 
at  either  extreme  of  possible  measurements. 

In  tests  made  with  a  number  of  speakers,  it  was  found  that  at 
the  usual  inch  from  the  telephone  transmitter  their  voices  caused 
an  average  variation  of  pressure  on  the  plate,  §  819,  of  11-5  dynes 
per  sq.  cm.  From  the  known  elasticity  and  density  of  air  it  is 
calculable  that  the  energy  in  ergs  per  sq.  cm.  falling  on  the  surface 
per  second  is  1/41-5  of  the  (pressure)^,  3-2  ergs.  Assuming  the  same 
for  every  sq.  cm.  of  the  hemisphere  of  2-5  cm.  radius,  i.e.  of  2-n  X 
2-52  =  39  sq.  cm.  area,  into  which  the  mouth  is  speaking  at  dia- 
phragm distance,  gives  3-2  x  39  =  125  ergs  per  second  as  the  energy 
output  in  ordinary  speech. 

If  you  are  familiar  with  wireless,  perhaps  you  would  rather  call 
this  12-5  microwatts  of  power;  for  loud  public  speaking  it  may 
increase  twenty-fold,  to  a  quarter  milliwatt. 

What  does  125  ergs  amount  to  ?  By  burning  a  match  under  a 
tobacco  tin  with  an  ounce  of  water  in  it,  you  can  collect  125  cals., 
with  half  the  stick  left.  Now,  1  cal.  is  42  million  ergs,  and  figuring 
it  out,  you  will  find  that  if  you  have  talked  for  2  hr.  a  day  all  your 
life,  you  have  emitted  as  much  energy  as  that  match. 


I 


328  SOUND  [§  420 

Carrying  on  a  conversation  at  arm's  length,  in  the  absence  of 
strengthening  echoes,  this  125  ergs  per  second  gets  spread  over 
a  hemisphere  of  a  metre  radius,  of  which  your  ear-drum  forms  part ; 
the  area  of  this  hemisphere  is  2iv  X  100^  =  62,500  sq.  cm.,  so  that 
2/lOOOths  erg  per  second  passes  out  per  sq.  cm.  (0-0002  micro- 
watt). 

With  that  we  must  make  shift  as  a  standard  of  loudness,  and  it  is 
called  55  decibels  ('  above  threshold  '). 

A  sound  ten  times  louder,  like  your  neighbour's  loud-speaker, 
would  be  described  as  65  decibels ;  one  ten  times  weaker — as  the 
conversation  would  be  at  10  ft.  distance,  since  10  ft.  =  3-1  m.,  and, 
applying  the  inverse  square  law,  just  as  in  §§  475,  721,  1/3- 1^  is 
about  1/10 — would  be  45  db. 

This  is  just  simply  a  common  logarithmic  or  slide-rule  scale  ;  add 
10  db.  means  multiply  by  10 ;  add  1  db.  means  multiply  loudness 
by  the  tenth  root  of  10,  which  has  log  0-1  and  is  factor  1-26.  Log  2 
is  0-3,  so  that  3  db.  louder  means  twice  as  loud. 

The  justification  for  this  way  of  reckoning  is  that  25%  happens 
to  be  about  the  smallest  step  in  loudness,  up  or  down,  that  the  ear 
can  detect  with  confidence,  when  the  sounds  to  be  judged  follow  at 
intervals  of  a  second  or  two. 

It  is  always  a  '  geometrical  progression,'  a  percentage  increase ; 
just  as  a  musical  interval  is  always  a  percentage  increase  in  fre- 
quency, and  the  keyboard  a  6%  logarithmic  scale,  §  459. 

The  simplest  instrument  for  comparing  sounds,  physically, 
is  the  Rayleigh  disc,  a  l-cm.-diam.  galvanometer  mirror,  §  764, 
hung  by  a  quartz  fibre  at  45°  to  the  oncoming  stream  of  sound, 
which  it  turns  to  face,  just  as  the  soap  dropped  into  the  bath  always 
turns  flatways,  into  the  position  of  maximum  resistance.  Unfortu- 
nately, its  sensitivity,  for  general  use,  does  not  reach  down  beyond 
this  ordinary  conversational  2  milliergs/sq.  cm.  sec,  with  its  pressure- 
variation  of  about  0-3  dyne/sq.  cm.  and  actual  ampUtude  of  move- 
ment of  the  air  about  1/8  micron,  a  quarter  of  a  wave-length  of 
ordinary  light  (or  less  for  higher  pitches).     Contrast  Fig.  143. 

The  minimum  audible  sound  has  been  measured  lately  as  bringing 
only  a  400  millionth  of  an  erg  per  second  to  the  ear,  which  is  about 
the  amount  of  energy  receivable  as  light  from  a  candle  15  miles 
away.  This  would  be  visible  to  an  eye  10  times  the  area  of  your 
dark-adapted  eye,  or  as  big  as  your  ear-drum,  so  that  these  two 
surfaces  of  contact  with  the  outer  world  are  about  equally 
sensitive. 

On  the  subjoined  Loudness  Scale  this  minimum  occupies  some 
non-committal  position  between  '  threshold '  0  and  10  db.  It 
must  vary  with  different  ears  and  at  different  times,  and  a  scream 
or  shriek,  of  frequency  between  1000  and  4000,  is  a  hundred  times 
more  audible  than  a  masculine  roar. 

Now  that  more  confidence  is  being  placed  in  absolute  measure- 
ments, some  prefer  to  have  no  argument  about  '  threshold,'  but 
to  speak  definitely  of  '  phons.' 


§420] 


SOUND   TRAVEL 


329 


(in 


Noisy- 


tube 


trains. 
Public 


LOUDNESS    OF   NOISE 

decibels  above  threshold,'  or  '  phons  '  of  absolute  measure.) 
Limit  of  ear's  endurance. 


pU 
generally. 


130. 
120. 
110. 
100. 


90. 


80. 


Aero  engine  10  ft.,  boilermakers  hammering  plate  2  ft. 

Roland's  oliphant  at  Roncesvalles,  heard  10  miles. 
Very  loud  klaxon,  20  ft. 
Roadbreaking  drill,  20  ft. 
Lion  roaring,  20  ft. 

Busy  New  York  traffic.     Motor  horn  80  ft. 

The  Zoo  Parrot  House. 

Man  shouting,  2  ft. 

Loud  thunder. 

Busy  London  traffic. 

Yoiu*  neighbour's  loud  speaker. 

Your  loud  speaker. 

Conversation  at  arm's  length. 

General  sound  of  distant  London  traffic. 


Home. 


Rustling  leaves,  whisper. 

Quiet  garden. 

Mice. 

Threshold  of  audibility. 


Recipe  cum  grano  salis,  English  and  American  estimates  do  not  agree  too 
closely. 

You  see  how  enormous  the  scale  really  is ;  you  expect  to  hear 
little  noises  a  few  millionths  of  ordinary  conversation,  and  to  tolerate 
ten  million  times  its  loudness  before  having  to  cover  your  ears  and 
seek  shelter. 

That  is  the  trouble  with  microphonic  deaf-aids,  §  819 — they 
amplify  perhaps  100  times,  but  the  merely  '  hard  of  hearing  '  really 
wants  an  amplification  of  10,000. 

Noise  is  the  most  all-round  useful  of  warnings,  but  never  let  mere 
noise  impress  you.  Think  of  that  125  ergs  ;  and  reply  '  Two  bels  ' 
to  the  most  wrathful  shout.  Think  of  a  cricket,  or  a  wren  :  I  am 
sure  that  a  nightingale's  just  estimate  of  the  nasty  nothingness 
of  a  motor-bike  exhaust  would  be  a  jolt  to  the  pride  of  its  power- 
worshipping  possessor  ;  the  devastating  roar  of  a  600-h.p.  aeroplane 
engine  and  propeller  absorbs  only  l/50th  h.p. ;  the  loud  noises  of 
Nature,  thunder,  or  the  scarce-noticed  Niagara,  are  of  less  im- 
portance than  the  song  of  a  'skeeter  in  the  stilly  night,  and  no  more 
terrible  than  those  little  shivers  and  rustHngs  wherewith  she  some- 
times stands  our  hair  on  end. 

But  for  all  that,  nothing  is  more  wearying  than  the  incessant 
nagging  reiteration  of  a  senseless  and  useless  noise.  It  batters  at 
your  ears  all  the  time,  it  is  as  much  in  contact  with  you  as  a  hard 
bed,  it  wraps  round  you  like  uncomfortable  clothing,  there  is  no 
turning  away  as  from  an  unpleasant  sight.     Reflecting  walls,  which 


330  SOUND  [§  420 

prevent  noises  going  away,  are  responsible  for  a  great  deal :  think 
how  railway  arches  deafen  you  as  they  roar  past  the  open  window. 
Let  us  glance  at  Architectural  Acoustics,  and  see  how  this  pest  is 
being  controlled. 

§421.  Architectural  Acoustics.  In  the  old  days  only  now 
passing  away,  public  halls  and  concert-rooms,  such  as  it  will  doubtless 
fall  to  your  lot  to  speak  or  sing  in,  used  to  enjoy  reputations  for 
'  bad  acoustics,'  much  as  some  patients  enjoy  bad  health  :  it  seemed 
that  little  could  be  done  to  alleviate  their  troubles.  They  put  a 
broad  reflecting  sounding-board  over  the  pulpit,  to  keep  the  voice 
from  wandering  away  to  the  vaults  above — that  was  common  sense  ; 
they  stretched  wires  from  point  to  point — that  was  senseless  super- 
stition ;  sometimes  they  piled  horses'  skulls  in  the  corners,  or  packed 
them  tight  under  the  floorboards,  between  the  joists.  Mostly  they 
adjured  the  speaker  to  *  throw  his  voice  ' — '  hat  the  legs  of  the 
Hegyptian  lady  standing  hon  the  pedestal,  sir  ;  you  will  be  'eard 
perfectly,  sir ;  thank  you,  Sir ' — a  feat  beyond  our  best  bowlers  ; 
the  fact  being  that  the  practised  speaker,  listening  to  his  own  ears, 
learnt  to  avoid  bad  spots,  and  to  adjust  the  loudness  of  his  voice 
and  his  speed  of  delivery. 

How  noisy  reverberations  really  could  be  smothered  needed 
little  observation  :  it  jumps  at  you  every  spring-cleaning,  when 
the  curtains  and  the  pictures  come  down  and  the  carpets  come  up, 
and  the  '  hollow  house  resounds  with  many  a  groan,'  as  we  used 
to  translate  it.  Yet '  hangings  '  on  the  scale  demanded  by  halls  were 
condemned  as  inartistic,  and  of  course  they  cost  a  lot  and  get 
dusty  and  faded ;  and  the  trouble  went  on  until  Sabine  began  his 
experiments,  with  this  century. 

Let  us  avoid  generalities,  and  take  a  convenient  concrete  instance. 
In  a  particular  room  it  was  observed  that  the  murmuring  reverbera- 
tions of  a  sharp  '  bark  '  remained  perceptible  for  3 J  sec,  by  which 
time  they  blended  with  a  noise  in  the  distance  estimated  at  a 
thousandth  the  strength  of  the  bark. 

The  sound  dies  down  in  the  general  logarithmic  curve  of  decay, 
which  we  met  in  §  231,  always  to  the  same  fraction  of  itself  in  the 
same  interval ;  in  this  case  to  l/7th  in  the  first  second,  to  l/49th 
by  the  end  of  the  next,  to  1/7^  or  1 /350th  at  the  third,  and  to 
1/(350  X  70-5)  =  1/1000  in  the  3-5  sec.  Fig.  148  shows  this  curve 
falling  from  7  to  1  and  l/7th  in  the  two  seconds. 

This  room  is  55  ft.  long,  so  that  a  sound-wave  started  at  one  end, 
travels  down,  is  reflected  by  the  end  wall,  and  comes  back,  110  ft. 
in  a  tenth  of  a  second.  It  is  reflected  at  the  near  end  wall,  and 
repeats  its  journey  again  and  again.  In  bare  rooms  with  soHd 
painted  walls  the  microphone  has  recorded  the  passing  impulse 
more  than  100  times ;  it  is  just  like  the  surging  to  and  fro  with 
which  you  occasionally  succeed  in  slopping  water  over  the  end  of 
the  bath. 

In  the  figure,  that  means  that  to  the  left  of  the  thin  dotted  line 


§421] 


SOUND   TRAVEL 


331 


at  0-1  sec.  the  sound  of  a  voice  has  travelled  down  the  room  and 
returned.  This  echo  falling  inside  the  tenth  of  a  second  is  not 
objectionable — and  most  of  it  came  back  sooner  than  this,  from 
side  walls  and  ceiling — it  is  the  difference  between  indoor  and  out- 
door speaking ;  the  speaker's  voice  comes  back  to  his  ears  and 
reassures  him  what  a  fine  fellow  he  is,  and  it  doubles  the  loudness 
to  the  hearer  without  causing  any  dragging  confusion.  But  those 
that  follow,  ajter  0-1  sec,  are  a  nasty  noisy  nuisance,  and  merely 
get  in  the  way  of  the  next  syllable. 

If  you  fill  up  Fig.  148,  under  the  curve,  with  vertical  lines  l/20th 
sec.  apart,  they  represent  the  loudnesses  of  the  sound  in  its  successive 


or  • 

K 

6 

B1 

0  / 

Z   i 

ii    1 

6  r 

'Z 

S 

E 

C 

o 

N 

0 

S 

7 

$ 

\ 

\ 

0) 

^ 

</> 

\\ 

y 

if 

i  \ 

z 

1    \ 

o 

3 

1 

V 

D 

\ 

o 

i 

\ 

J 

1 

\ 

1 

\ 

^ 

^ 

iK 

"^ 

^ 

-^ 

f 

■ 

z 

s 

E 

c 

o 

N 

0 

s 

A 

A 

A 

A 

A 

A 

A 

A 

/^ 

f^ 

f 

A 

f" 

f 

A 

I 

r 

/ 

A 

f 

f 

< 

1 

\ 

h 

/ 
1 

\ 

\ 

1 
1 

\ 

1 

\ 

1 
1 

\ 

1 
1 

\ 

f 

\ 
\ 

1 
1 

\ 

\ 

1 

1] 

^h 

s^h 

/ 

ij\ 

i\ 

K 

/ 

h 

^ 

I^ 

/  > 

1 

p-N.u;^ 

^ 

^ 

Fig.  148. 


Fig.  149. 


journeys  down  and  up  the  room,  and  you  will  find  that  each  is 
10%  shorter  than  the  one  before  it.  Measuring  the  first  two,  to  the 
left  of  the  dotted  line,  they  add  up  to  13,  and  that  is  the  useful 
sound ;  measuring  and  adding  up  the  lengths  of  all  the  rest,  they 
amount  to  about  4-7  times  as  much  {i.e.  the  area  to  the  right  is 
4-7  times  that  to  the  left  of  the  dotted  line),  and  they  are  the  useless 
roar  with  which  succeeding  syllables  have  to  contend.  When  you 
speak  in  the  room,  five  ghostly  interrupters  of  equal  stren^h 
chorus  your  every  word,  at  dragging  intervals  :  five  leeches  hang 
on  your  lips. 

Fig.  149  shows  the  effect  better.  Ordinary  speech,  excluding 
stops  for  breath,  emits  five  syllables  per  second ;  they  are  the 
succession  of  impulses,  height  2,  each  rising  to  full  value  in  about 
1/7  sec,  and  then  dying  away  along  its  bit  of  the  log  curve  of  Fig. 
148.     The  second  syllable  piles  on  top  of  the  decaying  first,  the 


332 


SOUND 


[§421 


Fig.  150. 


third  on  top  of  both,  and  so  on ;  so  that  at  any  moment  the  total 
volume  of  sound  in  the  room,  the  upper  graph,  is  got  by  adding 
up  the  heights  of  all  the  individual  syllable  curves  on  that  vertical 
line  (check  this),  and  after  each  addition  it  decays  along  the  bit 
of  the  log  curve  at  that  loudness  level. 

You  see  the  effect ;  the  message  you  seek  to  convey  is  only  a 
succession  of  saw-teeth  on  the  top  of  a  huge  stack  of  reverberant 
blah,  and  taking  the  ideal  as  Fig.  150,  an  intensity  variation  of 

2:1,  you  will  see  that  this  occupies 
only  the  space  between  5  and  6,  and 
five  times  as  much  rubbish  lies  be- 
neath it. 

It  is  no  use  talking  louder,  the 
teeth  get  twice  as  high,  but  so  does 
the  heap  of  waste,  and  the  ratio 
remains  unaltered.  The  only  thing 
to  do  is  to  talk  half  as  fast,  and  then 
you  get  the  dotted  jag,  as  you  can 
easily  check,  by  leaving  out  every 
alternate  impulse.  In  that  way,  and 
that  way  only,  you  can  he  heard 
(apart  from  mere  whispers  into  the 
ear),  and  that  is  the  way  you  must 
adopt  in  difficult  conditions  any- 
where :  speak  v-e-r-y  s-1-o-w-l-y  until  you  are  weary,  and  your 
audience  asleep. 

Look  now  at  the  left-hand  curve  of  Fig.  148,  and  you  see  it  dies 
down  to  half  in  l/20th  sec,  1/4  in  1/lOth  (dotted  line)  to  a  six- 
teenth by  the  first  fifth  (solid  line),  and  after  that  doesn't  matter, 
for  the  next  syllable  has  the  air  to  itself  all  but  a  sixteenth  part. 
The  result  is  Fig.  150  (on  double  the  scale),  which  can  fairly  be 
described  as  ideal. 

What  can  be  done  to  improve  Fig.  149  ?  The  reverberation 
must  be  absorbed  somewhere,  and  the  only  where  is  in  the  surfaces  ; 
it  is  they  that  must  put  the  brake  on  the  moving  air  particles  and 
quieten  them  by  friction.  All  absorption  of  energy  means  F  X  5, 
§61,  force  x  distance.  A  hard  unyielding  surface  is  merely  a 
nodal  surface,  Fig.  137,  and  air  particles  next  to  it  cannot  move ; 
consequently  it  can  do  almost  nothing  to  absorb  sound  energy. 
The  moving  molecules  farther  out  must  be  reached  out  to — the  first 
wavy  line  of  Fig.  137,  and  that  means  a  thick  layer  of  material,  of 
thickness  actually  comparable  with  the  length  of  a  sound-wave, 
say  a  hundredth  of  it,  into  which  the  molecules  may  be  squeezed 
by  the  wave-motion,  and  from  which  they  can  escape  only  with  friction 
— a  shock-absorber,  exactly  comparable,  in  miniature,  with  the  hy- 
draulic brake  of  a  field  gun,  with  its  long  run  bach.  To  expect  a 
hard  surface  to  absorb  the  sound  is  to  expect  a  9-in.  brick  wall  to 
take  the  kick  of  such  a  gun,  butted  back  against  it. 

In  exactly  the  same  way  a  shingle  beach  gradually  breaks  and  soaks 


§421]  SOUND   TRAVEL  333 

in  the  energy  of  the  waves,  and  is  the  best  of  all  defences  to  the 
cHfFs  behind  it. 

A  cellular  material  exactly  like  honeycomb,  made  of  asbestos, 
lines  the  walls  of  some  '  talkie  '  studios,  and  is  extremely  effective  in 
quietening  noises  unwanted  on  the  film,  but  is  unsightly  and  ex- 
pensive, and  all  sorts  of  thick  porous  materials — most  of  these,  in 
fact,  used  for  heat  insulation — serve  very  well  as  sound-suppressors. 
'  Cabot  quilts  '  stuffed  with  Newfoundland  eel-grass,  thick  hair- 
felt  faced  with  cheese-cloth,  and  quite  indistinguishable  from 
plaster  ceiling — for  the  ceiling  is  often  the  most  convenient  place, 
large,  unwanted,  and  keeping  clean — or  simply  a  thick  soft  asbestos 
plaster,  price  and  all  considered,  the  best  of  all. 

The  Unit  of  Absorption  of  Sound  is  a  square  foot  of  open  window, 
through  which  it  escapes,  and  nothing  comes  back — a  unit  apt  to 
take  on  a  minus  value  in  city  streets.  Some  acoustic  absobption 
COEFFICIENTS  of  various  materials,  for  ordinary  vocal  frequencies, 
are  as  follows  : 

Open  window 

Brick  wall 003 

Solid  waxed  floor     . 
Painted  plaster  on  brick 
Window  glass     . 
Matchboard,  lino,  etc.  . 
Carpets  and  rugs 
Cretonne  curtain 

We  reckoned,  however,  that  the  echo  returned  10%  diminished 
from  the  end  wall,  and  here  it  is  only  1-5%  ?  But  going  down  the 
length  of  the  room  the  large  waves  of  sound  of  course  spread  out, 
like  Fig.  131,  and  scrape  along  the  side  walls,  floor,  and  ceiling, 
and  meet  various  obstacles,  just  as  the  wash  of  a  river  steamer 
slops  along  the  banks,  upsetting  the  tea-things  in  your  punt,  nibbUng 
out  weak  places  in  the  bank,  and  wasting  its  energy  in  miscellaneous 
mischief  ;  it  is  all  this  that  adds  up  to  the  10%  loss. 

Hence  it  does  not  matter  much  where  the  absorbent  material  is 
put,  it  is  bound  to  come  into  use.  But  it  does  matter  how  much  ; 
and  for  this  formulae  have  been  worked  out,  based  on  wide  experience, 
backed  by  a  modicum  of  theory. 

For  instance,  one  calculates  first  the  desirable  reverberation 
period  in  seconds,  T  =  0-65  +  0-125  ^  (volume  of  room  in  cu.  ft.) 
for  speech,  or  0-75  and  0-175  for  music.  For  the  room  under 
discussion,  these  are  0-955  sec.  and  1-175  sec,  the  longer  for  music. 

Then  the  Absorbing  Power  which  will  bring  this  about  is 
Volume  of  room  in  cu.  ft.  -H  20T 
which  for  this  room  gives  14,500  -^  (20  X  1-175)  =  760  units  for 
speech  ;   or  620  for  music. 

~  Its  total  area  of  waxed  floor,  hard  plaster  walls  and  ceiling,  all 
on  soHd  backing,  and  windows,  is  5000  sq.  ft. ;  crediting  this 
'  hospital  finish  '  with  an  average  absorbency  of  2%,  gives  100  units, 


1-00 

Inch-thick  hair  felt      .      . 

0-6 

003 

Akoustolith  tile 

0-4 

002 

Pax  felt,  dumboard,  etc. 

0-4— 

0015 

0025 

Audience,  per  person  . 

.     5 

01 

Wooden  chair 

.     017 

015— 0-3 

Upholstered  chair 

.      3 

015 

Cushion        .... 

.     2 

334  SOUND  [§  421 

and  plain  painted  benches  contribute  another  40,  on  formula ; 
a  total  of  140  actually  existent,  leaving  620,  or  4-5  times  as  much, 
to  be  provided — ^near  enough  to  the  4-7  we  found  in  Figs.  148,  149. 
To  do  this,  plaster  5/6ths  of  the  ceiling  with  asbestos  pax-felt,  and 
the  room  is  transformed. 

This  redoubtable  room,  the  effort  of  an  honoured  architect 
fresh  from  triumphs  with  the  noisiest  of  quayside  station  waiting- 
halls,  acoustically  contemporaneous  as  it  is  with  the  Chislehurst 
Chalk  Caves,  or  the  chambers  of  the  Great  Pyramid,  might  serve 
well  enough  as  the  recreation-room  of  a  renowned  rugger  team. 
For  be  it  noticed  that  the  hard  non-absorbent  germ-proof  finish, 
otherwise  desirable  in  hospitals,  has  the  very  smallest  absorptive 
power  for  sound,  and  all  the  rattling  and  slamming  in  which  such 
places  seem  to  excel  has  to  roam  around  until  it  can  snuggle  into 
the  beds. 

While  speech  in  it  is  but  the  ticking  of  a  wrist-watch  in  the  swish 
of  the  sea,  its  constant  ba-aa-ing,  as  of  a  lamb-fold  at  weaning  time, 
may  be  no  detriment  to  jubilation  and  j azz .  But  that  anyone  should 
be  seized  of  the  hilarious  hallucination  that  it  is  suitable  for  the 
instruction  of  youth,  in  sober  matters  of  natural  philosophy,  is  an 
attitude  of  mind  paralleled  only  by  that  of  the  tank-spoiler  of  §  172 — 
here,  fortunately,  there  is  no  need  to  go  overseas,  only  to  the  near 
eastern  end  of  a  minor  metropolitan  borough — and  forcibly  reminds 
one  of  the  ancient  Spanish  proverb  :  '  To  the  man  of  small  under- 
standing, it  is  easier  to  bear  with  evil  than  with  good.' 

And  it  has  served  us  philosophically,  it  has  been  our  Shocking 
Example. 

On  the  other  hand,  a  large  audience  is  so  absorptive  that  often 
the  difficulty  is  to  get  the  sound-wave  to  go  the  whole  way  :  that 
nowadays  is  a  question  of  '  loud-speakers.'  A  special  case  crops 
up  with  large  crowded  talkie-halls  :  if  the  loud-speaker  is  behind 
the  screen,  the  front  seats  are  deafened,  and  the  distant  back  seats 
hear  everything  quite  perceptibly  later  than  they  see  it.  What 
is  done  is  to  set  up  several  loud-speakers  down  the  length  of  the 
building,  and  to  pad  everything  out  with  the  maximum  of  absorbency, 
so  that  distant  loud-speakers  are  inaudible. 

Overdone,  as  by  thick  oriental  carpets,  plush  seats,  heavy  cur- 
tains, etc.,  the  stuffy  silence  becomes  that  of  the  dusty  tomb, 
words  drop  dead  off  your  dry  lips,  you  rub  your  ears  and  look  round 
at  a  loss ;  starting  with  awed  whispers,  you  finish  by  shouting,  in 
a  vain  effort  to  hear  some  sort  of  response,  sonie  sign  of  life — and 
you  rush  off  for  a  drink,  something  with  the  tinkle  of  ice  in  it,  or 
even  a  teaspoon. 

To  sound-insulate  room  from  room,  partitions  are  made  of  alternate 
thick  absorbents,  such  as  2,  in.  of  eel-grass,  or  slag-wool,  and  hard 
reflecting  sheets  of  uraHte,  etc.  A  mere  chink,  such  as  a  door 
left  open  a  thousandth  part,  can  pass  an  immerffee  amount  of  sound. 


SOUND   TRAVEL  336 


EXAM   QUESTIONS,   CHAPTER   XXVII 

The  questions  cover  the  first  part  of  the  chapter,  and  of  course  involve 
reference  to  the  preceding  one. 

From  §  420  on  is  a  brief  notice  of  a  science  very  much  in  the  making,  but 
one  which  no  medical  man,  unless  in  the  depths  of  the  country,  can  now 
afford  to  ignore. 

1.  Describe  experiments  to  show  that  soimds  are  transmitted  by  means 
of  the  air.  Explain  the  mechanism  of  the  process.  In  what  way  is  it  affected 
when  both  source  and  receiver  axe  under  water  ?     (  X  4) 

2.  What  is  the  physical  nature  of  Soiuid -waves  ?  Give  some  account  of 
experiments  in  illustration.  Explain  the  change  of  direction  of  soimd  travel- 
ling against  the  wind.     (  X  2) 

3.  Explain  the  motion  of  air  transmitting  a  musical  note.  Make  diagrams 
showing  the  variation  of  displacement  (1)  with  position,  at  a  given  instant, 
(2)  with  time,  at  a  given  point. 

4.  Describe  the  chief  characteristics  of  wave  motion. 

What  differences  are  there  between  soimd-waves  and  light-waves  ?  Why 
can  sound  turn  comers  ?     (  X  2) 

5.  Distinguish  between  waves  of  sound,  waves  on  a  string,  and  waves  of 
light.  Show  how  to  calculate  the  speed  of  travel  of  one  of  them  or  how  to 
find  it  experimentally.     (  X  3) 

6.  Explain  what  happens  if  a  gas  is  expanded  more  quickly  than  in  a 
Boyle's  Law  tube. 

7.  How  is  sound  propagated  in  a  gas,  and  on  what  physical  properties  of 
the  medium  does  the  velocity  depend  ? 

Why  did  error  arise  when  gas  pressiu-e  was  taken  as  a  measure  of  one  of 
these,  without  further  consideration  ?  What  was  the  nature  of  the  correction  T 
( X  3) 

8.  Describe  how  the  velocity  of  soiuid  in  air  may  be  measured  accurately. 
Assuming  the  pulse  beats  75,  show  that  the  distance  in  miles  of  a  lightning 
flash  =  pulse  beats  between  flash  and  thunder,  divided  by  6. 

9.  How  has  the  velocity  of  sound  been  determined  ? 

A  man  stands  at  100  ft.  from  a  series  of  steps  which  are  2  ft.  wide ;  explain 
what  he  observes  after  clapping  his  hands  once. 

10.  How  is  the  velocity  of  sound  in  air  affected  by  change  of  temperature  ? 
Given  that  at  0°  it  is  1090  ft.  per  sec,  calculate  it  in  the  air  of  a  tube  railway, 

20°  C.  and  77  cm.  of  mercury. 

1 1 .  Calculate  the  maximum  speed  of  an  impulse  along  a  train  brake-pipe 
which  contains  air  at  6  atmos.  and  10°  C. 

12.  Why  and  how  is  the  speed  of  sound  affected  by  temperature  ?  A  watch 
was  set  to  a  signal  gim  5  km.  away;  find  the  error  to  the  nearest  0-01  sec., 
t  =  15°  C. 

13.  How  does  V  depend  on  t  and  p  ?  Air  being  4/5  nitrogen,  calculate  V 
in  oxygen .     At  what  temperature  would  it  be  equal  to  that  in  air  at  0°  C.  ? 

14.  If  the  speed  in  hydrogen  at  N.T.P.  is  1270  m/s,  calculate  it  at  13*  C. 
and  2  atmos. 

15.  How  is  the  pitch  of  a  note  affected  by  relative  motion  of  source  and 
observer  ?     Give  an  optical  analogue. 


336  SOUND 

16.  How  is  the  pitch  of  a  note  heard  by  a  stationary  observer  affected  by 
the  motion  of  the  source  ? 

An  engine  runs  through  a  station  blowing  a  whistle  of  true  frequency 
600.  The  note  heard  by  an  observer  on  the  platform  has  a  frequency  of  650 ; 
what  is  the  speed  of  the  train,  and  what  is  heard  as,  and  after,  it  passes  ? 
(V  =  1100)  (X  3) 

17.  Explain  the  apparent  change  of  pitch  of  a  moving  source  of  sound. 

A  locomotive  sounding  a  whistle  of  frequency  800  approaches  from  400 
to  300  ft.  distance  from  an  observer  in  2  sec.  Calculate  the  frequency  of  the 
note  he  hears,  and  the  fractional  increase  in  the  intensity  of  the  sound. 

18.  A  car  drives  straight  towards  a  cliff  at  30  m.p.h.  and  the  driver  sounds 
a  horn  of  frequency  500.  What  frequency  echo  does  he  hear,  and  what  notes 
are  heard  by  pedestrians  (o)  in  front  of  the  car,  (6)  behind  it  ?     (  X  2) 


CHAPTER  XXVm 
PITCH,  AND  STRINGS 

PITCH 


The  Frequency  or  number  of  complete  vibrations  per  second, 
of  a  musical  note,  is  the  physical  measure  of  its  Pitch. 

§431.  Comparison    of    near    frequencies.     Beats.     It   was   ex- 

plained  in  §  383  (6)  that  a  compound  harmonic  motion  resulting 
from  two  S.H.M.'s  not  very  unlike  was  characterized  by  its  variable 
ampUtude.  When  in  phase,  both  pull  together,  and  their  amplitudes 
are  added ;  presently  one  gains  half  a  period  (180°  of  phase),  they 
pull  opposite  ways  and  their  amplitudes  are  subtracted,  another 
half-period  gain  brings  them  into  phase  once  more,  and  so  on.  The 
tides  were  given  as  an  instance  ;  every  fortnight  the  solar  tide  gains 
one  period  (12  hr.)  on  the  lunar,  and  there  is  one  set  of  spring  and 
one  of  neap  tides. 

Consider  a  particle  of  air  near  the  ear,  affected  by  the  joint 
action  of  the  air  waves  coming  from  two  sources  of  sound,  not  quite 
of  the  same  pitch.  Both  combine  in  driving  it  in  and  out  of  the 
ear,  and  its  amplitude  of  motion  increases  and  decreases  oncSy 
every  time  one  source  gains  a  whole  vibration  on  the  other.  Loudness 
being  proportional  to  square  of  amplitude,  this  means  that  the  sound 
swells,  and  softens,  once,  or  one  Beat  is  heard. 

First  acquaintance  with  Beats  is  best  made  by  sounding  together 
two  near  notes  on  a  harmonium ;  they  are  heard  as  a  tremolo 
varying  from  2  or  3  per  second  to  a  rapid  burr-r.  Slow  beats  fill 
the  air  when  a  twin-engined  aeroplane  flies  over,  their  frequency 
the  difference  in  revs,  of  the  two  engines. 

Beats  enable  the  sustained  notes  of  any  musical  instruments  to 
be  tuned  together  to  within  one  vibration  per  second.  Count- 
ing them  gives  the  arithmetic  difference  between  the  number  of 
vibrations  during  the  time  of  listening  (and  reducing  to  1  sec., 
between  their  frequencies).  If  one  frequency  is  actually  known, 
adding  or  subtracting  the  rate  of  beating  gives  the  other.  The 
faster  vibrator  can  be  identified,  because  gradually  loading  it  with 
specks  of  wax  (or  slowing  it  in  any  appropriate  manner)  slows  the 
beats  down  to  zero  when  the  notes  come  exactly  into  tune,  and  then 
increases  them,  as  the  now  overloaded  spring  gets  farther  and  farther 
away  below  the  other  in  pitch.  Per  contra,  loading  the  slower 
vibrator  increases  the  rate  of  beating  straightaway. 

337 


338 


SOUND 


[§432 


§  432.  Comparison  of  frequencies  nearly  in  simple  ratio  1:1, 
1 :  2, 1  :  3,  etc.    Lissajou's  figures. 

In  §  384  the  curves  obtained  by  combining  S.H.M.'s  at  right 
angles  were  described.  When  the  swinging  harmonograph  pen- 
dulums are  replaced  by  vibrating  tuning-forks,  the  pen  and  link- 
work  have  to  be  superseded  by  an  inertia-less  ray  of  light.  To 
compare  two  forks,  one  prong-end  on  each  is  ground  flat  and 
polished,  one  fork  is  fixed  vertical  and  the  other  horizontal ;  and 
a  ray  from  a  pinhole  with  a  lens  in  front  is  reflected,  from  both 
in  succession,  to  a  focus  on  a  screen.  Fig.  151,  left.  The  vertical 
fork  sounding  alone  draws  the  spot  out  into  a  vertical  line  of  light, 
which  the  horizontal  fork  converts  into  a  figure  like  those  in  §  384. 
If  the  ratio  is  not  quite  exact,  the  figure  slowly  changes  shape, 


Fig.  151. 


through  one  complete  cycle  of  phases  for  each  whole  vibration 
gained,  as  in  Beats.  Large  standard  forks,  electrically  maintained. 
Fig.  155,  such  as  are  nowadays  used  to  control  the  frequency  of 
alternating-current  supplies,  and  radio  stations,  with  extreme 
accuracy,  can  be  tuned  in  this  way  within  one  '  beat '  in  a  day. 

A  small  lens  mounted  on  a  large  fork  constitutes  the  Vibration 
Microscope.  Looking  through  it  at  a  silvered  speck  on  a  string, 
say,  little  Lissajou's  figures  appear,  and  enable  vibrations  to  be 
studied,  Fig.  151,  right. 

§  433.  Measurement  of  frequency  of  vibration.  The  number  of 
complete  vibrations  per  second  of  a  vibrating  body,  or  the  pitch  of 
a  rapid  vibrator,  can  be  found  directly  by  chronographic  methods, 
or  stroboscopically.  Both  are  purely  mechanical;  for  acoustic 
methods  dependent  on  calculation  see  §§  438,  442,  etc. 

Chronograph  methods.  The  simplest  way  is  that  described  in 
§  47,  Fig.  4.  Drop  a  smoked  plate  in  front  of  a  pointer  attached 
to  the  vibrating  body  ;  then,  knowing  g,  n  is  calculated. 

More  elaborately,  m  Fig.  152  (fork  being  tested),  the  plate  is 
replaced  by  a  rotating  smoked  paper  drum,  and  close  beside  the 


§433] 


PITCH,   AND   STRINGS 


339 


marking  point  is  another,  displaced  electrically  every  second  by 
a  pendulum  which  touches  a  globule  of  mercury  at  the  middle 
of  every  swing  and  completes  an  electric  circuit. 

To  get  the  frequency-ratio  of  two  vibrators  let  them  mark  side 
])y  side  on  a  plate  or  drum  moved  at  any  speed,  e.g.  by  hand, 

A  disadvantage  is  that  attached  pointers  load  and  slow  the 
vibrations. 

A  sung  note  can  be  received  in  a  gramophone  sound-box,  and 
the  needle  arranged  to  mark  the  smoked  drum.  Phonograph 
and  gramophone  grew  out  of  this,  in  fact :  by  way  of,  first,  a  soft 
tinfoil  coating,  soon  superseded  by  hard  '  wax  '  (wherewith  the 
phonograph  survives  as  the  Dictaphone),  and  ultimately  this  wax 
mould  has  been  reproduced  in  the  familiar  hard  discs. 


Fig.  152. 


Fig.  153. 


Fig.  154. 


Fig.  155. 


The  stroboscope.  Attached  near  the  ends  of  a  large  tuning-fork, 
Fig.  155,  are  two  little  overlapping  plates,  with  a  slit  in  each,  so 
that  both  can  be  seen  through  only  when  in  mid-swing.  In  each 
whole  vibration,  therefore,  two  brief  gUmpses  can  be  obtained  of 
any  moving  object,  such  as  a  sounding  string,  or  a  revolving  wheel. 
Watching  the  wheel,  suppose  that  between  gUmpses  each  spoke 
moves  forward  exactly  into  the  place  of  its  predecessor  (so  that 
an  eight-spoke  wheel  is  revolving  once  in  eight  glimpses),  then, 
all  the  spokes  being  alike,  it  always  looks  alike — there  is  never  any 
sign  that  it  has  moved  at  all.  If  revolving  rather  slower  than  this 
'  synchronous  '  speed,  the  spokes  would  be  always  gUmpsed  just 
before  they  had  arrived  the  full  eighth,  and  the  wheel  would  appear 
to  be  rotating  backwards,  fast  or  slow  according  as  the  defect  in 
speed  was  substantial  or  sUght.  If  revolving  above  the  synchronous 
speed,  the  wheel  appears  to  rotate  forwards,  slow  or  fast. 


340  SOUND  [§  433 

You  see  the  whole  performance  when  a  car  starts  on  the  cinema 
screen.  Out  of  a  blur  appears  a  wheel  running  backwards  :  detected 
in  this  ridiculous  trick,  it  slows,  stops,  and  runs  faster  and  faster 
forwards.  Watch,  and  at  double  the  speed  the  whole  performance 
is  gone  through  again,  each  spoke  moving  forward  now  two  spaces 
between  successive  exposures  of  the  camera. 

A  standard  fork  fitted  with  these  plates  and  maintained  in  motion 
precisely  like  any  electric  buzzer,  Fig.  370,  is  a  very  exact  '  strobo- 
scopic  '  means  of  examining  and  controlling  the  speed  of  machinery. 
By  a  minute  adjustment  of  the  throttle,  any  particular  action  in 
an  engine,  running  at  thousands  of  revolutions,  can  be  watched, 
forwards  or  backwards,  at  leisurely  slowness  ;  and  if  the  axle  carries 
a  disc  with  various  rings  of  dots,  a  concentric  square,  pentagon, 
hexagon,  etc.,  it  can  be  held  steadily  to  a  variety  of  known  speeds. 
Anything  which  keeps  step  with  the  fork  is  seen  at  rest. 

A  sounding- string  can  have  its  length  or  tension  altered  until 
nearly  in  unison  with  the  stroboscope  fork,  and  its  fluctuating 
movement  can  be  watched.  Every  complete  fluctuation  or  '  cycle  ' 
of  movement  means  that  one  motion  has  gained  one  complete 
period  over  the  other,  and  one  heat  will  he  heard. 

§  434.  Fig.  153  is  a  cross-section  of  a  Syren,  such  as  is  employed 
on  t.b.d.'s  and  yachts,  and  for  fire-alarms,  etc.  Steam  or  com- 
pressed air  admitted  to  the  outer  shell  blows  the  inner  drum  round 
at  a  great  pace,  and  a  violent  combined  puff  issues  every  time  all 
the  slits  coincide,  as  shown,  into  the  little  end  of  a  great  conical 
*  trumpet,'  whereby  its  expanding  energy  is  spread  to  the  outer 
air.  As  a  noise-producer  it  was  for  long  unequalled,  but  its  ululating 
howls  and  whoopee  shrieks,  in  those  situations,  are  plain  evidence 
why,  as  a  measurer  of  musical  pitch  in  physical  laboratories,  it  has 
been  relegated  to  a  back  shelf  :  its  speed  is  too  variable  and  difficult 
of  control,  although,  in  small  form,  fitted  with  a  revolution  counter, 
it  was  formerly  much  in  vogue  for  that  purpose. 

Our  Alderney  neighbour  is  a  little  trying  at  times,  as  with  28  lb. 
wind  pressure  whirring  his  6-in.  syren  at  800  r.p.m.,  he  breathes 
out  the  continuous  effort  of  a  25-h.p.  oil-engine  in  brief  200-h.p. 
sighs  into  the  mist,  but  the  birds  of  the  Bass  Rock  never  lifted  a 
feather  when  the  7-ft.-wide  trumpet-mouths  of  a  similar  fog-signal 
were  planted  right  in  their  midst. 

The  Syren's  younger  rival  is  the  Diaphone,  fitted  on  motor-ships 
and  cruisers.  Essentially  it  is  a  syren  which  reciprocates  instead 
of  rotating.  Fig.  154,  simplified  from  a  drawing  kindly  supplied 
to  me  by  Messrs.  Chance,  the  well-known  makers  of  lighthouse 
equipment,  is  a  section  of  their  6-in.  G  pattern,  supplied  as  having 
a  range  of  5  miles,  though  it  has  been  heard  at  35.  The  diagram 
shows  the  encircling  air-casing  which  was  omitted  from  Fig.  153. 
The  moving  part  is  a  light  gun-metal  casting  which  can  perhaps 
be  best  described  as  a  low  top-hat,  but  open  at  the  top  and  closed 


§436]  PITCH,   AND   STRINGS  341 

at  the  bottom.  The  broad  brim  acts  as  piston  in  the  very  short 
cylinder  on  the  left ;  and  when  supplied  with  pressure-air  by  the 
IJ-in.  pipe,  at  the  top,  it  immediately  oscillates  right  and  left  with 
a  frequency  of  90  per  second,  and  a  stroke  of  about  l/5th  in.  : 
the  inlet  and  exhaust  ports  for  this  engine  action  are  not  shown. 

The  cyHndrical  part  of  the  hat  is  cut  completely  into  rings  by 
ten  circumferential  saw-cuts,  so  that  it  is  held  together  only  by 
eight  ribs  (five  shown),  and  at  mid-stroke  these  narrow  cuts  pass 
opposite  ten  similar  saw-cuts  in  the  cylinder,  so  that  180  times  per 
second  the  35-lb.  pressure  air  blows  freely  through  all  of  them, 
at  23  cu.  ft. /sec,  and  out  through  the  yard-long  iron  trumpet  towards 
the  right,  about  16  h.p.  providing  4  sec.  of  blast  per  minute. 

Whereas  a  syren  takes  time  to  run  up  to  its  high  speed,  this 
diaphone  piston  jumps  into  its  stride  so  quickly  that  it  can  be  used 
for  Morse,  and  its  note  is  a  perfectly  steady  180  (a  rather  sharp  F), 
a  square-shouldered  raucous  note  like  H,  Fig.  120,  more  insistent 
and  penetrating  than  a  sine-wave,  until  at  the  end,  by  the  cam- 
mechanism  cutting  off  the  driving  air  before  the  4J-in.  sounding 
supply,  it  slows  down  in  a  characteristic  grunt.  This  is  the  most 
deadly  of  noise-machines  known,  and  at  Trinity  House  they  assure 
me  that  the  crop  of  complaints  from  the  whole  neighbourhood 
when  one  is  installed  in  this  country  is  never-failing,  and  always 
affords  them  complete  satisfaction. 


STRINGS 

§  435.  Hubal,  Jubal,  and  Tubal  Cain  were  the  Makers  of  Music. 
The  one  winded  the  ox-horn  at  his  lips,  his  brother  strung  its  sinews 
dowTi  its  skull,  and  the  anvil  of  Tubal  rang  to  the  blows  of  the 
mighty  smith  :  behold  them  stand  in  the  square  at  Copenhagen, 
the  first  jazz  band  in  bronze. 

Our  sinews  shall  be  Strings  of  gut  or  thin  wire,  supposedly  per- 
fectly flexible,  uniform,  and  stretched  with  a  force  quite  unaffected 
by  their  vibration.  For  better  music  in  the  bass  the  strings  are 
loaded  by  a  wire  wrapping  which  does  not  spoil  their  flexibility. 
Thick  wires  are  very  unmusical,  gut  stretches  a  great  deal ;  thin 
wires  are  of  most  use  in  the  laboratory. 

As  everyone  knows,  their  musical  vibrations  are  transverse; 
whether  in  one  plane,  or  like  a  skipping-rope,  does  not  matter  in  the 
least  (cf.  pendulum).     They  can  be  studied  visually  thus  : 

§436.  Melde's  experiment.  To  a  strongly  vibrating  prong 
is  attached  a  long  horizontal  thread  of  thick  white  crochet  cotton, 
stretched  over  a  pulley  at  the  far  end  by  50  gm.  or  more.  The 
transverse  waves  sent  running  along  the  string  are  reflected  at 
the  pulley,  and  the  two  equal  wave-trains  running  opposite  ways 
set  up  a  stationary  wave-motion,  dividing  the  string  into  a 
succession  of  nodes  and  loops,  as  in  Figs.  135  and  136.    At  first 


342 


SOUND 


[§436 


the  motion  is  unsteady  and  dodges  about,  but  after  careful 
adjustment  of  the  length,  shows  well-defined  segments  and  steady- 
nodes,  becomes  more  ample  (resonance),  and  then  the  average 
length  from  Node  to  Node  =  Half-length  of  Running  Wave. 

The  ends  of  any  string  must,  of  course,  be  very  nearly  fixed 
nodal  points. 

Now  gradually  increase  the  pull  on  the  string,  and  after  an 
interval  of  unsteady  quivering  it  will  settle  down  to  steady 
vibration  with  one  less  segment  (B,  Fig.  156,  weight  increased 
about  one-half).  Putting  more  weights  on  the  pan  causes  this  to 
be  repeated,  segments  disappearing  one  by  one. 


Measurements  of  lengths  and  weights  will  show  that 

Length  of  segment  varies  as  square  root  of  stretching  pull,  e.g.  to 
get  segments  of  double  length  the  pull  is  quadrupled. 

Now  loosely  twist  four  threads  together  so  as  to  get  a  string 
4  times  as  massive,  and  hang  on  the  same  weight  used  to  pull 
one  thread.     The  segments  shorten  to  half  their  length,  C,  hence 

Length  of  segment  varies  inversely  as  square  root  of  mass  per  cm. 
Hang  on  4  times  the  weight,  and  segments  resume  original  length, 
D  ;  of  course  they  do,  it  is  just  4  original  strings. 

Finally,  if  the  experiment  is  made  with  double  the  frequency, 
as  in  E  (by  turning  the  fork  sideways  so  that  it  gives  its  natural 
frequency,  hitherto  it  has  been  acting  in  '  push-pull '  with  half- 
frequency,  moving  full  to  the  left  every  mid-swing  of  string),  the 
segments  are  doubled  in  number  and  therefore  halved  in  length. 


§438]  PITCH,   AND   STRINGS  343 

Putting  all  this  together 

Length  of  Segment  oc  -  J ^^ 

7i  v  mass  per 


mass  per  cm. 


and  by  weighing  a  length  of  the  thread  to  get  its  mass  in  grammes 
per  centimetre,  and  reckoning  pull  in  dynes,  it  will  be  found  that  the 
constant  of  proportionaUty  is  \,  and 


1     /P 


2nA/^    ^'     ^ 


1    /? 
2l^lm' 


Millions  of  Melde  strings  are  in  full  swing  as  you  read  these  words, 
for  as  the  carriage  of  the  spinning-mule  in  the  cotton-mill  moves 
out  its  5-ft.  stroke  from  the  spindles  (at  7000  r.p.m.),  the  lengthening 
quivering  threads  keep  breaking  into  more  and  more  bellying 
'  ventral  segments.' 

§  437.  Theoretical  deduction  of  this  expression.  By  §  394  the  speed 
at  which  transverse  waves  run  on  a  stretched  string  is  V  =  \/P/m. 
Now  V  =  waves  per  second  x  length  of  running  wave,  §  391 

=  waves  per  second  x  twice  length  from  node  to  node, 
§404. 


y  =  2nl=  V¥J^.         ,.  .  =  i-^^|. 


The  Frequency  of  a  String  is  the  square  root  of  [stretching  pull 
in  dynes  -f-  mass  of  I  cm.  of  string]  divided  by  twice  the  length  of 
one  vibrating  segment. 

§  438.  The  Sonometer,  or  Monochord,  is  the  rudimentary  string 
instrument  you  use  in  the  laboratory  in  studying  these  laws  of 
strings  by  ear  ;  don't  attempt  to  study  this  paragraph  without  it. 
Over  two  sharp-edged  bridges  near  the  ends  of  a  long  sound-box  a 
thin  wire  is  stretched  by  a  spring-balance  or  by  weights.  A  third 
bridge,  a  little  taller,  can  be  placed  under  the  wire  to  partition  off 
any  measured  length  of  it.  A  second  wire  stretched  on  wrest  pins 
is  a  treacherous  nuisance,  and  should  be  got  rid  of.  Here  the  wire  is 
plucked,  and  becomes  the  driver,  and  the  sound-box  is  the  driven 
resonator  which  gives  out  the  sound.  For  a  wire  stretched  between 
two  heavy  weights  on  the  ground  '  cuts  through  '  the  air  almost 
noiselessly.  The  bridges  which  transmit  the  wire's  motion  to  the 
board  are  therefore  not  exactly  (though  quite  nearly  enough  for 
us)  fixed  nodes. 

Only  such  vibrations  can  persist  on  a  string  as  have  the  bridges 
for  nodes.  All  others  die  out  forthwith.  Put  the  movable  bridge 
at  J,  pluck  the  shorter  section,  and  the  longer  vibrates  also,  giving 
its  octave.  Put  the  bridge  at,  say,  I/tc,  and  the  incommensurate 
longer  piece  will  not  take  up  any  motion. 


344  SOUND  [§  438 

Provided  with  two  or  three  forks  of  known  frequencies  the  laws  can 
be  studied  thus  : 

(1)  n  oc  1/^,  frequency  is  inversely  as  length  of  vibrating  segment. 

A.  If  the  wire,  plucked  not  far  from  one  end,  is  touched  lightly 
at  the  middle  point,  this  is  induced  to  become  a  node,  and 
the  fundamental  is  choked  out,  leaving  the  octave  prominent ; 
the  string  vibrating  in  two  halves.  Touched  at  J,  the  twelfth 
sounds  out  (G  in  octave  above  C),  at  J  the  double  octave,  and 
so  on. 

In  all  that  follows  the  string  is  assumed  to  be  vibrating  in  one 
piece,  from  fixed  to  movable  bridge,  and  I  becomes  that  whole 
length. 

B.  Lengths  in  tune  with  the  various  forks  will  be  found  inversely 
proportional  to  their  vibration  numbers  ;  see  §  459. 

Tuning  is  tested  by  slowing  out  of  beats,  or  by  a  little  paper 
jockey  jumping  off  when  the  wire  is  exactly  in  tune  and  is  resounding 
to  the  fork  pressed  on  the  sound-board.  With  a  vertical 
instrument  you  have  to  listen  for  this  picking-up  of  sound,  most 
easily  by  using  a  stick  of  wood  as  '  stethoscope  '  between  board  and 
ear ;  practise  this. 

(2)  n  oc  \/P,  frequency  is  proportional  to  square  root  of  stretching 
force.  Tightening  strings  sharpens  their  pitch.  The  stretching 
weights  necessary  to  tune  the  same  length  of  the  same  wire  to 
different  forks  will  be  found  proportional  to  the  squares  of  their 
vibration  numbers. 

(3)  n  ex;  Vl/m,  frequency  is  inversely  as  square  root  of  mass 
per  centimetre. 

Different  wires  are  stretched  with  the  same  force,  the  same  fork 
is  used  for  all,  and  the  lengths  in  tune  with  it  are  measured.  Then, 
in  the  formula,  n  and  P  being  the  same  throughout,  ?\/w  should  be 
constant,  or  l^m,  e.g.  half  the  length  of  a  4  times  heavier  wire,  should 
give  the  same  note.  Cut  spare  bits  of  the  wires  and  weigh  them  ; 
m  =  weight  -4-  length  ;   so  check  the  relation. 

What  substance  the  string  is  made  of  does  not  matter  in  the  least, 
nor  how  the  mass  is  made  up,  nor  whether  it  is  round  or  square 
or  a  flat  ribbon.  It  is  only  the  mass  per  unit  length  that  counts  : 
compute  this  first,  do  not  potter  with  new  formulae  for  special 
cases. 

For  comparing  the  pitches  of  notes  produced  by  any  instruments 
use  the  monochord.  They  are  inversely  as  the  lengths  of  wire  in  tune 
with  them.  And  a  knowledge  of  P  and  m  will  further  enable  them  to 
be  calculated  absolutely,  using  the  whole  formula. 

§  439.  What  actually  happens  on  the  string  can  be  studied  by 
vibration  microscope,   §  432,  or  spark  photographs  :    the  pull  of 


§440] 


PITCH,   AND   STRINGS 


346 


the  finger,  or  drag  of  the  bow,  or  blow  of  the  hammer,  deflects  the 
strmg  into  two  straight  segments  XA,  YA,  Fig.  157.  Released 
the  hummock  at_A  breaks  into  two,  B,  C,  which  rush  opposite 
ways  at  speed  VP/m  :  these  become  DE,  FG,  HK  (having  now  both 
been  reflected  at  fixed  ends),  and  pass  each  other  again  at  L  (the 
corresponding  point  to  A),  and  so  on. 

The  general  effect  as  seen  by  the  eye  is  shown  below  ;  all  positions 
of  the  kinks  lie  on  the  two  smooth  curves,  and  delude  one  into  the 
entirely  false  idea  that  the  motion,  completely  contained  within  them, 
is  the  smooth  one  of  a  skipping-rope. 


Fig.  157. 

Notice  how  quick  must  be  the  change  from  a  steep  upward  pull 
of  the  wire  CY  to  a  downward  pull  EY  (and  FX  to  XH).  That 
means  that  the  bridge  Y,  and  the  thin  elastic  sound-box  or  belly  of  the 
instrument,  is  being  pulled  up  and  down  by  an  abruptly  changing 
force,  is  being  asked  to  move  more  like  the  top  right-hand  zigzjig 
than  a  smooth  S.H.M. 

It  is  this  broad  surface  that  gives  out  the  sound  to  the  air ;  the  thin 
wire  itself  has  no  grip  at  all  on  a  bulk  of  air  :  it  is  the  old  sound-box 
that  constitutes  the  valuable  violin,  not  the  new  fiddlestring. 

The  sound-board  is  massive,  and  cannot  be  accelerated  rapidly 
over  the  sharp  tops  ;  it  is  also  flexible,  and  ripples  flow  out  over  it 
from  the  bridge,  as  over  water  :  the  resultant  motion  given  out  to 
the  air,  for  you  to  hear,  is  something  like  the  middle  figure  on  the 
right ;  modulated  from  the  sharp  dragging  bow  on  the  string  above  it. 

§  440.  This  is  a  Compound  Harmonic  Curve,  and  it  was  explained 
in  §  383  how  such  curves  can  be  analysed  into,  or  built  up  from, 


346  SOUND  [§  440 

series  of  Simple  Harmonic  curves  of  wavelengths  1,  J,  J,  J,  ^,  etc. 
It  is  as  if  one  took  a  length  of  soft  iron  wire  and  bent  it  first  into 
the  smooth  strong  sine  curve  of  the  Fundamental  S.H.M.,  then  went 
over  it  again  twice  as  closely,  giving  it  pinches  with  finger  and  thumb 
corresponding  to  the  octave,  or  '  First  Overtone,'  or  '  Partial,'  or 
'  Harmonic  ' ;  then  smaller  fingers  added  the  modification  of  the 
Second  Overtone,  stout  pliers  put  in  the  bends  of  the  Third  Overtone, 
smaller  ones  the  diminutive  alteration  due  to  the  fourth,  and  so  on. 
The  curve  has  been  copied  below,  and  you  see  how  the  1,  2,  3  com- 
ponent S.H.M.'s  build  it  up. 

Acoustically  this  means  that  by  using  resonators,  §441,  tuned 
to  notes  of  1,  2,  3,  4,  5  .  .  .  times  the  lowest,  fundamental,  fre- 
quency (that  which  we  calculated  above),  so  as  to  pick  up  and 
exaggerate  their  importance,  notes  of  these  frequencies  can  be  dis- 
covered in  the  compound  note  of  the  instrument. 

Or,  alternatively,  that  by  sounding  these  simple  pure  tones  all 
together,  in  appropriate  loudness,  the  complex  note  characteristic 
of  the  violin  (or  whatever  musical  instrument  was  used)  can  be  built 
up.  This  was  actually  done  with  tuning-forks  before  1870  ;  it  can 
be  done  much  more  easily  nowadays  by  feeding  a  loud-speaker 
from  a  number  of  different  '  oscillating  circuits  '  at  once,  tuned  to 
these  frequencies,  and  suitably  energized. 

Consequently  it  is  customary  to  say  that  a  Musical  Tone  consists 
of  a  Fundamental  and  a  number  of  Overtones.  When  the  frequencies 
of  the  overtones  are  2,  3,  4,  5,  or  any  integral  number  of  times 
that  of  the  fundamental — and  on  a  string  they  have  to  be, 
because  you  can't  fit  3^,  or  fractional  bits  of  segments,  between 
fixed  bridges — they  are  called  harmonic  overtones,  or  simply. 
Harmonics.  There  are  so  few  instances  of  anharmonic  overtones 
(one  is  the  shrill  ring  of  a  hard-hit  fork  6J  times  fundamental, 
distinctly  recognizable  and  harmless)  that  Overtones  and  Harmonics 
are  commonly  employed  as  synonymous. 

In  '  strings,'  '  wood-wind,'  and  organ  pipes,  the  fundamental 
is  the  loudest  dominant  note,  and  the  rest  embroidered  on  it  produce 
the  distinctive  Quality  which  tells  the  cognoscenti  what  instrument 
is  sounding.  In  '  brass  '  the  fundamental  is  seldom  heard,  and  one 
or  other  overtone  can  be  given  predominant  intensity  by  suitably 
hard  blowing,  §  450  ;  in  bells  the  fundamental  is  heard  only  when 
'  muffled,'  §  454.  In  a  piano  the  objectionable  sixth  and  seventh 
are  kept  down  in  intensity  by  the  hammer  hitting  the  wire  at  about 
that  fraction  of  its  length  (just  where  either,  sounding  by  itself, 
Melde  fashion,  would  have  a  node),  and  the  '  tinny  '  sixteenth, 
seventeenth,  etc.,  by  the  soft  felt  face  of  the  hammer  striking  a 
blow  broader  than  the  whole  width  of  these  little  waves. 

You  see,  what  really  happens,  in  any  case,  is  one  single  mechanical 
movement,  as  simple  as  the  rise  and  fall  of  the  tide  on  your  last 
summer  holiday.  What  combined  to  cause  that  movement,  or  what 
can  be  read  into  it,  or  made  out  of  it,  may  be  very  complicated  indeed, 
yet  of  an  ordered  complexity. 


PITCH,   AND   STRINGS  347 


EXAM   QUESTIONS,   CHAPTER   XXVIII 

This  is  largely  a  laboratory  chapter  :  see  Fig.  156  done;  don't  learn  up 
details  of  instruments  you  never  use.  §§  439,  440  are  to  help  fill  the  gap  which 
is  apt  to  yawn  between  fiddlestrings  and  music. 

1.  Define  amplitude,  wave-length,  frequency.  Describe  beats,  explain 
them,  and  give  experimental  illustrations. 

2.  Describe  a  method  of  determining  the  frequency  of  a  note. 

If  you  were  given  two  tuning-forks  of  nearly  the  same  pitch,  and  knew 
the  frequency  of  one,  how  would  you  find  that  of  the  other  ?     (  X  2) 

3.  Explain  the  terms  :    pitch,  beats,  resonance. 

Describe  carefully  how  the  pitch  of  a  note  may  be  determined  experi- 
mentally.     (  X  2) 

4.  Describe  one  method  of  finding  the  frequency  of  a  tuning-fork. 

A  fork,  when  sounded  with  one  of  288,  gives  4  beats  per  second,  and  when 
loaded  with  a  piece  of  wire  again  gives  4.  How  do  you  account  for  this,  and 
what  was  the  unknown  frequency  ? 

5.  A  column  of  air  and  a  tuning-fork  produce  4  beats  per  second  when 
sotmding  together,  the  fork  giving  the  lower  note,  air  at  16*  C.  At  10**  C. 
they  produce  3.     Find  frequency  of  fork. 

6.  Calculate  the  velocity  of  sound  in  a  gas  in  which  two  waves  of  lengths 
1  and  1-01  m.  produce  10  beats  in  3  sec. 

7.  Six  pipes  are  of  successively  higher  pitch,  and  the  sixth  is  the  octave 
of  the  first.  First  and  second  beat  nine  times  per  second,  second  and  third 
seven,  next  eight,  nine  and  seven.     Find  their  frequencies. 

8.  Describe  a  siren,  and  explain  how  it  can  be  used  to  determine  the  pitch 
of  a  note. 

9.  How  would  you  make  a  permanent  record  of  the  movement  of  the  central 
part  of  a  stretched  wire  ? 

10.  Explain  by  diagrams  what  notes  may  be  produced  by  a  transversely 
vibrating  string  stretched  between  two  supports. 

How  would  you  show  separately  the  first  three  of  these,  and  how  prove 
their  being  all  three  present  if  the  string  is  plucked  at  one-fourth  its  length  ? 
(X2) 

11.  Describe  Melde's  experiment  of  vibrating  strings.  How  can  you  get 
two  frequencies  from  the  fork  ? 

12.  Explain  why  a  string  is  scarcely  audible  unless  a  soimding-box  be 
provided.  How  do  the  shape  and  size  of  the  box  influence  its  audibility  ? 
How  do  you  measure  loudness  in  theory  and  in  practice  ?     (  X  2) 

13.  Describe  any  experiments  which  may  be  used  to  ascertain  the  laws  of 
transverse  vibration  of  strings. 

14.  Two  strings  otherwise  equal  have  densities  1'3  and  21-8;  find  the  ratio 
of  their  frequencies.     (  X  2) 

15.  Two  strings  of  the  same  length  and  diameter  are  of  materials  of  density 
1-21  and  9-0,  respectively.  Compare  their  tensions  in  order  that  the  note 
of  the  second  string  may  be  the  octave  below  that  of  the  first. 

16.  If  the  addition  of  10  lb.  to  the  tension  on  a  wire  and  a  decrease  of  10% 
in  length  raise  the  pitch  by  three  halves,  find  the  original  tension.     (  X  3) 

17.  State  the  laws  of  vibration  of  strings.  Two  strings,  identical  except 
in  diameter,  are  knotted  together  and  stretched ;  they  vibrate  with  the  knot 
as  a  node,  and  with  twice  as  many  segments  on  the  thick  string  as  on  the 
thin.     Compare  their  diameters. 


348  SOUND 

18.  Two  forks  sounded  together  give  5  beats  per  second;  one  is  in  unison 
with  a  length  of  96  cm.  and  the  other  with  97  cm.  of  a  monochord  string. 
Find  their  frequencies. 

19.  Explain  how  to  ascertain  an  unknown  note  by  the  sonometer.  A 
wire  weighing  0-006  gm.  per  cm.  is  vibrating  in  two  segments  on  a  length 
of  60  cm. ;  the  load  on  it  is  5  kgm.,  calculate  the  frequency  of  the  note. 

20.  With  6-kgm.  load  a  wire  is  making  200  vibrations  per  sec.  With  what 
load  would  one  of  half  the  length  make  240  ? 

21.  Calculate  the  frequency  of  a  65-9-cm.  length  of  wire  of  total  mass 
0-333  gm.  stretched  with  5-kgm.  wt. 

22.  Given  a  tuning-fork  and  a  stretched  wire  of  variable  length,  how  would 
you  find  the  note  emitted  on  plucking  a  bicycle-spoke  or  a  hack-saw  ?  How 
does  the  note  change  as  the  spoke  is  tightened  ? 

23.  A  steel  wire  1  m.  long  is  stretched  1  mm.  and  plucked.  Calculate 
n,  given  Y.M.  2  x  10^^^  d  7-8  gm./c.c,  diam.  0-5  mm. 

24.  A  ship's  foretopmast  stay,  40  m.  long,  is  of  wire  rope  3  kgm.  to  the 
metre.  A  jerk  sent  up  it  returns  to  the  hand  in  exactly  1  sec.  Calculate 
its  average  tension. 

25.  A  3-5-m.  length  of  ship's  anchor  chain,  from  hawse-pipe  to  capstan, 
vibrates  six  times  per  second  when  struck.  The  2-25-in.  chain  weighs  500  gm. 
per  cm.     Calculate  the  pull  on  it. 


PRACTICAL  QUESTIONS 

Using  a  vertical  sonometer  wire,  find  the  weight  of  the  bag  of  weights 
(compare  with  known  weight ;  or  sometimes  calculate  right  out). 

Compare  the  masses  per  cm.  of  two  wires  or  compare  the  densities  of  their 
metals. 

Calculate  the  mass  per  cm.  of  a  wire. 


CHAPTER   XXIX 
ACOUSTIC  RESONANCE.      PIPES 


§441.  Acoustic  Resonance.  In  §§  385,  386,  which  please 
re-read  at  once,  it  was  pointed  out :  (1)  that  anything  elastic  can 
be  compelled  to  vibrate  at  any  rate  and  to  any  extent  we  please, 
provided  plenty  of  force  is  used,  and  (2)  that  when,  and  only  when, 
the  periodicity  of  the  force  applied  agrees  nearly  with  the  natural  time 
of  free  swing,  a  small  force  will  gradually  work  up  a  large  motion. 

Now,  the  air  in  every  jar,  jug,  bottle,  box,  etc.,  has  a  natural 
period  of  its  own,  in  which  it  will  vibrate,  bouncing  in  and  out  like 
a  jack-in-the-box  spring.  One  has  only  to  drop  a  loose-fitting 
bung  into  a  tall  gas- jar,  and  try  to  hurry  it  down,  or  punch  down 
the  handle  of  the  bicycle-pump  when  the  valve  is  stuck,  and  one 
gets  a  clear  idea  of  the  elastic  kick  of  the  imprisoned  air.  The 
rebound  is  very  much  more  rapid  when  only  the  light  air  in  the  mouth 
of  the  vessel  has  to  be  moved,  instead  of  the  heavy  solid  piston. 

It  can  be  made  visible.  Make  up  a  resonance  tube,  §  442,  to  the 
note  of  a  good  fork,  and  lay  it  on  the  stage  of  a  microscope.  A 
couple  of  inches  from  the  open  end  make  a  hole,  into  which  your 
low  power  looks  ;  close  to  it,  but  horizontal,  make  another  hole,  into 
which  a  lens  slides,  and  concentrates  an  intense  light,  from  sun,  or 
bare-wire  lamp,  on  to  smoke  filUng  the  tube,  Fig.  274  I.  Focussing 
on  the  brilliant  point  of  the  cone  of  light,  you  see  the  smoke  particles 
as  drifting  shining  dots  ;  holding  the  fork  to  make  the  tube  resound, 
they  become  dashes,  lengthwise  of  the  tube,  plainly  indicating  the 
motion  of  the  air  containing  them.  This  is  how  Fig.  143  was  ob- 
tained. 

The  rapid  bouncing  vibration  makes  a  musical  note,  the  note 
of  its  own  to  which  the  vessel  will  resound  most  strongly  when  sung 
to,  which  can  sometimes  be  ehcited  by  blowing  across  the  mouth, 
which  tells  you  how  the  filling  of  the  jug  under  the  tap  is  progressing, 
for  the  note  rises  as  the  air-space  is  reduced,  i.e.  as  the  driving  spring 
gets  shorter  and  less  '  cushy.' 

Thin  glass  tumblers,  and  gas  globes,  have  been  shattered  by  their 
intense  resonance  to  the  powerful  sound  of  a  great  bell,  or  even  a 
great  voice. 

Per  contra,  partly  closing  the  mouth  of  the  resonating  cavity 
lowers  the  note,  for  it  takes  longer  to  push  the  air  past  the  obstruction. 
Your  own  mouth-cavity  is  a  resonator  to  the  vocal  cords,  and  the 
change  of  its  note  can  be  heard  as  you  scratch  your  cheek  with  a 
finger-nail,  and  slowly  open  and  shut  the  mouth.  I  am  told  that 
this  experiment  fails  with  smooth  cheeks ;  but  sing  the  vowels,  or 
go  and  tease  the  hoolock  gibbons  at  the  Zoo. 

349 


350  SOUND  [§441 

In  a  small  room  one  particular  low  note,  the  natural  note  of  the 
resounding  room — will  come  out  very  loud  as  you  sing  or  hum  down 
the  scale  ;  it  is  this  flattering  sonority  that  encourages  vocal  effort 
in  the  bathroom. 

Acoustic  resonance  is,  of  course,  not  peculiar  to  air  cavities  only. 
A  fork  pressed  on  the  sound-box  of  a  string  instrument  sets  the 
whole  box  into  slight  vibration  :  if  one  of  the  strings  is  of  the  same 
frequency  (or  double,  or  treble,  etc.)  the  tremor  of  the  bridge  will 
provide  the  necessary  periodic  push,  and  the  string  vibrates  visibly 
and  audibly.     This  was  how  you  tuned  your  sonometer,  §  438. 

Solid  contact  is  unnecessary  :  open  the  piano,  depress  the  forte 
pedal  to  lift  the  dampers,  sing  any  note,  and  the  instrument  answers 
that  same  note.  The  air- waves  you  produced  set  the  sound-board  in 
vibration,  but  only  the  corresponding  strings  took  up  and  stored 
energy  from  it,  thereafter  to  be  returned  to  it,  and  thence  to  the  air. 
There  may  be  an  improvement  in  musical  quality,  because  the  octave 
string  also  vibrates  a  little. 

Broad  Resonance.  Now,  the  piano  contains  only  a  few  dozen 
definite  notes,  but  intermediate  notes  are  resounded  to,  though  the 
frequencies  of  the  strings  on  either  side  of  them  may  be  a  dozen 
per  second  wrong. 

But  it  was  pointed  out  in  §  388  that  when  the  resonator's  motion 
was  '  damped,'  resonance  was  neither  so  strong  nor  so  sharp,  but 
occurred  fairly  well  over  a  long  range  of  frequencies. 

A  broad  sounding-board  is,  of  course,  intended,  and  admirably 
adapted,  to  give  out  quickly  to  the  air  the  energy  of  the  blow  on  its 
strings ;  therefore  its  motion  quickly  dies  down,  it  is  '  damped  by 
Radiation  of  Energy'  and  this  explains  why  resonance  was  wider 
spread. 

Since  a  broad  board  radiates  sound-waves  powerfully,  it  ought 
(1)  to  pick  them  up  easily,  (2)  over  a  wide  range  of  frequencies, 
and  (3)  should  therefore  be  able  to  emit  many  different  notes  when 
properly  excited.  A  thin  drawing-board  carried  along  a  city  street 
trembles  at  every  loud  noise,  while  the  immense  variety  of  Chladni's 
figures,  §453,  shows  the  truth  of  (3).  It  is  very  essential  that 
the  receiving  diaphragms  of  microphones  and  sound-boxes,  and  the 
reproducing  discs  or  cones  of  telephones  and  loud-speakers,  should 
have  no  very  determined  resonance  of  their  own,  or  the  resultant 
sound  may  be  badly  distorted.  An  old  violin  owes  much  of  its 
excellence  to  the  equable  response  of  the  seasoned  sounding-box  to 
all  notes. 

Among  air  cavities,  open  boxes  must  emit  their  energy  fast,  and 
therefore  resound  broadly,  as  do  wide-mouthed  sea-shells  to  the 
roar  of  the  breakers.  The  resonators  with  small  mouths  employed 
for  analysing  sound,  and  long  narrow  pipes,  disperse  their  contained 
energy  much  more  slowly,  and  therefore  resound  more  precisely. 

With  the  latter,  indeed,  and  a  pure  steady  source  of  sound  such 
as  a  Knipp's  silica  singing-tube,  an  acoustic  paradox  can  be  demon- 
strated.    Perfectly  sharp  acoustic  resonance  would  imply  no  damp- 


442] 


RESONANCE,   PIPES 


351 


ing  at  all,  therefore  no  radiation  of  sound,  no  increased  loudness ; 
but,  on  the  contrary,  must  be  inaudible:  as  the  tube  is  slowly 
lengthened,  in  the  midst  of  increasingly  loud  response  to  a  steady 
pure  note,  there  comes  a  momentary  lull,  the  position  of  exact 
equahty  of  frequency.  For,  as  you  see  from  Fig.  122,  if  you  want 
the  pendulum  to  swing  in  exactly  the  frequency  natural  to  its  length, 
its  upper  end  must  be  a  fixed  point,  so  you  have  no  means  of  making 
it  go  ! 

The  only  form  of  resonating  air-cavity  that  can  be  dealt  with  in 
detail  in  this  book  is  a  straight  tube. 

§  442.  The  Resonance  Tube.    The  resonance  tube  in  the  labor- 
atory is  usually  a  long  vertical  glass  tube  an  inch  or  more  in  bore, 
and  the  movable  stopping  is  the  surface  of 
water  which  can  be  run  in  or  out  to  any  level. 
Fig.    158.     Consider  the  action  on  it  of  one 
prong  of  a  fork. 

The  prong  starts  at  the  top  of  its  swing  to 
drive  air  before  it,  and  therefore  to  send  a 
compression  down  the  pipe.  The  action  in- 
creases, up  to  mid- swing,  when  the  prong  is 
chasing  the  air  fastest,  and  then  gradually 
diminishes  again.  Accordingly,  the  densest 
part,  the  '  crest,'  of  a  compression,  leaves  the 
fork  at  midswing,  travels  to  the  stoppered  end 
of  the  pipe,  is  instantly  reflected,  and  returns. 
The  prong  has  gone  to  the  bottom  of  its  swing, 
and  is  now  moving  up.  If  the  reflected  '  crest ' 
reaches  the  prong  just  at  mid-swing  up,  the 
two  '  crests  '  are  exactly  added  together  ;  the 
fastest  outrush  of  air  particles  from  the  pipe 
coincides  exactly  with  the  hardest  pull  up- 
wards of  the  prong  on  the  air.  The  two 
combine  to  drag  air  out  of  the  pipe.  The 
next  down-swing  of  the  prong  therefore 
drives  compressed  air  into  a  partial  vacuum 
waiting  for  it.  Like  a  swing  pushed  always 
at  the  right  moment,  the  air  in  the  pipe  is 
gradually  excited  to  more  and  more  violent 
motion,  and  resounds  strongly  to  the  note  of  the  fork. 

As  in  §  387,  it  takes  perhaps  a  hundred  vibrations  to  work  up 
strong  resonance,  and  the  reflected  '  crest '  and  the  mid- swing 
position  of  the  upgoing  prong  must  very  closely  coincide.  For 
if  the  crest  gets  up  1%  too  soon,  the  new  push  from  the  fork  follows 
1%  late,  on  the  next  pulsation  it  would  be  2%  late,  and  so  on, 
to  50%  late.  Then  the  fork  is  in  direct  opposition  to  the  vibrating 
air  in  the  pipe,  and  begins  to  wipe  out  its  previous  work,  for  the 
next  fifty  swings,  then  to  rebuild  it  for  fifty,  and  so  on.  This  im- 
perfect resonance  never  gains  much  strength.    Loud  resonance. 


Fig.  158. 


i 


352  SOUND  [§442 

then,  means  that  a  sound-wave  travels  down  and  up  the  pipe  while 
the  fork  moves  from  mid-swing  down  to  mid-swing  up,  i.e.  makes 
half  its  complete  vibration.  During  the  upper  half  of  its  swing, 
prong  and  pipe  are  occupied  spreading  compression  out  into  the 
atmosphere. 

.*.  Sound  travels  twice  the  length  I  (see  below)  of  the  air  tube, 
in  half  the  periodic  time  of  the  prong. 

Or  4Z  in  the  periodic  time  of  the  prong,  which  makes  n  vibrations 
per  sec. 

.'.  Speed  of  sound  =  n  X  4Z ;  frequency  X  4  times  length  of  tube. 

This  length  of  tube  requires  a  Mouth  Correction ;  why,  can  be 
seen  from  a  simple  analogy.  When  slowly  drifting  down  a  corridor 
out  of  a  crowded  hall,  you  cannot  dash  off  at  full  speed  the  moment 
you  cross  the  threshold,  there  is  still  a  press  of  people  round  the  door, 
and  a  little  more  delay ;  it  is  as  if  the  corridor  were  a  little  longer, 
and  then  perfect  freedom.  Add,  to  the  length,  J  the  diameter,  if  the 
tube  stands  out  in  free  air  with  a  thin  edge,  2/5  if  it  ends  flush  with 
a  wall. 

On  the  one  hand,  you  see  the  analogy  between  this  experiment  and 
that  of  Fig.  144  ;  on  the  other,  you  recognize  now  that  the  fork  is 
simply  calling  forth  the  natural  note  of  the  resonating  air  cavity, 
which  has  the  same  pitch  as  itself.  Since,  in  all  wave-motion. 
Speed  of  travel  =  frequency  X  wave-length,  V  =  nL,  §391,  you 
see  that  the  length  of  waves  emitted  by  a  plain  pipe  stopped  at 
one  end  is  4  times  the  (corrected)  length  of  the  pipe. 

Turn  to  Fig.  135.  The  stopped  end  is  M,  the  open  end,  where 
of  all  places  the  air-motion  is  most  free,  is  A,  the  first  antinode  ; 
the  pipe  is  J  wave-length  long.  Work  the  single  length  NA  of  Fig. 
137,  and  you  see  how  the  air  pulses  in  and  out  of  the  mouth. 

Suppose,  now,  we  lower  the  water-stop  to  3  times  the  length. 
Resonance  to  the  same  note  will  occur  again,  because  the  extension 
is  a  ^a// wave-length,  and  the  crest  will  just  travel  along  it  and  back 
in  one  period  of  the  note,  and  will  catch  the  prong  midway  of  its 
second  swing  up.  And  another  half-wave-length  extension  will 
produce  a  5  times  longer  tube  that  can  resound  to  the  same  note, 
the  crest  travelling  inside  for  two  extra  periods  and  catching  the 
prong  on  its  third  swing  up,  and  so  on.  In  Fig.  137  work  the 
lengths  NANA  and  NANANA,  and  you  see  exactly  what  is  going 
on  inside  the  tube  thus  extended. 

These  half -wave-lengths  need  no  correction.  The  speed  of  sound 
does,  for  temperature,  if,  as  very  commonly  in  exams,  you  are  asked 
to  use  this  method  to  give  its  value  at  0°,  using  a  known  fork.  You 
see  easily  enough  how  to  reverse  the  calculation,  and  use  the 
Resonance  Tube  as  a  means  of  measuring  the  Pitch  of  any  note  to 
which  you  can  hear  it  resound,  knowing  the  speed  of  sound  in  the 
room.  Also  how  to  compare  different  forks  by  its  aid  ;  frequencies 
are  inversely  as  (corrected)  lengths.  Or  if  we  choke  the  mouth, 
and  ask  you  for  the  '  mouth-correction,'  it  is  the  difference  between 


§443]  RESONANCE,    PIPES  353 

the  first  length  and  half  the  (unaffected)  distance  from  first  to  second 
resonance  positions. 

Turning  again  to  Figs.  135, 137,  the  open  end  must  be  an  antinode  ; 
it  can  be  any  A,  i.e.  the  tube  may  be  1/4  or  3/4  or  5/4,  etc.,  wave- 
lengths long.  The  original  note  is  the  Fundamental  of  the  shortest, 
the  first  harmonic  of  the  next,  the  second  harmonic  of  the  next,  and 
so  on,  these,  of  course,  having  fundamentals  of  1/3,  1/5,  etc.,  the 
frequency  we  are  using. 

§443.  A  Kundt  Dust  Tube,  Fig.  163,  is  made  by  scattering  fine 
cork-dust  along  a  long  wide  glass  tube.  This  has  a  loosely  fitting 
adjustable  plug  at  one  end,  and  into  the  other  projects  the  end, 
armed  with  a  light  disc-piston,  of  a  brass  or  wooden  rod,  clamped 
firmly  in  the  middle,  and  stroked  towards  the  other  end  with  a 
rosined  rag,  as  if  to  pull  it  longer.  This  sets  it  into  longitudinal 
vibration  (like  a  concertina,  gripped  amidships),  and  it  produces  a 
loud  piercing  note.  Evidently  it  has  a  node  in  the  middle  and  an 
antinode  of  freest  motion  at  each  end ;  its  length  AA  is  therefore 
half  the  wave-length,  in  brass  or  wood,  of  the  high  note  n  it  emits, 

and  Speed  of  sound  in  material  of  rod  =  n  x  twice  length. 

=  also  VE/D  by  §  396, 

so  that  Young's  Modulus  can  be  calculated  if  the  density  be  measured 
also. 

With  a  little  manipulation  of  the  movable  plug,  a  length  will 
be  found  (as  in  Melde's  experiment)  at  which  the  tube,  resounding  to 
the  rod,  breaks  into  stationary  wave-motion,  blowing  the  cork-dust 
first  into  large  oval  cross-striated  patches  at  the  antinodes,  and 
finally,  with  sufficient  power,  into  quiet  heaps  at  the  nodes  (side 
view).  Each  is  then  a  half -wave-length  of  the  note  in  Air,  and 
your  tube  is  Fig.  135  from  M  to  (very  nearly)  an  N. 

The  way  to  get  an  average  is  this  :  fix  a  scale  alongside  and 
record  the  middle  of  each  clear  patch,  as  near  as  you  can  judge. 
Suppose  you  get  fourteen  readings  ;  take  first  from  eighth,  second 
from  ninth,  and  so  on,  giving  seven  differences,  each  the  length  of 
seven  segments ;  add  them  together  and  divide  by  49.  If  fifteen, 
throw  out  the  middle  reading,  take  first  and  ninth,  giving  seven, 
each  of  eight  segments  ;  add  and  divide  by  56.  You  see  you  get 
the  mean  of  49  or  56,  instead  of  only  13  or  14.  In  this  way  no 
reading  is  used  twice,  and  this  is  always  the  correct  way  oj  utilizing 
all  readings  in  a  series. 

_,,  length  of  rod  _  speed  of  sound  in  rod 

Segment  length  in  tube  ~~  speed  of  sound  in  air 

The  Dust  Tube  can  be  used  for  Notes  beyond  Audible  Frequency ; 
and  also,  in  a  slightly  modified  form,  for  different  gases,  and  was 
so  used  for  Argon,  §  415  ;  but  we  will  employ  another  variant,  due 
to  G.  D.  West. 

N 


354 


SOUND 


[§444 


§  444.  West  mounts  a  little  shrill  whistle  on  a  long  dry-air  supply- 
tube  and  pushes  it  up  the  long  glass  tube,  corked  at  the  far  end. 
Silence  positions,  sharply  defined,  are  found  along  the  pipe  :  they 
are  evidently  J  wave-length  apart  of  the  whistle  note. 

I  have  utiUzed  this  for  comparing  the  speeds  of  sound  in  different 
gases  as  follows.  Take  another  whistle  and  blow  it  with  the  various 
gases,  coal  gas,  COg,  SOg,  HgS,  NHg,  oxygen,  even  hydrogen,  though 


159 


163 


^ 


L 


'm//////'''''~__ 


rE 


IB 


iii'-'iiiiini-iinii-a 


Figs.  159—163. 


this  makes  a  poor  squeak — anything  you  can  get  dry  from  the 
chemistry  lab.  For  Hot  Air  use  breath  through  a  copper  coil  in 
boiling  water.  To  make  sure  the  gas  is  unmixed  at  the  mouth  of 
the  whistle,  keep  it  inside  an  open  test-tube  all  the  time. 

The  whistles  are  adjustable,  as  shown  full  size  in  Fig.  159,  the 
plug  sliding  up  and  down  the  little  '  stopped  organ-pipe.'  Keep 
the  gas  whistle  fixed,  i.e.  use  the  same  gas  wave-length  throughout ; 


§446]  RESONANCE,    PIPES  355 

tune  the  air  whistle  to  it  for  each  gas,  and  then  measure  its  wave- 
length in  the  tube.  Puzzle  out  for  yourself  that  the  gas  speeds  are 
inversely  as  these  wave-lengths. 

The  little  experiment  suffers  in  accuracy  from  the  insensitivity 
of  the  ear  in  tuning  high  notes. 

§  445.  The  wave-length  (and  thence  the  pitch,  S/L)  of  one  of 
these  whistles  can  also  be  measured  out  in  the  open  as  in  Fig.  161. 
Set  it  up  a  foot  or  so  from  the  wall,  and  in  line  between  hold  a 
Sensitive  Flame,  a  pinhole  gas-jet  supplied  with  gas  from  a  gas  bag 
on  which  somebody  sits,  so  as  to  give  2  or  3  times  the  mains  pressure 
(or  you  can  draw  down  several  nozzles,  preferably  of  hard  glass, 
until  you  find  one  sensitive  at  m.p.).  This  flame  shortens  and 
roars  (left)  at  a  high  note,  but  towards  the  wall  finds  points  half -wave- 
length apart  at  which  it  remains  long  and  quiet.  This  is  evidently 
Fig.  135  at  its  simplest. 

Another  sort  of  flame  is  the  Manometric  Flame,  which  has  been 
utilized  in  many  ways.  A  little  tambour,  made  by  tying  up  a 
finger-ring  of  metal  in  thin  rubber  tissue,  so  as  to  make  it  a  flat 
box  with  elastic  top  and  bottom.  Fig.  162,  is  mounted  at  the  end  of 
twin  tubes  long  en^pugh  to  reach  down  an  organ-pipe  under  investiga- 
tion. One  admits  gas  to  the  capsule,  and  the  other  takes  it  away 
to  a  pinhole  burner.  At  a  node,  towards  which  air  rushes  from  both 
sides  at  once.  Fig.  137,  the  changes  of  pressure  force  the  elastic 
drum-heads  in  and  out,  varying  the  drum's  internal  capacity,  and 
driving  the  gas  out  in  puffs.  The  Httle  flame  sings,  and  its  reflection 
in  a  moving  mirror  (or  as  you  swing  your  eyes  across)  is  a  drawn- 
out  band  of  light  with  a  jagged  saw-tooth  edge. 

Or  a  solid-back  manometric  capsule  can  be  mounted  with  a  mouth- 
piece to  receive  spoken  sounds.  Fig.  162  left ;  or  can  form  the 
sound-box  of  a  gramophone. 

§  446.  The  Interference  Tube,  Fig.  160,  can  be  used  for  measuring 
longer  wave-lengths.  The  note  is  played  in  front  of  the  upper 
T-piece  of  a  sort  of  double  trombone  slide,  and  its  sound  travels 
round  both  ways  to  the  lower  T-piece,  whence  a  rubber  tube  leads 
to  the  ear.  Both  slides  are  at  first  pushed  home,  and  it  is  the  same 
distance  round  either  way  ;  as  one  is  pulled  out  the  sound  weakens 
and  ceases.  This  means  that  one  path  is  now  half  a  wave-length 
longer,  so  that  crest  and  trough  obliterate  each  other  at  the  ear  : 
thus  the  wave-length  is  4  times  the  distance  the  slide  has  been  moved. 
Other  notes  get  through  ;  it  is  a  '  wave-trap.' 

In  listening  for  aircraft,  a  pair  of  large  ear-trumpets  face  forwards, 
and  equal  tubes  lead  from  them  to  the  ears.  When  facing  the  source, 
the  two  are  in  phase,  and  the  sound  is  much  louder  than  when  turned 
aside,  so  that  one  ear  is  at  the  end  of  a  longer  path. 


356 


SOUND 


[§447 


PIPES 

Musically  speaking,  most  wind  instruments  are  pipes,  and  from 
a  physical  point  of  view  a  pipe  is  a  resonance  tube  provided  with 
some  means  for  producing  a  commotion  in  the  air  at  one  end  of  it. 

§447.  Pipes  and  how  they  are  blown.  (1)  The  ancient  Pan-pipe 
was  a  row  of  hollow  reeds  of  graduated  lengths,  stopped  by  the 
stem  '  knots  '  at  their  lower  ends  and  made  to  sound  by  blowing 
across  the  open  tops.  Nowadays  one  occasionally  uses  a  key  in 
the  same  way,  and  the  winter  wind  uses  a  keyhole.  Flutes  and  fifes 
are  uniform  tubes  open  at  the  far  end,  and  with  a  large  side  hole  at 
the  near  end,  merely  blown  across. 


Fig.  164. 


(2)  In  the  '  flue  '  pipes  of  the  organ,  Fig.  164  (A),  and  in  most 
whistles,  there  is  the  well-known  '  mouth,'  up  across  which  blows 
a  flat  stream  of  air,  from  a  narrow  slit  inside  the  lower  lip,  to 
impinge  on  the  thin  wood  or  metal  edge  of  the  upper  lip.  '  Stopped  ' 
organ-pipes  are  closed  at  the  top  by  an  adjustable  plug ;  '  open ' 
pipes  are  open  at  the  top.  Steamboat  whistles  are  stopped  pipes 
with  double  mouths,  railway  whistles  and  factory  bulls  usually 
have  mouth  all  round,  to  get  most  noise. 

(3)  In  the  '  reed  '  pipes  of  the  organ,  Fig.  164  (B),  there  is  a 
*  reed  '  consisting  of  a  narrow  elastic  metal  tongue  almost  closing 
the  narrow  slot  through  which  the  wind  is  suppUed. 


§448]  RESONANCE,    PIPES  367 

The  tongue  either  swings  in  and  out  of  a  slot  slightly  larger 
than  itself  (free  reed  C),  or  flaps  down  on  to  a  smaller  slot  (beating 
reed  D),  thus  permitting  the  wind  to  issue  in  periodic  puffs.  For 
such  reeds  in  miniature,  dissect  a  toy  mouth-organ.  The  reed 
has  a  note  of  its  own,  and  the  natural  frequencies  of  tube  and 
attached  reed  must  be  about  the  same,  or  resonance  is  defective, 
and  the  pipe  '  speaks  '  badly.  Harmoniums  etc.  have  one  common 
sound-box  instead  of  individual  pipes. 

Clarionets,  oboes,  and  bassoons  have  '  reeds '  of  split  cane. 
Stretched  membranous  '  vocal  cords,'  with  the  resonant  pharynx 
and  mouth,  produce  the  human  voice.  E  is  a  rough  model  larynx 
constructed  of  two  pieces  of  thin  sheet  rubber  tied  over  the  cut 
end  of  a  pipe,  so  as  to  leave  a  narrow  slit  between  them.  A  resonance 
tube  (dotted)  can  be  added. 

(4)  The  lips  are  the  vibrating  reeds  for  brass  instruments. 

(5)  Tubes  can  be  sounded  by  a  flame  burning  inside  them  ; 
F  has  a  paper  tuning-slide  at  the  top.  Listen  to  the  deep  booming 
of  the  chimney  when  you  are  '  drawing  up  '  the  fire  with  a  newspaper. 

Note. — A  reed  is  practically  a  stopped  end ;  it  is  only  a  small 
aperture,  and  there  is  a  wall  of  compressed  air  behind  it. 

One  can  understand  metal  reeds,  but  how  is  it  that  blowing 
contrivances  which  of  themselves  make  only  a  feeble  irregular 
noise — a  very  '  dry  whistle  ' — can  call  forth  loud  musical  notes 
from  the  tubes  ? 

Any  fluid  flowing  through  a  narrow  orack  at  more  than  a  very 
slow  speed  sets  up  eddies.  It  is  these  that  make  the  dry  whistling 
sound  :  they  are  heard  and  seen  when  a  flat  gas  flame  is  turned 
too  high  and  flares.  These  eddies  mean  local  variations  of  speed 
and  pressure,  §  120,  the  flat  blade  of  air  sways  and  sends  a  puff  of 
compression  up  the  pipe.  This  surges  up  and  down,  the  large  mass 
of  air  begins  to  pulsate,  and  soon  alternately  blows  the  thin  stream 
of  wind  away  from  the  mouth,  or  sucks  it  in  in  puffs  at  times  to  suit 
itself,  taking  up  the  energy  of  the  puffs  to  produce  its  own  note ; 
just  as  a  heavy  pendulum  takes  energy  when  it  pleases  from  the 
scape- wheel,  and  keeps  its  own  time. 

A  badly  aimed  stream  or  misshapen  upper  lip  of  course  en- 
feebles this  action ;  steamboat  whistles  are  often  husky  on  this 
account,  even  after  clear  of  water. 

How  preponderant  is  the  control  of  the  resonator  anyone  can 
feel  in  whistling  a  tune.  The  lips  remain  fixed,  while  the  tongue 
is  busy  all  the  time  altering  the  size  and  shape  of  the  resounding 
cavity. 

§  448.  From  what  has  been  said  in  §  442  it  will  be  clear  that 
in  a  sounding-pipe  the  air  is  acted  on  by  waves  running  both  up  and 
down,  and  is  therefore  in  the  state  of  Stationary  Wave  Motion 
described  in  §  404. 

It  will  also  be  pretty  plain,  from  what  has  just  been  said  about 


358  SOUND  [§  448 

a  flapping  stream  of  air,  and  by  analogy  with  the  mechanics  of  §  383, 
that  these  waves  are  by  no  means  likely  to  be  the  simple  sine  waves 
of  an  S.H.M.,  but  that  upper  partials  will  be  detectable  in  the  tones 
emitted  by  pipes,  in  proportions  varying  widely  with  their  individual 
shape  and  character.  We  must  consider  the  possibilities  of  these 
in  two  or  three  cases. 

Stopped  pipes.  Fig.  165.  Taking  these  first,  the  stopped  end 
is  a  Node  of  no  motion  and  the  open  mouth  an  Antinode  of  maxi- 
mum motion,  the  wind  blowing  to  and  fro  most  freely  there 
(0-4  cm.  motion  has  been  observed  by  the  aid  of  smoke  at  the  end 
of  a  pipe  125  cm.  long,  and  see  Fig.  143). 


- 

•              — 

—              .              — 

. 

\ 

FOURTH       HARMONIC.        FREQ:  9 

— 

. 

—                   .                  —                   . 

— 

A 

N 

THIRD        HARMONIC.       FREQ:  7 
A                          N 

A 

N 

— 

. 

— 

— 

A 

SECOND      HARMONIC.       FREQ:  5 
N                                          A 

N 

— 

— 

A 

FIRST         HARMONIC.       FREQ:  3 

N 

— 

FUNDAMENTAL.       FREQUENCY  1 
Fig.  165. 

If  no  other  nodes  are  present,  the  pipe  is  now  sounding  its  lowest 
or  fundamental  note,  and  the  wave-length  of  this,  4 AN,  is  four 
times  the  length  of  the  pipe.  | 

Then  in  §  442  it  was  pointed  out  that  a  tube  3  times  as  long  could  1 
give  the  same  note,  having  now  an  extra  node  and  antinode  at  I 
the  thirds  of  its  length.     That  is,  it  is  giving  a  note  the  wave-length  ' 
of  which  is  four 'thirds  the  length  of  the  stopped  pipe.     This  is  called  its 
first  overtone,  and   its   frequency   is   evidently   3   times   that   of 
the  fundamental,  since  wave-length  x  frequency  =  constant  Speed.  | 
It  is  a  harmonic  overtone,  §  440,  for  the  ratio  of  frequencies  is  a  • 
small  integral  number,  and  it  lies  in  the  harmonic  scale  (§459) 
containing  the  fundamental  (G  in  the  octave  above  the  fundamental 
C).     Indeed,  all  the  overtones  of  plain  pipes  are  harmonics. 

It  was  further  pointed  out  that  a  tube  5  times  as  long  could 
give  the  same  note,  having  now  two  extra  nodes  and  two  extra 
antinodes,  at  the  fifths  of  its  length.  That  is,  it  is  giving  a  note 
the  wave-length  of  which  is  four-fifths  the  length  of  the  stopped  pipe,  its 
second  overtone,  5  times  the  pitch  of  the  fundamental. 


§449]  RESONANCE,    PIPES  359 

So  one  can  go  on,  as  in  Fig.  165,  dividing  up  the  stopped  pipe  into 
any  odd  number  of  equal  parts,  keeping  the  stopped  end  a  node 
and  the  open  an  antinode,  putting  in  alternate  nodes  and  anti- 
nodes  along  the  pipe  and  producing  successive  harmonics  of 
frequencies  1,  3,  5,  7,  9,  and  any  odd  number  of  times  that  of  the 
fundamental. 

The  full  natural  tone  of  the  pipe  results  from  the  complex  air 
motion  which  contains  them  all  as  its  simple  harmonic  components 
or  Partials. 

In  Fig.  137  a  stopped  pipe  extends  from  one  end  N  to  any  A ; 
work  the  diagram  and  watch  the  air  movements. 

§  449.  Open  pipes,  Fig.  166,  which  are  tubes  open  at  both  ends, 
must  have  antinodes  at  both  ends,  and  the  simplest  stationary 


FOURTH 

HARMONIC. 

FREQ:  5 

— 

• 

— 

— 

•               -^ 

• 

- 

A 

N 

THIRD 
A 

HARMONIC. 
N 

FREQ:  4 
A 

N 

A 

• 

— 

• 

— 

A 

SECOND 
N 

HARMONIC. 
A 

FREQ:  3 
N 

A 

• 

— 

• 

— 

A 

FIRST 

HARMONIC. 
N 

FREQ: 2 

A 

— 

• 

— 

FUNDAMENTAL.     FREQUENCY  1 
Fig.  166. 

wave  motion  possible  in  them  has  therefore  a  node  in  the  middle 
of  the  pipe.  Such  motion  is  possible,  for,  as  explained  in  §404, 
reflection  can  take  place  from  a  loose  or  open  end.  The  pipe 
acts  like  a  couple  of  stopped  pipes  of  half  its  length,  put  bottom 
to  bottom.  The  wave-length  of  the  fundamental  of  an  open  pipe, 
4AN,  is  therefore  twice  the  length  of  the  pipe,  so  that  unstopping  a 
pipe  raises  its  pitch  an  octave,  and  vice  versa.  Blow  across  any  bit 
of  tubing,  and  try  it. 

The  next  possible  motion,  got  by  putting  in  one  extra  node  and 
antinode,  has  an  antinode  in  the  middle  and  nodes  at  the  quarters, 
its  wave-length  is  twice  half  the  length  of  the  pipe,  and  its  frequency 
twice  the  fundamental. 

In  the  next,  there  is  again  a  node  at  the  middle,  and  the  pipe  is 
again  like  a  stopped  pipe  standing  on  its  own  reflection ;  the 
frequency  is  three  times  the  fundamental. 


360  SOUND  [§  449 

So  one  can  go  on,  as  in  Fig.  166,  dividing  up  the  open  pipe  into 
any  even  number  of  equal  parts,  keeping  both  ends  antinodes, 
putting  in  alternate  nodes  and  antinodes,  and  producing  successive 
harmonic  overtones,  of  frequencies  1,  2,  3,  4,  5,  and  any  number 
of  times  that  of  the  fundamental.  The  presence  of  the  even 
harmonics  gives  the  open  pipe  a  fuller  musical  tone  (§440)  than 
the  stopped  pipe  :  compare  the  8-ft.  open  diapason  with  the 
4-ft.  stopped. 

These  long  laboured  explanations  can  be  summed  up  very  briefly  : 
cut  off  successive  lengths  of  Fig.  135,  paying  heed  to  the  last  two 
lines  of  §  404,  and  work  Fig.  137  for  all  it  is  worth. 

§450.  Wind  Instruments  and  Organ  Pipes  alike  are  Resonance 
Tubes ;  some  are  identical,  such  as  tin  whistles,  or  flutes,  but  generally 
there  is  this  difference  between  them.  Organ  pipes  are  bulky  and 
contain  a  large  mass  of  air  and  are  blown  gently ;  the  tube  of  a 
wind  instrument  is  narrow,  and  is  blown  harder.  One  lad  blows 
one  church  organ,  or  one  bugle.  The  organ  pipe  sticks  to  its 
fundamental  vibration,  and  its  overtones  are  embroidered  on  it : 
it  takes  a  good  breath  to  blow  the  octave  on  one  as  big  as  your  arm  : 
the  little  mass  of  air  in  a  band  instrument  leaves  its  fundamental 
far  behind,  and  breaks  into  higher  and  higher  harmonics  quite 
readily  under  the  compulsion  of  increasing  pressure. 

It  can  be  shown  that  a  conical  pipe,  whether  open  or  stopped  at 
the  small  end,  has  the  full  series  of  harmonics  of  an  open  pipe. 
Hence  the  tapering  form  of  practically  all  reed  instruments,  including 
'  the  brass,'  where  the  lips  form  the  reed  ;  for  the  reed  is  almost  a 
stopped  end. 

(a)  The  '  cheery  simple  compass  of  few  notes  '  of  a  Bugle  or  Post 
horn  consists  of  the  first  five  harmonics,  into  which  the  conical 
pipe  breaks  by  harder  blowing. 

The  most  perfect  example  of  this  is  the  long  French  Horn,  which 
gives  the  sixteen  notes  got  by  multiplying  the  fundamental  (herein 
called  C  for  simplicity)  by  the  natural  numbers  from  1  to  16,  as 
follows  : 


9/8       5/4       4/3        3/2  5/3         15/8 


iatonic  scale     . 

C 

D 

E 

F 

G 

A 

B 

1st  octave 

1 











2nd      „ 

2 

— 

— 

— 

3 





3rd      „ 

4 



5 



6 

—     7 



4th      „ 

8 

9 

10 

11 

12 

13  14 

15 

5th      „ 

16 

— 

— 

— 

— 



11  is  a  trifle  too  sharp  and  13  a  trifle  too  flat,  7  and  14  are  A 
sharps. 

(6)  Per  contra,  one  depends  primarily  on  alteration  of  length 
in  Fifes  and  Flutes,  which  are  virtually  open  tubes  extending  from 


§450]  RESONANCE,    PIPES  361 

the  mouthpiece  to  the  first  open  hole  ;  but  blows  harder  to  reach  the 
next  octave  in  the  smaller  instrument. 

(c)  Trombones  depend  largely  on  length  change,  and  brass 
piston  instruments  have  their  tubes  temporarily  lengthened  by  crooks 
brought  into  circuit  by  pressing  the  piston  valves.  This  enables  the 
gaps  in  the  natural  trumpet  scale  to  be  filled  up  without  going  beyond 
the  seventh  harmonic. 

The  broad  flaring  bell-mouth  of  brass  instruments  spreads  the 
energy  gradually  over  a  large  area  of  air,  and  also  smoothes  away 
unwanted  high  overtones  ;  and  the  metal  itself  resounds  and  vibrates, 
bell-fashion. 


EXAM   QUESTIONS,   CHAPTER   XXIX 

Resonance  is  of  extreme  importance.  Omit  §§  444,  445,  446,  unless  you 
make  practical  acquaintance  with  the  instruments.  The  whole  chapter 
hangs  on  Figs.  135  and  137  :   use  them  throughout. 

1.  Explain  amplitude,  and  timbre.  How  would  you  trace  waves  of  sound  ? 
Define  and  explain  acoustic  Resonance,  distinguishing  between  that  of  a 
board  and  of  a  cavity. 

2.  What  effect  has  a  resonance  box  on  its  fork,  as  regards  loudness  and  time 
of  sounding  ?     What  length  of  box  would  suit  a  256  fork  ? 

For  what  other  pitch  would  this  box  be  suitable  ?     Explain.     (  x  3) 

3.  Explain  the  meaning  of  the  terms  Wave-length  and  Frequency.  What 
relation  is  there  between  wave-length  and  frequency  ?  How  may  the  wave- 
length of  sound  waves  in  air  be  measured  ? 

4.  Explain  what  is  meant  by  Resonance,  giving  instances.  Calculate  the 
minimum  length  of  a  cylindrical  column  of  air  which  will  resound  to  256  per 
sec,  and  describe  the  mode  of  vibration  of  the  air  in  the  colunm.     (  X  3) 

5.  Explain  the  term  Resonance,  and  give  illustrations. 

A  tube  2  m,  long  is  filled  with  water.  A  timing-fork  of  frequency  448  is 
sounded  over  the  upper  end  as  the  water  slowly  drains  out.  At  what  positions 
of  the  water  surface  will  resonance  occur  at  1 0°  C.  ?  Where  will  these  positions 
move  to  at  20°  C.  ? 

Explain  how  observations  with  such  a  tube  may  be  used  to  find  the  '  end 
correction.'     (  X  2) 

6.  If  to  a  348  fork  there  are  two  resonance  positions  in  a  tube,  63  cm.  apart, 
what  is  the  speed  of  soimd  ? 

7.  At  what  lengths  of  a  4-cm,  diameter  tube  would  you  expect  resonance 
to  a  300  fork  ?     (  x  2) 

8.  Explain  how  increase  of  temperature  changes  the  pitch  of  the  note 
given  by  (a)  a  tuning-fork,  (6)  an  organ  pipe,  (c)  a  piano. 

9.  A  tube,  26  cm.  long,  is  closed  at  one  end  by  a  cork,  and  at  the  other  by 
a  piston.  This  is  pushed  in  until  the  pressure  of  the  air  has  risen  to  3/2  of 
its  value,  when  the  cork  pops  out. 

Calculate  the  frequency  of  the  sound  produced,  ignoring  the  end-correction. 
(V  =  340.) 

10.  What  experiments  show  that  note  of  definite  pitch  corresponds 
to  waves  of  definite  length  in  air  ?  How  do  you  explain  the  change  of  note 
heard  as  a  jug  is  filled  under  the  tap  ? 


362  SOUND 

11.  What  do  you  understand  by  'stationary  waves'?  How  may  they 
be  produced  ?  Give  examples,  and  show  how  the  frequency  of  a  tuning- 
fork  may  be  determined  by  their  use. 

12.  Show  how  '  stationary  waves  '  are  formed.  What  is  the  lowest  note 
in  unison  with  them  in  a  tube  81  cm.  by  4  cm.  diameter  ? 

13.  Distinguish  between  progressive  and  stationary  wave-motion,  (o)  on 
a  string,  (6)  in  a  tube.     Under  what  condition  can  one  set  up  the  other  ? 

14.  A  2-ft.  open  tube  resoimds  to  a  530  fork;  calculate  its  nodal  and 
antinodal  positions.     V  =  1120  ft. /sec. 

15.  What  relation  exists  between  the  speed  of  travel  of  sound  in  a  gas, 
the  pitch,  and  the  wave-length  ? 

A  shrill  whistle  is  blown  inside  a  tube  corked  at  the  far  end ;  as  the  whistle 
is  moved  along  the  tube  the  sound  is  heard  much  louder  at  a  number  of 
positions,  which  are  the  same  whether  in  air  or  in  coal-gas.  Explain  this, 
and  state  what  difference  is  observable  with  the  different  gases. 

16.  When  A  sounds  its  fundamental  B  resounds,  but  A  does  not  resound 
to  B's  fundamental.     Which  is  the  higher,  and  why  ? 

17.  Forks  of  frequencies  130,  260,  520,  780,  and  1040  are  successively  held 
over  the  open  end  of  a  stopped  pipe,  which  resounds  best  to  the  260.  What 
happens  with  the  others  ? 

18.  How  would  you  compare  the  velocity  of  sound  in  air  with  that  in 
hydrogen  and  carbon  dioxide  ?     (  X  4) 

19.  Ditto,  at  frequencies  of  (a)  about  250,  (b)  5000  or  more. 

20.  On  what  properties  of  the  medium  of  transmission  does  the  speed  of 
sound  depend  ? 

Describe  some  method  of  comparing  the  speeds  of  sound  in  a  solid  and  in 
a  gas.     (  X  2) 

21.  Describe  Kundt's  dust-tube  method  of  comparing  the  velocities  of 
soimd  in  a  gas  and  in  a  solid,  explaining  how  the  rod  clamped  at  its  middle 
vibrates.     (  X  4) 

22.  Describe  the  vibration  of  the  air  in  closed  and  open  pipes,  explaining 
how  overtones  are  produced  and  co -exist  with  the  fundamental. 

How  does  the  wind  whistle  in  a  keyhole  ?     (  X  3) 

23.  Explain  how  the  frequency  of  an  '  organ-pipe  '  depends  on  the  dimen- 
sions of  the  pipe. 

If  a  pipe  sounds  256  when  the  temperature  of  a  hall  is  15°  C,  what  will 
be  the  frequency  at  20°  C.  ?     (  X  2) 

24.  Describe  the  construction  of  an  '  organ  pipe,'  and  give  the  relation 
between  the  lowest  pitches  obtainable  from  an  open  pipe,  a  stopped  pipe, 
and  a  stopped  pipe  blown  with  COg.     (  X  2) 

25.  Contrast  '  open  '  and  '  closed  '  pipes,  and  calculate  the  ratio  of  their 
lengths  for  the  third  overtone  of  the  open  to  be  of  the  same  pitch  as  the  second 
of  the  closed.     (  X  2) 

26.  A  slightly  conical  pipe  is  closed  alternately  at  little  and  big  ends;  how 
would  this  affect  the  frequency  of  its  fundamental  ? 

27.  Why  is  the  quality  of  a  string  different  from  that  of  an  organ  pipe 
closed  at  one  end  ? 

28.  To  what  is  ascribed  the  difference  of  quality  or  timbre  of  musical 
notes  sounded  on  different  instrimients  ?  How  can  the  explanation  be  tested 
experimentally  ? 

29.  Distinguish  between  the  loudness,  pitch,  and  quality  of  a  note,  and 
explain  their  physical  characteristics.     (  X  4) 

30.  Calculate  Young's  modulus  for  a  rod  1-72  m.  long,  density  8-5,  which, 
gripped  at  the  middle  and  stroked  lengthwise,  gives  1000  vibrations /sec. 


RESONANCE,   PIPES  363 


PRACTICAL  QUESTIONS 

Compare  two  forks,  and  load  the  faster  until  they  differ  by  2  vibrations 
per  second. 

Compare  two  forks  by  the  resonance  tube. 

Find  the  frequency  of  a  fork  by  ditto,  given  speed  of  soimd  at  0**  C. 
Find  speed  at  0°  C.  given  known  fork,  or  find  temperature  of  room. 
Find  the  mouth  correction  of  a  badly  choked  resonance  tube. 
Find  the  speed  of  sound  in  CO,. 


CHAPTER  XXX 
COMPLEX  VIBRATORS 


§451.  The  longitudinal  vibrations  of  rods  and  wires  are  the  only 
ones  that  lend  themselves  to  simple  theoretical  treatment.  They 
have  been  referred  to  in  §  443.  The  rod  is  held  in  the  middle,  or 
the  rod  or  wire  clamped  firmly  at  one  or  both  ends,  and  wiped 
lengthwise  with  a  wet  leather  or  rosined  cloth,  when,  without 
any  visible  vibration,  it  emits  an  unmusical  shriek.  Like  the 
air  in  a  pipe,  it  is  in  lengthwise  oscillation,  '  concertina  fashion,' 
for  a  pellet  hung  in  contact  with  the  flat  free  end  dances  off  when  it 
sounds,  Fig.  163.  The  shuddering  motion  of  rubber  tubing  pulled 
through  wet  fingers,  and  the  wet  and  dry  rings  left  on  it,  evidence  a 
slower  vibration  of  the  same  sort,  and  glass  tubing  can  be  set  into 
such  violent  motion  that  it  shatters  into  rings. 

The  thickness  of  the  wire  or  rod  makes  no  difference  to  the 
pitch,  each  square  millimetre  of  cross-section  (of  any  shape)  looks 
after  itself,  and  a  thick  rod  can  be  regarded  merely  as  a  bundle 
of  thin  ones  each  giving  the  same  note. 

Clamped  points  are  nodes.  Free  ends,  or  the  middle  point  when 
clamped  at  both  ends,  are  antinodes.  The  wave-length  in  the 
material  is  4AN  as  usual,  e.g.  bar  clamped  at  end  has  w.l.  =  four 
times  length,  and  the  harmonics  of  a  stopped  pipe ;  wire  clamped 
at  ends  has  w.l.  =  twice  its  length,  and  full  series  of  harmonics. 
The  speed  of  travel  of  the  longitudinal  disturbance  =  of  course 
the  speed  of  sound  in  the  material 

=  ^/  {Young's  moduliis  -^  density),  and  =  frequency  x  wave-length. 

For  instance,  the  Quartz  piezo-electric  plate  of  §§  157,  802,  837, 
vibrates  to  get  thicker  and  thinner  like  a  rubber  heel  when  jumped  on, 
concertina  fashion,  i.e.  has  its  wave-length  twice  its  thickness 
merely,  2  x  J  cm. ;  its  modulus  of  elasticity  in  that  direction  is 
6-8  X  1011  and  its  density  2-65. 

.-.  Speed  =  V(6-8  x  1011/2-65)  =  500,000  cm./sec. 

and  dividing  by  wave-length,  0-5  cm. 

Frequency  =  1,000,000  per  second. 

§452.  Transverse  vibrations  of  bars.  From  the  days  when  we 
essayed  tunes  on  a  row  of  pins  driven  to  different  depths  in  the 
desk,  we  have  all  been  familiar  with  the  sonorous  transverse  vibra- 
tion of  '  bars.'  In  the  little  clockwork  musical-box  there  was  a 
whole  row  of  them  in  a  '  comb  '  plucked  by  pegs  in  a  revolving 
barrel.     One  is  the  mainspring  of  that  curious  instrument,  the  Jew's 

364 


§453]  COMPLEX   VIBRATORS  865 

harp.  Worked  by  wind,  thin  '  bars  '  form  the  reeds  of  the  mouth- 
organ,  harmonium,  concertina,  etc.,  and  with  the  addition  of  reson- 
ance pipes,  of  the  clarionet,  the  organ,  etc. 

The  alternating-current  Frequency  Meter  presents  a  row  of  teeth 
in  a  black  mouth.  They  are  the  ends  of  steel  reeds,  tuned  to 
successive  frequencies,  49,  50,  51,  etc.,  all  in  line  above  an  electro- 
magnet carrying  the  A.C.  This  pulls  on  all,  but  only  the  one  in 
tune  works  up  to  a  strong  vibration,  and  its  whitened  end  draws 
out  into  a  conspicuous  long  tooth. 

The  modern  drawing-room  clock  chimes  tunefully  on  long  rods  of 
hard  bronze  struck  near  the  end  by  leather-faced  hammers. 

All  these  are  bars  clamped  at  one  end  and  free  at  the  other,  but 
in  the  tuning-fork  two  equal  bars  balance  each  other's  motion, 
and  clamping  is  unnecessary  :  a  bit  of  wax  stuck  on  one  leg  destroys 
the  balance,  and  the  fork  spends  its  energy  in  shaking  the  hand, 
and  soon  stops.  Undamped  also  are  the  straight  bars  of  the  har- 
monicon,  supported  (not  too  rigidly)  at  the  nodes  of  their  funda- 
mental vibration,  about  one-fifth  length  from  either  end — even  a 
bar  of  wood  develops  some  musical  talent  then,  while  Miss  Waller's 
magic  baton  of  '  dry  ice,'  §  272,  will  set  any  bit  of  metal  ringing 
loud  and  clear  as  a  bell,  provided  its  supports  touch  it  only  near 
nodal  points  of  minimal  motion. 

Additional  nodes  are  present  in  bars  sounding  overtones,  and 
can  be  demonstrated  by  scattering  sand  on  the  horizontal  bar. 
When  sounded,  the  sand  gathers  at  the  quiet  nodes.  In  this  way 
a  node  can  be  found  about  one-third  way  down  a  tuning-fork 
prong,  when  the  shrill  first  overtone,  more  than  six  times  faster 
than  the  fundamental,  is  ringing.  The  clock  gong  is  struck  near 
its  root,  and  overtones  ripple  along  it ;  its  fundamental  vibrations 
when  the  free  end  is  plucked  may  be  slow  enough  to  count. 

The  Overtones  of  bars  are  not  Harmonics,  for  they  are  not 
in  the  simple  ratios  2,  3,  4,  5,  etc.,  times  the  fundamental,  §440 ; 
none  the  less  the  sound  may  be  harmonious,  like  the  blacksmith's 
anvil. 

It  is  easy  to  see  that  the  thicker  and  stififer  [Young's  modulus] 
the  bar  is,  at  the  fixed  part,  the  greater  will  be  the  elastic  forces 
called  into  play  by  a  slight  bend.  This,  and  lightness  in  the  free 
moving  parts,  means  rapid  vibration.  Filing  a  fork  near  the 
tip  raises  its  pitch  ;  near  the  base,  lowers  it.  On  a  large  scale  all 
this  is  of  interest  to  engineers,  bridges  and  ships  under  '  live  ' 
forces  are  vibrating  bars,  and  turbine  discs,  propeller  blades,  etc., 
moving  at  high  speeds,  may  break  from  excessive  plate  vibration. 

§453.  Plates.  The  vibrations  of  plates  are  very  complex,  and 
numerous  patterns  of  nodal  lines  can  be  obtained  by  scattering  sand 
on  them  when  vibrating.  These  Chladiii*s  figures  are  usually  demon- 
strated on  a  square  metal  plate,  clamped  conveniently,  and  bowed 
somewhere  on  the  edge,  while  another  point  is  touched  by  the  finger 
to  induce  it  to  remain  a  node.     A  few  figures  are  given  in  Fig.  167, 


366 


SOUND 


[§453 


and  many  more  in  Tj^idall's  Sound.  Each  has  its  own  note. 
They  depend  on  where  the  plate  is  bowed  and  touched  and  how  it 
is  supported.  A  very  simple  mode  of  vibration  of  a  plate  fixed 
at  the  centre  produces  a  nodal  cross,  the  alternate  quadrants 
moving  opposite  ways.  The  S.W.  figure  shows  practically  the  1/5 
nodal  points  of  bars,  mentioned  above  ;  the  plate  may  be  supposed 
cut  into  strips,  either  way.  A  uniform  disc  supported  at  points 
on  a  circle  two-thirds  its  diameter,  and  struck  in  the  middle,  acts 
as  a  gong,  with  this  as  a  nodal  circle.  Jingling  coins  are  vibrating 
in  one  or  other  of  these  ways. 


^ 

\ 

^^ 

%■ 

i 

1 

%. 

/^■• 

'\. 

.#• 

■■v.^< 

1 

I     M     I 


Fig.  167. 


It  is  because  the  thin  mica  or  metal  diaphragms  of  telephones 
and  gramophones  can  vibrate  in  this  wide  variety  of  different  ways 
that  they  are  able  to  take  up  and  reproduce,  fairly  faithfully,  a 
sufficient  range  of  frequencies  of  ordinary  speech  ;  they  do,  however, 
hang  on  and  over-emphasise  particular  notes  of  their  own. 

The  stiff  card  cone  of  the  loud-speaker  leads  on  to  the  next  para- 
graph. 

§  454.  Gongs  and  bells.  The  ordinary  dinner-gong  is  a  plate 
with  a  turned-up  edge,  the  stiffness  and  extra  weight  of  which 
bring  a  nodal  circle  out  to  the  suspending  string  holes. 

Bells  can  be  looked  upon  either  as  deeply 
'  dished '  plates,  or  as  short  cylinders.  When 
struck  in  the  ordinary  way,  the  circular  mouth 
becomes  elliptical,  and  vibrates  between  this  and 
an  ellipse  at  right  angles  to  it.  Fig.  168.  Four 
points  90°  apart  are  moving  radially,  and,  since 
the  outer  arcs  are  longer  than  the  flat  inner 
arcs,  the  points  at  45°  have  to  move  tangentially. 
Hence  the  tangential  drag  of  the  wet  finger  on  a 
tumbler  rim  evokes  its  note.  These  45°  positions 
are  nodal  '  meridians  '  (diameters  in  plan) ;  pellets  hung  in  contact 
with   the   bell  there   are   not   driven   off.      Besides   this    motion 


Fig.  168. 


§456]  COMPLEX   VIBRATORS  367 

characteristic  of  a  cylinder,  the  bell  has  also  nodal  circles  like  a  plate, 
the  whole  rim  flapping  up  and  down,  and  making  the  bell  alter- 
nately shorter  and  taller. 

The  five  partials  of  the  best  modem  English  bells  appear  to  run 
as  near  the  frequency  ratio  0-25  :  0-5  :  0-6  :  0-8  :  1  as  the  founder  can 
get  them.  The  highest  is  the  loudest  after  the  usual  hard  blow,  and 
gives  the  bell  its  name.  The  lowest  two  are  heard  in  a  muffled  peal. 
The  0-25  has  4  nodal  meridians,  the  0-5  a  nodal  circle  in  addition, 
and  the  1  has  8  meridians.  The  beating  heard  as  the  sound  dies 
away  originates  from  accidental  irregularities  in  the  bell. 

Chimes  of  vertical  steel  Tubes,  8  diam.  long,  slung  on  a  rope 
through  two  holes  about  one-eighth  below  the  top,  and  struck 
on  the  top  edge,  cost  little  more  than  a  solitary  monotonous  church 
bell,  and  sound  very  well,  only  the  ringers  will  play  hymns. 

§455.  Membranes.  Membranes  are  to  plates  what  strings  are 
to  bars,  their  power  of  vibration  is  due  to  the  tension  put  upon  them, 
and  not  to  natural  stiffness.  Their  vibration  bears  some  general 
resemblance  to  that  of  plates,  and  can  be  studied  experimentally 
by  scattering  sand  on  them  in  the  same  way.  The  blow  of  a  drum- 
stick on  the  drum-head  can  be  likened  to  the  fall  of  a  drop  into  a 
teacup  ;  circular  ripples  flow  out,  and,  reflected  at  the  fixed  circular 
edge,  return  and  produce  nodal  circles  as  the  disturbance  continues. 
The  soft  stroke  of  a  '  muffled  '  stick  smothers  out  short  waves  and 
dulls  the  tone.  In  Sedley  Taylor's  Phoneidoscope  a  soap  film  is 
stretched  over  a  cup  sung  into  through  a  short  speaking-tube,  and 
shows  the  different  and  beautiful  nodal  patterns  in  brilUant  colours 
for  every  note. 

A  membrane  with  free  edge  which  you  know  quite  well  is  the  grass- 
blade  between  the  thumb -knuckles  of  the  hands,  closed  as  in  supplica- 
tion. Its  squall  when  blown  on  is  probably  pitched  by  their 
resonating  cavity. 

These  two  types  of  membrane  will  be  recognized  in  the  Ear  and 
the  Vocal  Cords. 

§  456.  The  Voice.  Stretched  across  the  windpipe  are  two  mem- 
branes of  fine  elastic  fibre,  the  vocal  cords,  very  roughly  imitated 
by  the  strips  of  rubber  in  Fig.  164  E.  When  breathed  through 
and  tightened  so  that  the  '  glottis  '  between  them  becomes  a  narrow 
slit,  they  vibrate.  For  high  notes  they  are  very  tight,  and  only  the 
thin  edges  quiver.  The  two  resonating  cavities  of  the  pharynx 
and  mouth,  divided  by  the  tongue,  control  the  pitch,  and  the  tongue, 
teeth,  and  lips,  the  articulation,  of  the  sound. 

Shouting  with  wide-open  mouth  means  over-blowing  and  strain. 
A  Megaphone  becomes  useful  now,  for  in  it  a  narrow  conical  mass 
of  air  first  receives  all  the  energy  formerly  spread  out  almost  spheric- 
ally ;  its  vibrations  have  therefore  much  greater  amplitude,  and  at 
the  nearly  nodal  reed  end  (§450)  provide  a  greater  back  pressure. 
This  supports  the  vocal  cords,  and  enables  them  to  be  blown  very 


368  SOUND  [§  456 

hard  without  injury.  You  work  harder.  The  directive  action  of 
a  megaphone  is  limited,  for  the  5-ft.  waves  of  a  man's  voice  diffract 
widely  from  its  1-ft.  aperture,  §  401. 

Those  unfortunates  who  have  lost  their  vocal  cords  need  no  longer 
suffer  total  loss  of  speech.  An  '  Artificial  Lar3mx  '  has  been  con- 
trived for  them  ;  a  pipe  like  a  bent  forefinger  is  hitched  into  one 
corner  of  the  mouth  ;  it  has  a  reed  in  its  end,  blown  by  a  little  bellows 
carried  under  the  arm,  and  sounding  F  sharp,  or  thereabouts. 
In  this  monotone  the  user  talks,  with  tongue  and  teeth  and  lips, 
almost  as  well  as  any  man  with  a  pipe  in  his  mouth. 

In  vocalization,  some  hold  that  the  vocal  cords  do  not  vibrate 
as  membranes  with  free  edge,  but  act  simply  as  a  valve,  opening 
and  closing  with  a  succession  of  '  glottal  snaps  ' — that  manly 
crackliness  in  the  voice  which  is  so  reassuring  to  the  relatives  and 
so  priceless  at  the  bedside — and  that  the  resonating  cavities  vibrate 
to  these  sharp  impulses  as  an  electric  bell  vibrates  to  its  hammer, 
producing  short,  heavily  damped  trains  of  waves,  of  frequencies 
controlled  by  shape  and  aperture  of  the  cavities, 
and  adding  up  to  the  varied  tones  of  the  voice. 
The  resonating  cavities  are  three,  intercon- 
nected as  suggested  in  Fig.  169.  Above  the 
vocal  cords  is  the  pharynx  PH  separated  by  the 
large  moving  mass  of  the  tongue  from  the  mouth 
M,  more  or  less  open  except  in  humming  ;  and 
above  the  palate  is  the  nasal  cavity  N  normally 
open  at  the  end  except  in  a  bad  cold — the  effect 
of  that  is  easily  heard  experimentally  by  pinching 
your  nose  ;  it  suppresses  '  nasals.'  That  of  the 
labial  aperture  you  can  test  by  scratching  the 
Fig.  169.  cheek,  §  441  ;  while  you  will  quite  likely  observe 

that,  after  listening  for  some  time  to  a  piece  of 
music,  your  tongue  has  taken  up  some  set  position  in  your  mouth, 
dependent  on  the  key-note,  varying  the  two  volumes  of  pharynx 
and  mouth  resonators,  and  ready  for  you  to  join  in. 

Sir  Richard  Paget  has  not  only  drawn  up  a  chart  showing  the 
resonances  to  two  notes  in  each  of  the  many  individual  EngUsh 
vowel  sounds,  but  has  also  made  models  capable  of  sounding  single 
vowels  when  their  simple  reeds  are  blown,  and  has  even  succeeded, 
by  sounding  a  note  into  the  linked  hollows  of  his  two  skilful  hands, 
in  making  their  cavities  vaguely  articulate  '  I  do  like  London.' 
See  his  '  Human  Speech.' 

The  consonants  are  not  mere  beginnings  and  endings  of  vowel 
sounds.  As  you  know,  many  deaf  mutes  manage  cacophonous 
vowels,  the  Scottish  weaver  describes  his  three  qualities  as  '  oo,' 
'  aw  oo,'  and  '  aw  aye  oo,'  and  in  the  familiar  speech  of  your  own 
countryside,  consonants  are  largely  missing.  They  are  a  higher 
development,  and  are  sounds  of  much  higher  frequency,  up  to 
6000  for  S,  and  it  is  because  commercial  telephony,  from  electrical 
limitations,  shuts  down  at  about  frequency  2400,  that  it  still  depends 


§457]  COMPLEX   VIBRATORS  369 

to  such  a  large  extent  on  the  human  power  of  guessing  from  scanty 
clues.  In  everyday  conversation,  indeed,  we  hear  three-tenths, 
or  less. 

§  457.  The  Ear.  In  the  Ear  the  air-waves  fall  upon  a  stretched 
tympanic  membrane  T,  Fig.  170,  of  area  1-3  sq.  cm.  In  structure 
and  tension  this  resembles  a  garden-spider's  web,  and  it  can  be 
punctured  locally  in  the  same  way  without  destroying  its  usefulness, 
i.e.  the  hearing.  Attached  to  it  is  the  handle  of  the  hammer-bone, 
which  articulates  with  the  anvil-bone  ;  these  two  (together  weighing 
50  mg.)  reduce  the  amplitude  of  the  motion  to  two-thirds,  and  trans- 
mit it  through  a  stirrup-bone  to  the  membrane  covering  the  '  oval 


lSooo^°^Ji9oo^'>°°    'o?**    *^°  »oo     too       too 

Fig.  170. 


"^  307nm  ^ 


window  '  0,  of  area  0-028  sq.  cm.,  every  square  mm.  of  which  there- 
fore gets  3/2  X  1  •3/0-028  =  70  times  the  force  of  the  original 
motion. 

The  dotted  line  passes  through  the  attachments  of  the  bones  to 
the  skull ;  these  are  not  in  the  plane  of  the  paper.  Their  ligaments 
maintain  their  contact ;  and  also  a  slight  pull,  through  the  handle, 
upon  T  ;  and,  that  this  may  not  be  interfered  with  by  changes  of 
atmospheric  pressure,  the  Eustachian  tube  E  passes  to  the  pharynx, 
swallowing  movements  in  which  open  it  from  time  to  time,  so  putting 
both  sides  of  T  in  communication  with  the  atmosphere.  Driving 
up  a  long  hill,  the  reduction  in  barometric  pressure  brings  T  outwards, 
slacking  the  bones,  and  you  think  how  very  silently  the  car  is 
climbing,  until  a  chance  swallow  suddenly  restores  noises  :  running 
downhill  fast,  you  swallow  again,  to  relieve  a  pressing  pain  in  the 
ears.  E  is  apt  to  get  stopped  by  swelling  during  a  cold  in  the  head, 
and  then,  if  you  are  unwise  enough  to  blow  your  nose  violently 
(thereby risking  blowing  infective  material  through  E  into  the  middle- 
ear,  where  it  incubates  and  causes  intense  pain,  and  must  be  let  out 


370 


SOUNt) 


t§457 


by  puncturing  T  low  down,  lest  it  eat  away  0  and  R  and  leave  a 
dry  deaf  singing  ear),  T  blows  out,  but  the  bones  unlock  and  save 
O  from  destruction,  and  then  you  must  swallow  or  squeak  to  put 
things  right. 

The  Cochlea  is  a  whelk- shell  structure,  of  which  two  half -turns 
are  shown  in  section  in  Fig.  171,  and  the  whole  length,  unrolled,  in 
Fig.  170.  It  is  full  of  liquid,  which  takes  up  the  intensified  force 
at  O  ;  R  is  the  pressure-relieving  '  round  window.'  Along  the 
cochlea,  and  dividing  it  into  two  compartments,  stretches  the  long 
narrow  tapering  Basilar  Membrane,  shown  diagrammatically  in 
edge  and  in  plan,  Fig.  170,  in  enlarged  cross-section  in  the  lower 

Fig.  171,  and  just  identifiable  at  the 
outer  edge  of  the  shelves  in  the 
upper  figure. 

It  is  composed  of  joined  parallel 
fibres  which  are  all  on  the  stretch, 
like  the  wires  of  a  piano.  Like 
them,  they  resound  locally  to  the 
vibrations  pervading  the  liquid : 
though  exceedingly  small  and  light, 
they  are  so  loaded  with  liquid  and 
with  the  structures  shown — practi- 
cally of  the  same  density  as  the  liquid 
—that  for  them  (1/2  I)  X  VC^M 
has  the  ordinary  range  of  frequency. 
The  plan  of  the  membrane  in  Fig. 
170  is  marked  with  its  known  fre- 
quencies of  response,  in  man. 

Fig.  171,  below,  shows,  in  black, 
how  the  '  rods  of  Corti '  rest  on 
the  basilar  membrane  like  the 
rafters  of  a  long  weaving-shed 
roof ;  packed  on  to  them  are  the  ranks  of  inner  and  outer  '  hair- 
cells,'  nerve  cells  which  present  tactile  hairlets  at  their  upper  ends, 
and  from  their  lower  ends  send  nerve  fibres  inwards,  these  all 
running  side  by  side  in  the  great  central  '  columella,'  shown  on  the 
right,  to  the  brain.  Lying  loosely  over  the  hairs  of  this  '  organ  of 
Corti,'  like  a  duster,  is  the  limp  '  covering  membrane.' 

Undoubtedly  what  happens  is  that  particular  short  sections  of 
the  basilar  membrane  are  excited  to  resonant  vibration,  their 
tactile  hairs  are  thereby  tickled  against  the  lax  covering  membrane, 
and  this  stimulus  is  sent  to  the  brain.  Local  experimental  destruc- 
tion of  the  nerves  of  the  organ  of  Corti  causes  deafness  to  some 
particular  range  of  frequency. 

Incidentally,  the  rest  of  the  inner  ear  is  our  organ  of  balance. 
A  tube,  full  of  liquid,  with  a  bulb  at  one  end,  is  bent  into  a  semi- 
circle. In  the  bulb  is  a  prominent  bank  of  hair  cells,  embedded  in 
mucus,  in  which  are  scattered  calcareous  granules  called  otoliths. 
Rotation  of  the  head  in  the  plane  of  the  semicircular  canal  causes 


Fig.  171. 


§468]  COMPLEX   VIBRATORS  371 

the  liquid,  by  its  inertia,  to  '  wash  '  against  one  or  other  side  of  this 
bank,  disturbing  the  hair-cells  there.  Each  ear  contains  three 
canals  at  right  angles,  in  charge  of  the  three  component  axes  of 
rotation,  but  when  one  ear  is  destroyed  I  find  that  the  other  takes 
a  year  or  two  to  gain  complete  control. 

Prawns  etc.  have  sand  grains  as  otoliths  in  their  otocysts,  at  the 
base  of  the  small  feelers,  and  shed  them  when  they  moult.  Being 
given  only  iron  filings  with  which  to  replace  them,  they  have  been 
persuaded  to  turn  into  all  sorts  of  positions  by  dragging  on  these 
with  a  magnet. 

§  458.  Concord  and  Discord  have  a  physical  basis  which  we  must 
inquire  into,  though  their  complete  discussion  soon  becomes  psycho- 
logical :  the  halting  harmonies  of  jazz  please  many  who  yawn  over 
the  sweetest  symphonies. 

In  the  first  place,  it  is  to  be  noticed  that  the  first  note  must  be 
hanging  in  the  air,  or  in  the  memory,  when  the  second  is  struck. 
Without  this,  odd  notes  are  often  not  unpleasant.  A  child,  slowly 
picking  out  a  tune  note  by  note,  and  striking  a  wrong  one,  hurts 
nobody  :  played  at  speed,  the  same  blunder  is  intolerable.  The 
Arab  has  evolved  a  musical  scale  different  from  ours,  and  one 
presenting  more  opportunities  for  discord,  but  the  desert  music 
of  the  tent  door,  and  the  bagpipes  on  the  brae,  can  charm  us  by 
those  same  imperfect  harmonies  which  unfit  them  for  the  prolonging 
echoes  of  the  aisle  or  the  concert-room.  And  again,  if  these  are  too 
long,  one  is  no  more  pleased  than  with  a  pianist  who  treads  too  much 
on  the  forte  pedal. 

The  basis  is  not  the  old  dictum  that  the  simplest  pitch  ratios, 
2:1,  3:2,  etc.,  produce  the  best  concords  '  because  numerical 
simplicity  is  charming.'  Logically,  that  would  give  us  orchestras 
exclusively  composed  of  tuning-forks,  the  charm  of  which  would 
fade  in  five  seconds. 

The  fact  is,  not  only  is  every  note  in  music  complete  with  overtones, 
but  between  notes  there  are  Difference  Tones.  To  hear  these,  the 
very  best  way  is  to  get  two  tin-whistles,  and  paper  over  all  the  holes 
except  the  third  and  fourth  on  each.  Put  both  in  your  mouth  at 
once,  leave  only  the  third  holes  open,  and  blow  :  whirring  beats 
will  be  heard,  for,  though  nominally  the  notes  are  identical,  exact 
tuning  is  not  to  be  had  at  the  price.  But  now  open  the  third  hole 
on  one  and  the  fourth  on  the  other,  and  you  will  hear  a  growl,  as 
the  beats,  now  too  fast  to  hear  as  a  tremolo,  blend  into  a  note. 
Referee's  double  whistles  work  in  this  way,  and  some  yachts'  whistles ; 
the  heavy  note  gives  body  to  the  sound. 

The  frequency  of  this  difference  tone  is,  just  like  beats,  the 
difference  of  those  of  the  primary  notes  between  which  it  is  produced. 
You  can  prove  this  on  a  harmonium,  or  piano  with  forte  pedal 
down,  by  sounding  a  high  C  and  G,  when  the  C  below  sounds  out, 
one  being  twice  and  the  other  three  times  its  frequency  (and  try 
others). 


^n  SOUND  [§  458 

The  shrieks  and  howls  which  are  our  neighbours'  chief  joy  in 
wireless  are  the  difference  tones  between  the  incoming  signals  and 
the  receiver's  own  oscillations,  both  of  course  of  high  radio-frequency, 
and  therefore  quite  inaudible.  The  first  high  shriek,  10,000  or 
more,  is  faint  because  the  primary  frequencies  are  far  apart,  and  also 
because  the  ear  is  not  very  sensitive  there  ;  then  comes  a  descending 
and  increasing  roar  until  close  tuning  quiets  it  (cf.  §  441),  then  the 
reverse  process  as  one  passes  off  to  another  station. 

Difference  Tones  occur,  then,  whenever  we  have  notes  or  har- 
monics of  some  power,  and  not  far  apart.  Take  two  chance  notes, 
write  down  their  harmonics,  and  look  for  smallish  differences  which 
will  give  rise  to  them. 

200  400  600  800  1000         1200 

276  552  828  1104 

DifEerence      .  .  76  48  28  96 

Ignoring  the  last,  as  far  and  faint,  leaves  three  ugly  low  notes  which 
fit  in  nowhere,  neither  they  nor  their  multiples.  One  reasonably 
supposes  that  their  low  grumbling  distracts  the  ear,  whether  in  the 
organ  of  Corti  or  in  the  brain,  and  spoils  its  appreciation  of  the 
two  notes,  much  in  the  same  way  as  the  mutter  of  conversation 
spoils  one's  enjoyment  of  a  concert. 

For  contrast,  take  two  notes  of  the  diatonic  scale 


c 

4/3  C  =  F  . 

261 
348 

522 

843 

696 

1044 
1044 

1305 
1392 

1566 

1827 
1740 

Difference 

87 

143 

0 

87 

87 

Here  C  is  the  second  and  F  is  the  third  harmonic  of  the  difference 
tone  87  itself,  and  concord  is  veritably  built  upon  it,  even  the  aberrant 
143  is  very  nearly  5/3  of  87,  or  A  in  the  key  of  C  87. 

§  459.  The  Musical  Scale.  So  that  taking  a  numerically  simple 
ratio,  4/3,  while  there  is  nothing  ethical  about  it,  has  certainly 
brought  us  luck.  Come  on,  here  is  an  octave  which  wants  filling 
with  notes  ;  let  us  have  a  round  dozen.  And,  as  we  have  just  seen. 
Music  is  a  matter  of  Ratios  ;  arithmetical  differences  are  the  wolves 
that  lie  in  wait  for  the  unwary. 

Take  6  in.  of  squared  paper,  Fig.  172  ;  each  J-in.  must  represent 
6%  more  frequency  as  you  go  to  the  right,  for  at  6%  compound 
interest  (strictly  5-95)  money  doubles  itself  in  a  dozen  years. 

Fora;i2  =  2  12  log  a;         =  log  2 

12  X  00251  =  0-301  and  0-0251  =  log  1-06 

C  is  on  the  left,  c  =  2C  is  6  in.  to  the  right  of  it ;  where  is  the  next 
simplest  ratio,  3/2,  to  be  put  ?  in  how  many  years  does  money 
3/2  itself  ? 


§459] 
1.06"  =  3/2 


COMPLEX   VIBRATORS 


373 


2/ log  1-06  =  log  1-5 
y  X  00251  =0176        .'.y  =  l 
i.e.  3/2  is  at  the  seventh  half-inch,  G. 

Work  out  in  the  same  way  places  for  4/3  and  5/3  ;  for  5/4  (6/4  = 
3/2)  and  7/4  ;  for  6,  7,  8,  and  9  fifths  ;  for  7  and  11  sixths  ;  for  all 
the  sevenths,  the  unoccupied  eighths,  ninths,  tenths,  etc.  Here 
they  are  : 


1 

1 

. 

fi 

11 
10 

3 

>9 

10 

9 

3 

13 
9 

9 

i 

9 

2 

» 

If 

e 

? 

8 

1 

? 

? 

7 

ft 
7 

? 

1 
6 

»f 

c 

6 

? 

7 
? 

f 

1 

1 

3 

§ 

1 

3 

2. 

z 

F  G 

Fig.  172. 


You  see  the  4/3,  5/3,  and  5/4  jump  neatly  into  place  and  are  labelled 
FAE,  but  the  fifths,  sixths,  and  sevenths  are  very  contrary,  6/5  and 
7/6,  7/5  and  10/7,  11/7  and  8/5,  7/4  and  9/5  stand  competing  for 
four  of  our  musical  chairs.  By  the  eighths  we  have  got  desperate, 
and  down  come  D  and  B,  and  then  we  jam  the  others  into  half-way 
places,  invent  another  between  C  and  D,  dub  them  sharps  or  flats, 
and  paint  them  black  ;   and  there  is  an  Octave  on  the  piano. 

But  can  we  do  this  violence  to  our  true  ratios  ?  We  have  to, 
the  family  grows  too  big.  And  fortunately  the  latitude  Nature 
allows  in  everything  does  not  fail  us  here,  it  is  only  a  hypersensitive 
ear  that  is  offended  by  squeezing  F  sharp  and  G  flat  into  one  note, 
and  so  on.  Icily  perfect  exactness  is  as  uninteresting  in  music 
as  in  billiards  or  boxing  or  anjrwhere  else.  Make  a  tuning- 
fork  of  your  dinner-fork,  hit  it  either  side,  or  pinch  any  pair  of 
prongs,  and  it  rings  the  same  note  ;  when  you  look  at  them,  every 
prong  is  a  trifle  different  in  size  and  shape,  and  must  be  wanting 
to  vibrate  on  its  own,  yet  they  manage  to  agree. 

Thus  we  have  arrived  at  the  Diatonic  Scale,  with  frequencies  in 
the  ratios  to  the  keynote. 

CD  EFG  A  Br 

1         9/8        6/4        4/3        3/2        5/3       15/8        2 


do 


fa 


•    la 


do 


374  SOUND  [§  459 

for  instance,  in  the  New  Philharmonic  Pitch  now  in  vogue 


c 

D 

E 

F 

G 

A 

B 

c 

261 

294 

326 

348 

392 

435 

490 

622 

though  it  makes  no  difference  to  the  ratios  what  the  actual  frequency 
of  the  keynote  is  :   addition  and  subtraction  have  no  place  here. 

By  putting  in  sharps  or  fiats  (theoretically  16/15  or  15/16  the 
nominal  note)  in  the  *  places,  this  gets  back  to  the  complete 
Chromatic  Scale  we  have  already  contrived  for  the  keyboard. 

It  must  be  possible  on  the  keyboard  to  play  a  piece  of  music 
equally  well  in  any  key,  i.e.  starting  from  any  note  (though  the 
asterisks  are  all  black  in  the  key  of  C  only),  and  the  only  way  of 
effecting  this  is  to  make  all  the  intervals  equal,  i.e.  all  lie  on  the  half- 
inch  lines  on  the  diagram,  in  *  Equal  Temperament.^  How  near 
your  piano  is  actually  tuned  to  this  depends,  of  course,  on  your 
tuner,  and  on  what  has  happened  since  he  came — Nature  does  allow 
latitude — but  the  intention  is,  that  the  keyboard  is  a  '  logarithmic 
scale  '  in  which  each  note  rises  by  6%  very  nearly  ;  so  that  if  at 
your  birth  your  happy  father  invested  £33,  the  lowest  note  on  a 
concert  grand,  in  a  happier  land  where  interest  is  6%  and  income 
tax  is  2d.  in  the  £,  and,  happiest  of  all,  you  survive  to  the  good  age 
of  eighty-four,  your  heir-at-law  will  prove  for  the  top  note,  £4224. 

Kound  the  actual  Pitch  of  the  key-note  a  battle-royal  has  raged 
among  musicians  for  300  years.  By  1833  the  orchestra  had  pressed 
the  pianoforte  closely  to  its  heart,  and  then  found  that,  although 
carefully  tuned  to  the  wind  instruments  during  cold-morning 
rehearsals  in  the  empty  saloon,  it  was  much  too  flat  in  the  evening, 
when  a  crowded  audience  and  the  blazing  heat  of  the  new  brilliant 
coal-gas  had  run  up  the  temperature,  expanded  and  slackened  the 
wires,  and  simultaneously  sharpened  all  the  wind  proportionally 
to  ^^T.  So  the  piano  had  to  be  strung  up,  and  then  the  wind 
instruments  were  nowhere  in  the  mornings,  and  so  on,  and  of  course 
nobody  knew  a  numerical  value  for  the  frequency  in  those  days. 

Among  singers,  too,  '  the  favourite  tenor  of  the  King  of  Wurtem- 
berg  had  a  high  C  of  remarkable  power  and  quality  :  he  represented 
to  the  King  that  the  modern  pitch  imposed  too  great  a  strain  upon 
him  ;  H.M.  consented  to  sacrifice  the  necessary  number  of  vibrations, 
and  the  Stuttgart  pitch  was  invented.  But  no  sooner  was  the 
decree  pronounced  than  the  tenor  found  his  former  electrical  effect 
was  wanting  ;  he  begged  for  the  restoration  of  the  old  diapason,  and 
restored  it  was.' 

The  Royal  College  of  Music,  and  Kneller  Hall,  give  the  following 
frequencies  for  upper  C  in  Musical  Pitches  of  the  present  day  : 


Concert        ..... 

528 

A  =  440 

Crystal  Palace  Philharmonic 

538 

Covent  Garden     .... 

540 

A  =  450 

Diapason  normal  (Fr.) 

517 

New  Philharmonic  and  Military 

522 

and  this  last  gives  A  439,  adopting  Equal  Temperament,  and  B  flat,, 
important  for  band  instruments,  465,  tuning  them  at  68°  F. 


§460]  COMPLEX   VIBRATORS  376 

§460.  Supersonics.  The  lower  limit  of  frequency  at  which 
aerial  impulses  blend  into  a  musical  note  is  somewhere  between 
16  and  30,  or  practically  that  at  which  '  flicker  '  ceases  to  the  eye, 
§  614.  Below  that,  the  great  pedal  pipes  of  the  organ  merely  shake 
the  building,  and  the  two-stroke  motor  still  stutters. 

The  upper  limit  is  about  30,000,  rising  for  very  intense  sources, 
and  dependent  a  good  deal  on  the  individual  ear.  It  is  said  that 
the  difficulty  of  picking  up  a  foreign  language  increases  as  one  grows 
older  because  the  finer  edges  of  the  sounds  become  less  audible, 
and  that  deterioration  of  the  ear  in  this  respect  begins  at  the  age  of 
three.  The  limit  is  measurable  by  the  little  whistles  of  Fig.  161, 
the  chief  difficulty  being  to  distinguish  the  faint  shriek  in  the  midst 
of  the  rustling  hiss  of  escaping  air.  Their  frequency  is  then  33,000  -f- 
wave-length,  measured  as  in  Fig.  162 ;  the  shding  plug  can,  of  course, 
be  graduated  once  for  all  by  trial  experiments. 

But  above  this  must  lie  a  wide  range  of  frequency  made  accessible 
nowadays  by  converting  into  mechanical  movement  the  oscillations 
of  electrical  circuits,  §  833,  as  described  in  §  837,  etc.,  or  by  using 
the  oscillating  current  to  magnetize  an  iron  ring,  for  iron  changes 
shape  a  very  trifle  on  magnetization. 

The  quartz  or  iron  is  immersed  in  insulating  oil,  and  with  from 
2  to  10  h.p.  in  circuit,  sets  it  into  exceedingly  small  but  very  intense 
supersonic  vibration,  with  frequencies  from,  say,  50,000  up  to  a 
million. 

For  the  power  employed,  the  results  are  not  impressive,  but  they 
are  curious.  The  oil  silently  rises  in  a  little  heap,  even  in  a  miniature 
fountain,  like  big  rain-drops  make  in  a  puddle.  It  soon  gets  hot. 
Oil  drops  in  water,  contained  in  a  test-tube  dipped  into  the  vibrating 
oil,  become  shaken  into  a  milky  emulsion  :  per  contra,  Indian  ink 
has  been  reconverted  into  lamp-black  and  water.  Perhaps  the  most 
striking  experiment  is  this  :  a  2-ft.  glass  rod  is  stood  upright  on 
the  quartz  plate — there  is  nothing  to  see  or  hear,  but  the  rod  is  not 
nice  to  touch.  If  its  upper  end  is  now  softened  in  a  flame,  it  loses 
its  elasticity,  and  can  no  longer  reflect  back  the  vibrations  sent 
up  from  below — their  energy  remains  in  the  plastic  material,  and  it 
continues  to  soften  and  run,  the  rod  melting  away  like  a  guttering 
candle. 

For  these  supersonic  frequencies  are,  of  course,  approaching  those 
of  radiant  Heat,  §961,  and  a  sort  of  '  death -ray  '  experiment,  in 
which  a  mouse  exposed  between  supersonic  plates  lost  its  tail,  and. 
ailing  apparently  nothing,  died  within  a  day  or  two,  may  be  due 
to  its  being  baked. 

Under  water,  with  a  speed  of  travel  of  a  mile  per  second, 
a  frequency  of  half-a-million  gives  waves  l/8th  in.  long,  and 
a  fairly  broad  company  of  such  waves  will  travel  for  long  distances 
without  excessive  loss  from  diffraction  at  the  edges,  §400,  and  will 
be  sharply  reflected  and  definite  in  direction  within  a  degree  or  two, 
instead  of  the  '  somewhere  over  there  '  of  long  sound-waves.  Con- 
sequently they  have  been  developed  for  spotting  submarines ; 
the  victim  of  the  searching  beam  hears  nothing,  the  hunter  hetcro- 


376  SOUND  [§  460 

dynes  it  into  audibility  on  its  return.  Doubtless  they  are  also  being 
used  in  the  present  furious  campaign  for  the  complete  extermination 
of  whales  :  more  peaceably,  they  actuate  a  noiseless  depth -sounder, 
cf .  §  414. 


I 


EXAM   QUESTIONS,   CHAPTER   XXX 

A  chapter  which  appears  to  offer  some  respite  from  examiners,  and  may 
help  to  annoy  your  musical  friends. 

1.  How  would  you  show  that  there  are  upper  and  lower  limits  of  musical 
pitch  audible  to  the  human  ear  ? 

If  you  had  reason  to  believe  that  a  vibration  of  inaudibly  high  frequency 
were  pervading  the  air,  what  tests  would  you  make  to  investigate  this  ? 

2.  What  is  the  approximate  normal  range  of  audible  frequencies  of  a  musical 
note  ?  How  may  vibrations  of  high  frequencies  be  produced  and  detected  ? 
What  other  factors  besides  pitch  govern  the  audibility  of  a  sound  ? 


LIGHT 

CHAPTER  XXXI 
ILLUMINATION 


§471.  Space  contains  Matter  and  Energy.  Some  of  the  energy, 
as  we  have  seen,  is  attached  to  the  matter,  some  is  free,  or 
radiant — '  Radiation.' 

Matter  stays  fairly  still :  Radiant  Energy  flies  about  all  the  while 
at  the  highest  possible  speed,  3  X  lO^o  cm.  per  sec,  300,000  km., 
186,000  miles.  Always  it  possesses  a  wave-structure,  which  we 
can  detect  and  measure  :  the  waves  are  in,  and  of,  the  moving 
radiant  energy. 

Dead-cold  dark  empty  Space  contains  only  emptiness  :  do  not 
afflict  your  brain  with  any  other  supposition. 

Radiation  of  rather  high  quality  we  see,  as  Light, 

* .  .  .  gladdening  Light,  of  His  pure  glory  poured 
Who  is  the  Immortal  Father,  heavenly,  blest.' 

Its  most  noticeable  waves  are  about  a  fifty-thousandth  of  an  inch 
long,  half  a  micron.  A  pinhole  is  about  a  thirtieth  of  an  inch 
wide,  850  microns.  These  lines  of  print  are  7-5  to  the  inch, 
so  that  if  they  were  20  ft.  long  they  would  be  like  the  waves  of  light 
surging  through  a  pinhole.  Narrow  as  we  have  made  it,  this  '  ray  ' 
or  '  beam  '  of  light  is  still  a  broad  procession  flowing  forward 
straight,  §400,  Fig.  131,  fraying  only  a  little  at  the  very  edges. 
First  we  will  march  with  the  regiment — in  bulk.  Light  travels  in 
straight  lines — later,  we  shall  attend  to  the  straggling. 

§472.  In  the  Pinhole  Camera,  straight  rays  from  lighted 
objects  pass  through  a  small  hole,  and  form,  on  a  plate  beyond, 
an  inverted  picture  of  them.  For  the  light  that  each  part  of  the 
plate  gets,  from  the  one  small  patch  of  the  object's  surface  facing 
it  through  the  hole,  is  proportional  to  the  brightness  of  that  surface, 
and  hence  the  light  and  shade  and  colour  of  the  object  are  reproduced. 

A  small  hole  in  a  card,  and  a  candle  flame,  enable  one  to  show 
that  the  shape  of  the  hole  does  not  matter  much  :  the  sun  shining 
through  the  irregular  gaps  in  foliage  throws  rounded  patches  on 
the  ground  ;  or  quaint  crescents  when  in  partial  eclipse  ;  Fig.  173 
left,  below. 

The  roundness  of  the  dots  in  a  '  half-tone  block  '  is  due  to  the 

377 


378 


LIGHT 


[§472 


same  cause.  Half  an  inch  in  front  of  the  plate  in  the  camera  is 
a  screen  ruled  with  opaque  cross  lines,  perhaps  120  per  inch,  leaving, 
of  course,  square  transparent  spaces.  But  the  light  coming 
through  each  of  these  forms  an  image  of  the  bright  round  window 
of  the  camera,  the  lens. 

Too  large  a  hole  causes  '  penumbral '  haze ;  but  a  very  small 
hole  also  gives  a  hazy  picture,  the  spreading  at  the  edges,  §  400,  has 
begun  to  show  too  much. 

Using  a  pinhole  the  diameter  of  which  is  1 /300th  its  distance 
from  the  plate,  one  can  obtain  photographs  which  define  everything 
beyond  a  foot  from  the  pinhole  softly  but  clearly,  and  give  a  more 
pleasing  solidity  in  the  stereoscope  than  do  sharper  lens  photographs. 


V( 


IP^- —     ' 

j>L_J    PENUMBRAL 

lo   umbrXT 

fc=>z     ^O 

1 

^^^     SPACE.    ^(^ 

1 
1 

^^^"----^Q 

Fig.  173. 

3         ^^- 

§  473.  Shadows.  The  terms  transparent,  translucent,  and 
opaque  are  familiar.  Opaque  bodies  obstruct  light  and  cast  shadows. 
If  the  source  of  light  is  small,  the  shadows  are  sharp,  at  their  edge 
is  sudden  change  from  light  to  darkness  :  such  are  the  unpleasant 
contrasts  produced  by  a  bare-wire  lamp.  Usually  shadows  are 
softer,  with  hazy  edges.  The  shadow  on  the  dial  is  scarcely  as 
sharp  as  a  pencil  line,  the  shadow  of  the  eaves  can  be  traced  quite 
definitely  enough  with  a  stick  ;  its  edge  spreads  very  hazily  indeed 
when  a  thin  cloud  blurs  the  sun. 

In  Fig.  173,  L  is  a  broad  source  of  light,  and  0  a  circular  obstacle. 
Into  the  space  PQZ  no  light  from  L  enters  at  all ;  here  is  the  dark 
umbra,  to  an  eye  placed  in  it  L  is  quite  eclipsed.  Outside  the 
space  RPQS  there  is  no  shadow  at  all,  but  to  an  eye  in  the  inter- 
mediate spaces  ZPR  and  ZQS,  L  is  visible  in  part  behind  0,  and 
the  penumbral  shadow  here  gradually  deepens  towards  the  umbra. 
On  a  screen  in  the  dotted  position  there  is  a  central  uniformly 
dark  umbra  surrounded  by  a  penumbra  fading  until  it  vanishes. 

Beyond  Z  the  shadow  is  all  penumbra,  O  appears  smaller  than 
L  and  can  never  cover  it  entirely ;  such  is  a  little  far-away  cloud 
or  tree  in  front  of  the  setting  sun. 

The  little  patches  behind  the  eyes  show  the  fractions  of  L  they 
can  see.  The  total  brightness  in  the  shadow  at  each  eye  is 
proportional  to  the  apparent  area  of  the  visible  patch,  e.g.  the 


§475]  ILLUMINATION  379 

lowest  eye  on  the  left  is  in  rather  deep  shadow,  the  lowest  on  the 
right  hardly  shaded  at  all,  but  is  farther  from  L. 

Of  course,  all  shadow  is  a  question  of  contrast.     When  the  sun 
peeps  out,  it  is  not  darker  in  the  new  '  shadows  '  than  it  was  before. 


PHOTOMETRY 

§474.  "In  these  days  of  the  'rapid  dry-plate  '  and  nights  of 
high-pressure  gas  and  flame  arcs,  when  tinder-box  and  snuffers 
are  prized  '  antiques  '  and  a  farthing  rushlight  is  not  to  be  had  for 
a  sovereign " 

Thus  myself,  twenty-four  years  ago.  Now,  films  are  four  times 
as  fast  as  that,  high-pressure  gas  is  the  last  stand  of  the  gas  companies 
in  street-lighting ;  flame  arcs,  slain  by  lamp-cleaners'  wages,  have 
their  warm-glowing  glory  bottled  in  vapour  lamps ;  domestic 
electric  bulbs  give  five  times  the  light  for  the  same  current,  and  power- 
stations  generate  twice  the  current  from  the  same  coal ;  farthing 
rushlights  are  made  in  factories,  like  any  other  fake-antiques — 
and  who  among  you  knows  the  jingle  of  sovereigns  ?  I  said  that 
book  was  a  bit  out  of  date. 

Anjrway,  we  all  take  some  interest  in  Photometry,  the  measure- 
ment of  the  Brightness  of  Lighting  or  the  *  Intensity  of  Ulumina- 
tion.' 

It  is  the  useful  illumination  of  a  surface  that  is  in  question. 
When  a  surface  squarely  faces  a  '  standard  candle  '  1  ft.  away 
it  is  said  to  be  lit  with  Unit  Intensity — one  *  candle-foot.' 

A  lamp  that  at  1  ft.  lights  a  surface  with  intensity  50  candle- feet, 
and  would  require  the  (theoretically)  concentrated  light  of  50 
candles  to  replace  it,  is  of  50  candle  power,  50  c.p. 

The  Standard  Candle  has  gone  the  way  of  the  three  barley-corns, 
but  for  a  century  candle-makers  have  been  keeping  pretty  close  to  it, 
and  the  common  domestic  candle  is  no  bad  representative. 

The  legal  standard  is  the  Vemon-Harcourt  Lamp,  which  bums 
a  regulated  supply  of  vapour  of  pentane  (a  very  volatile  petrol) 
in  a  well -ventilated  room,  and  is  of  ten  candle-power.  Subsidiary, 
and  far  less  fastidious,  standards,  are  electric  lamps,  sparingly 
used,  and  compared  with  it  at  intervals. 

§475.  The  law  of  inverse  squares.  In  the  board  1  ft.  from  the 
candle  (Fig.  174,  top)  cut  a  3-in. -square  hole,  and  hold  up  behind 
it  another  board  twice  as  far  from  the  candle.  Light  travelling  in 
straight  lines  through  the  hole  marks  out  a  bright  patch,  twice  as 
broad  and  twice  as  high  as  the  hole,  or  four  times  its  area  :  at  3  ft. 
it  is  '  thinned  out '  over  nine  times  the  area,  and  so  on.  Hence  it 
would  take  a  9-c.p.  lamp  to  give  a  brightness  of  1  candle-foot 
at  3-ft. 


380 


LIGHT 


[§475 


Thus  the  brightness  of  illumination  of  a  surface  is  inversely 
proportional  to  the  square  of  its  distance  from  the  source  of  light  {of 
comparatively  small  size). 

Hence  the  '  lux,'  or  candle-metre,  is  almost  exactly  one-tenth 
of  the  British  candle-foot. 

Caution ^The  source  must  not  be  broad  compared  with  the 

distance,  or  the  law  fails ;  the  diagram  becoming  confused  like 
Fig.  173.  A  photographic  printing  frame  is  not  quite  four  times 
as  well  lit  at  3  in.  as  at  6  in.  from  a  gas  mantle.  A  sunlit  whitewashed 
wall  sends  nearly  as  bright  a  light  to  your  book  at  3*yd.  distance 
as  at  1  yd. ;  the  broad  sky  above  illuminates  equally  well  at  all 
altitudes ;  the  inverse  square  law  has  little  to  do  with  making 
toast. 

§476.  Oblique  Illumination.  Hold  a  card  to  face  the  light. 
Turn  it  obliquely,  as  in  Fig.  174,  left ;  the  shadow  it  throws  gradually 


Fig.  174. 


narrows — to  nothing  when  the  card  is  '  edge  on.'  Simultaneously 
the  lighting  of  its  surface  decreases — also  to  zero.  The  amount  of 
light  the  card  retains  is  proportional  to  the  width  of  its  shadow, 
i.e.  to  the  cosine  of  the  angle  of  incidence  of  the  light,  i.  This 
amount  has  to  be  spread  over  the  whole  surface,  and  hence  the 
brightness  of  illumination  of  a  surface  is  proportional  to  the  cosine 
of  the  angle  of  incidence  of  the  light. 

If  you  are  unfamiliar  with  cosines,  say  instead :  to  the  Sine  of 
the  angle  e  which  the  stream  of  light  m^kes  with  the  surface. 

The  converse  is  fairly  true,  the  intensity  of  radiation  in  any 
direction  from  luminous  surface  is  proportional  to  the  cosine  of  the 
angle  between  that  direction  and  the  normal  (or  sine  of  angle  with 
surface),  as  shown  by  the  arrows  in  Fig.  174,  right.  But  the  ap- 
parent area  of  the  surface,  seen  obliquely,  diminishes  in  just  the  same 


§  477]  ILLUMINATION  381 

proportion.  Therefore  it  appears  equally  bright  in  all  directions, 
foreshortening  of  area  compensating  reduced  emission.  For  in 
Fig.  174,  middle,  BQ  =  BZ  cos  i  (or  sine  e)  and  breadth  AC  =  AB 
cos  i  (or  sine  e).  A  sheet  of  paper  illustrates  this,  but  glazed  paper 
and  high  angles  must  be  avoided.  This  also  is  the  explanation  of 
an  opal  lamp-globe,  or  the  sun,  appearing  as  a  flat  evenly-luminous 
disc. 

The  whole  thing  is  usually  called  Lambert's  Cosine  Law,  as  it 
was  not  invented  by  Lambert,  is,  as  you  see,  not  necessarily  expressed 
by  a  cosine,  and,  finally,  is  somewhat  lacking  in  reliability. 

Taken  together  these  laws  give 

Brightness  of  illumination  {in  candle  feet)  =  candle  power  of  source  x 
cosine  of  angle  of  incidence  {or  sine  angle  with  surface)  -^  square 
of  its  distance  in  feet 

I  =  -&-  cos  *  or  I  =  ^-  sine  e 
a^  d^ 

and  for  perpendicular  incidence  the  cos  and  sine  =  1 . 

§  477.  Photometers.  Look  at  the  moon,  and  guess  how  many 
candles  at  a  foot  she  appears  equivalent  to.  Set  an  inexperienced 
amateur  to  photograph  an  '  interior  '  without  any  exposure  guide. 
Catch  chequered  sunshine  on  a  paper,  and  say  how  many  times 
brighter  is  the  sunlit  part  than  the  shaded.  Then  look  at  the  end 
of  this  chapter  :  do  we  agree  ? 

But  the  eye  can  judge  when  two  Illuminations  become  equals  vnthin 
1%,  and  this  is  the  foundation  of  all  Photometers,  the  instruments 
used  in  comparing  the  candle-powers  of  sources  of  light. 

For  accuracy,  in  all  photometers,  the  two  illuminated  patches  on 
the  *  screen  '  must  touch  each  other  along  a  fine  line,  no  distracting 
illumination  should  be  in  sight,  and  no  stray  light  should  reach 
the  screen. 

Then,  when  each  patch  is  lit  solely  by  its  own  lamp,  and  receives 
its  light  at  the  same  angle  (perpendicularly,  or  very  near  it) 

T     *  fi    X      X  I,         c.p.  of  first  lamp         Cj 
I,  of  first  patch  =  (distance)^  of  ditto  =  3? 

T      .  ,      ^  ,        c.p.  of  second  lamp       c, 

I,  of  second  patch  =  (distance)' of  ditto  =  ^ 

and  when  the  lamps  have  been  moved  to  and  fro  until  the  patches 
appear  equally  bright 

T_T       .^J_-^       .£i_^ 

The  candle-powers  are  directly  as  the  squares  of  the  distances  of 
the  lamps  from  the  '  screen.' 


382 


LIGHT 


[§477 


The  Rumford  '  Shadow  '  Photometer,  Fig.  175  R.  The  patches 
are  produced  as  the  shadows  of  a  rod  standing  in  front  of  a  white 
wall,  each  shadow,  of  course,  being  lit  by  the  other  lamp  only.  It  is 
a  domestic  contrivance  ;  equally  broad  shadows  mean  equal  angles, 
which  is  necessary  ;  they  should  just  touch  each  other,  while  one 
or  other  lamp  is  moved  to  bring  them  to  equal  depth.  The  brighter 
light  on  the  rest  of  the  wall  is  a  hindrance  ;  but  this  is  less  affected 
by  stray  light  than  other  photometers. 

The  Bunsen  '  Wax  Spot '  Photometer,  Fig.  175,  B.  The  screen, 
at  right  angles  to  the  line  joining  the  lamps,  is  made  locally  trans- 
lucent, and  adjusted  until  the  translucent  part  is  neither  brighter 


Fig.  175. 


nor  darker  than  the  more  opaque  part,  and  '  disappears.'  The 
translucency  means  that  most  of  the  light  from  the  right  passes 
through  and  is  lost,  and  its  place  has  to  be  taken  by  an  equal  amount 
of  light  coming  through  from  the  left. 

Of  translucent  spots  the  worst  is  made  by  a  greasy  dirty  finger, 
the  best  is  printed  on  clean  soft  white  printing -paper  by  a  clean 
hot  metal  stamp  rubbed  with  white  wax,  and  is  ring-  or  star-shaped. 

The  mean  of  readings  taken  on  both  sides  of  the  screen  must  be 
used. 

Elaborate  modifications  of  this  simple  screen  are  in  use,  and  per- 
mit five  times  greater  accuracy  after  500  times  more  practice. 

In  flicker  photometers  the  screen  is  oscillated  sideways  about  seven 
times  a  second,  so  that  the  eye  sees  alternately  its  right  and  left 
faces,  and  equality  is  obtained  when  '  flicker  '  ceases. 

Another  device  is  to  reduce  one  light  by  interposing  a  graduated 
thin  obscuring  wedge,  made  of  gelatine  and  lamp-black. 

At  first  sight  differences  of  tint  on  the  screen  are  very  perplexing, 
but  practice  overcomes  this  :  different  observers  will  come  to  agree 
within  2  or  3%  in  equalizing  even  signal-red  and  green. 


§  478]  ILLUMINATION  383 

Innumerable  patterns  of  photometers  are  in  use,  adapted  to  all 
sorts  of  purposes,  a  very  important  one  being  the  control  of  general 
surface  lighting,  both  indoors  and  out. 

Human  judgment  is  being  done  away  with,  the  eye  being  super- 
seded by  sensitive  '  photo-electric  cells  '  :  you  open  a  little  pocket 
box,  the  light  shines  on  the  oxidized  copper  disc  inside  the  lid,  and 
a  galvanometer  pointer  indicates  candle-feet  straightaway.  »See 
§984.  *  ^ 

§  478.  One  candle-foot  is  reckoned  just  adequate  for  reading 
fair  print  at  night,  but  recollect  that  paper  and  type  and  ink  have 
been  evolved  by  the  competitive  efforts  of  generations  of  skilful 
typographers,  striving  for  the  maximum  of  clear  contrast,  often 
under  miserable  conditions  of  lighting. 

If  you  find  the  compact  sharp-cut  modem  bold-face  type  of  this 
book  going  at  all  streaky  to  your  eyes,  they  are  astigmatic,  §  608, 
reluctant  though  you  may  be  to  admit  it :  it  may  be  only  temporary, 
but   consult   the    Eye    Department.    The   type-cutter,    however, 

was  no  mathematician,  his  H x  -^  /  =  are  too  thin  :   thicken 

them  in  as  you  go. 

The  paper  was  chosen  for  its  pre-eminent  opacity,  which  keeps 
the  clear  contrast  of  black  and  white  unsullied  by  anything  at  the 
back. 

This  is  the  type  in  which  your  examination  papers  in  all  subjects 
are  set  up  :  familiarity  with  it  here  should  do  something  to  reduce 
the  risk  of  mis-reading  questions,  a  common  blunder  and  a  costly 
one. 

For  many  ordinary  small  manipulations,  with  objects  presenting 
more  unusual  shapes  and  positions,  and  far  less  contrast  in  reflecting 
power,  more  light  is  essential  to  efficiency. 

Probably,  taking  the  year  round,  among  professional  people, 
artificial  illumination  doubles  the  activities  of  our  lives,  and  from 
being  just  another  thing  upon  which  the  tools  of  the  silversmith 
could  be  exercised,  the  Lamp  has  been  lifted  high  on  a  great  tripod 
of  science  and  art  and  engineering,  its  study  the  life-work  of  hundreds ; 
its  maintenance,  of  tens  of  thousands.  Your  efficiency  and  your 
eyesight :  guard  them  both  ! 

These  are  the  most  modest  values  of  illumination  in  candle-feet 
called  for  at  the  present  day  :  Streets  0-05,  corridors  0-5,  bedrooms 
1,  living-rooms  2,  offices  3,  bench-work  and  sewing  5,  fine  work  10, 
operating  tables  15  ;  all  without  glare  or  moving  shadows,  and  all 
just  as  well  doubled  locally. 

A  fair  ordinary  allowance  I  find  generally  in  use  in  America, 
in  refectories,  common-rooms,  class-rooms,  etc.,  with  walls  and  ceiling 
distempered  light  (which  makes  a  great  difference)  and  no  ob- 
scuring ornamental  obstructions  round  the  opal  lamps,  is  one  watt 
per  square  foot  of  floor  space. 

Actual  excess  of  illuminating  power  is  never  likely  to  be  the  cause 
of  the  occasional  complaint  that  an  artificial  light  is  '  too  bright.' 


384  LIGHT  [§  478 

But  harsh  Contrast  vexes  the  eye  exceedingly,  and  its  banishment 
is  a  great  part  of  the  art  of  modern  hghting.  We  have  measured 
the  iris,  and  found  it  more  contracted  when  looking  at  a  bare 
glow-lamp  than  at  the  same  lamp  with  a  white  card  behind  it,  though 
in  the  latter  case  the  eye  is  receiving  nearly  double  the  light. 

During  the  greater  part  of  an  average  bright  day  the  illumination 
on  the  table -top  in  the  middle  of  a  fairly  light  room  will  be  something 
like  20  candle-feet ;  outside,  it  may  be  10,000.  No  wonder  one 
is  blind  on  coming  indoors  suddenly  from  the  sunshine. 

Sunlight  varies  :  1  hr.  at  Aden  has  been  found  equivalent  to 
48  hr.  winter  sunshine  in  Manchester  ;  6  weeks'  allowance.  A  white 
cloud  may  be  much  brighter  than  blue  sky.  Your  exposure-meter 
(another  photometer)  will  tell  you  that  your  own  shadow  is  half  as 
bright  as  the  surrounding  sunshine,  which  means  that  about  half  the 
light  is  coming  from  the  whole  vault  of  sky  (§§  568,  956)  and  half 
direct  from  the  sun  ;  but  large  shadows  are  deeper. 

The  Moon  is  obliterated  by  the  daylight  sky,  and  is  therefore  less 
than  a  hundred-thousandth  of  good  daylight.  At  full,  at  about 
15°  altitude,  I  have  made  it  out  equal  to  a  candle  at  14  ft.,  1 /200th 
candle-foot ;  high  up  it  may  be  2  or  3  times  as  much ;  at '  half  moon,' 
only  a  tenth. 

And  then  we  all  talk  of  moonlight  as  bright  as  day  !  the  one 
1/200,  the  other  maybe  15,000 ;  a  three-million-fold  difference. 
Well,  that  is  about  the  average  daily  range  of  sensitivity  of  our 
eyes.  Not  the  extreme  range,  by  any  means  ;  the  earnest  star- 
gazer's  remarks  about  that  blinding  moon  are  oft  wafted  far  into 
the  stilly  night. 

During  the  day  it  takes  a  full  quarter  of  an  hour  in  a  dark  roorii 
for  the  eyes  to  quiet  down  to  night  sensitivity  :  epidiascopes, 
planetaria,  aquaria,  etc.,  are  apt  to  be  very  disappointing  indeed, 
and  it  is  much  better  to  put  off  such  shows  until  the  evening.  In 
the  100-second  eclipse  of  1932  I  saw  only  Arcturus,  deadly  dark  as 
it  seemed  to  be  :  by  night  one  sees  stars  100  times  less  brilliant. 
The  radium -sparkle  (not  ordinary  phosphorescence)  on  a  wrist- 
watch  may  be  no  brighter  than  the  twinkling  stars,  yet  in  the  early 
dawn,  when  the  room  seems  light  and  the  birds  are  tuning  up,  it 
is  still  glowing  plainly  :  evidently  birds'  eyes,  like  ours,  get  amazingly 
sensitive  overnight. 

One  calls  for  3  or  4  candle-feet — and  one  gets  them  most  simply 
by  going  close  to  a  candle — yet  things  have  been  done  with  less. 
Forerunner  of  the  miner,  with  his  diminutive  shrouded  Davy  lamp, 
was  Cromagnon  man,  of  the  caves  of  Dordogne,  on  the  limestone 
walls  of  which  he  cut,  with  graver  of  flint,  vivid  and  lively  figures 
of  lion  and  bison,  of  mammoth  and  reindeer  ;  his  lamp  a  scooped-out 
stone  whereon  grease  guttered  in  a  lichen  wick.  Nor  was  progress 
rapid  :  from  but  twenty  years  before  my  time  I  read  '  From  the 
tallow  dips  of  the  last  generation,  powerless  and  void  of  everything 
save  smell,  to  the  bright  stearic  acid  that  cheers  the  drawing-room 
nowadays,  what  a  development  !     In  the  iron  arteries  under  towns, 


§  478]  ILLUMINATION  385 

in  the  constellations  of  burners  that  rule  the  nights  of  favoured 
days,  rising  over  the  chaotic  oil-lamps  of  old,  what  a  creation  !  ' 

On  my  first  visit  to  London  little  yellow  points  of  light  glowed  in 
the  roof  of  an  exhibition  building  where  the  Science  Museum  now 
stands — smokeless,  flameless,  mystic,  wonderful — and  electric 
arcs  gleamed  strangely  white  in  the  heart  of  the  City.  At  home 
we  burned  paraffin — highbrows  stuck  to  colza — or  discussed  the 
rival  merits  of  bat's-wing  and  fish-tail  flat-flame  gas-burners.  Once 
in  a  while  we  went  to  an  oxy-hydrogen  limelight  lecture,  or  indulged 
in  the  fearful  thrill  of  magnesium  wire  at  half-a-crown  an  ounce. 
There  was  nothing  else. 

About  1892  came  a  pretty  greeny  incandescent  gas-burner — oh, 
so  fragile  !  and  then  the  chemistry  books  were  caught  out  over  the 
'  reddish  smoky  flame  '  of  acetylene.  Nernst  heated  little  sticks 
of  gas-mantle  stuff  electrically,  electric  lamps  you  lit  with  a  match  ; 
then  osmium  put  in  an  appearance  in — very  small — glow  lamps, 
soon  to  be  supplanted  by  tantalum,  and  ghostly  threads  of  rival 
tungsten,  and  the  frailty  of  them  was  a  horror  to  the  householder. 
Then,  in  your  time,  drawn  wires  of  single  crystals  of  tungsten,  glowing 
in  argon,  and  now  metal-vapour  lamps,  made  possible  by  the  dis- 
covery of  glasses,  the  vapours  do  not  corrode  :  the  Dark  Ages  were 
not  so  very  long  ago,  you  have  only  just  escaped  them  ;  use  your 
birthright,  let  neither  your  dinners  nor  your  diagnoses  be  deeds  of 
darkness — yet,  pray  you,  then,  switch  off  the  light ;  you  have  no 
labour  of  flint  and  steel  and  tinder-box  to  dread. 


EXAM   QUESTIONS,   CHAPTER   XXXI 

1.  Define  Intensity  of  Illumination  and  Illuminating  Power,  and  state  in 
what  units  they  are  measured  and  how  you  would  investigate  either. 

2.  What  factors  control  Intensity  of  Illumination,  and  how  would  you 
investigate  them  ?     (  X  2) 

At  what  angle  must  light  from  a  60-c.p.  lamp,  5  ft.  away,  strike  the  wall, 
to  give  an  illumination  of  1  candle-foot  ? 

3.  A  500-c.p.  lamp  is  slung  18  ft.  above  the  middle  of  a  roadway  48  ft. 
wide  from  path  to  path ;  calculate  the  intensity  of  illumination  on  the  surface 
of  the  footpath. 

4.  Give  description  and  theory  of  some  form  of  photometer,  stating^ the 
precautions  you  observed  when  using  it.  Lamps  of  15  and  30  c.p.  are  5  ft. 
apart ;  whereabouts,  on  or  near  the  line  joining  them,  do  they  give  equal 
illuminations  ?     (  X  4) 

5.  How  would  you  measure  the  candle  power  of  a  lamp,  and  iU  variation 
with  voltage  ?  ••  i 

The  screen  of  a  photometer  is  illuminated  by  a  30-candle-power  lamp 
placed  at  50  cm.  from  one  side.  On  the  other  side  of  the  screen  a  40-c.p. 
lamp  is  at  70  cm.  Where  should  a  20-c.p.  lamp  be  placet!  to  balance  the 
screen  ? 


386  LIGHT 

6.  The  light  from  lamp  A  produces  the  same  ilhmiination  as  that  from  B, 
at  twice  the  distance,  and  20%  obstructed  by  a  thick  glass;  compare  their 
candle  powers. 

7.  Two  lamps  give  equal  illumination  on  a  photometer  when  placed  at 
40  and  50  cm.  A  sheet  of  glass,  which  transmits  81%  of  the  light,  is  placed 
in  front  of  the  brighter  lamp.  How  far  must  the  weaker  be  moved  to  restore 
equality  ? 

8.  Describe  briefly  an  efficient  photometer.  How  would  you  use  it  to 
measure  the  percentage  of  light  stopped  by  a  sheet  of  stout  celluloid  ? 

A  candle  and  an  electric  lamp  are  fixed  150  cm.  apart,  and  balance  on  a 
screen  30  cm.  from  the  candle.  When  a  white  card  is  held  close  behind  the 
lamp,  the  screen  has  to  be  moved  5  cm.  nearer  the  candle ;  express  the  re- 
flecting power  of  cardboard. 

9.  At  30  cm.  to  the  right  of  the  photometer  screen  stands  a  candle,  10  cm. 
beyond  this  and  squarely  facing  it  is  a  plane  mirror,  capable  of  reflecting  light 
with  three-quarters  of  its  incident  intensity.  At  what  distance  to  the  left 
of  the  screen  must  an  equal  candle  be  placed  ?     (  X  3) 


Practically,  you  may  be  asked  to  measure  a  candle-power. 


CHAPTER  XXXII 
THE  REFLECTION  AND  REFRACTION  OF  LIGHT 


§481.  The  Laws  of  Reflection  were  worked  out  in  §§  403,  406; 
tliey  are : 

I.  The  incident  direction  (from  the  lamp),  the  perpendicular  or 
'  normal '  to  the  reflecting  surface,  and  the  reflected  direction,  lie  iji 
one  plane — ^which  means  that  the  whole  diagram  lies  flat  on  the  paper. 

II.  The  angles  of  incidence  and  reflection  are  equal.  These  (which 
are  the  angles  between  the  waves  and  the  surface)  are  the  angles 
i  i'  between  the  two  directions  and  the  '  normal '  in  Fig.  179. 

Call  these  directions  of  wave-travel '  rays  '  if  you  like,  but  recollect 
that  '  rays  '  the  actual  size  of  the  lines  in  that  diagram,  0-01  in. 
thick,  are  really  broad  processions  of  light  waves,  500  wave-lengths 
wide. 

These  laws  can  be  checked  with  the  aid  of  a  lamp,  a  little  cup  of 
water,  a  plumb-line  and  a  foot-rule  : 

I.  A  plumb-line  held  at  arm's  length  will  '  cut '  the  lamp  and  its 
reflection  in  the  level  water  ; 

II.  Putting  the  lamp  and  eye  at  the  same  height  on  opposite  sides 
of  the  room,  the  cup,  whether  on  floor  or  chair  or  table,  will  be  half- 
way across  the  room  when  you  see  the  reflection  in  it. 

Far  better  *  proof  '  of  the  laws  is  found  in  the  accuracy  of  the 
Sextant,  etc.,  in  everyday  practice. 

§  482.  It  follows  at  once  that  if  the  mirror  be  tilted  through 
an  angle,  the  reflected  ray  swings  through  double  the  angle  :  the 
reflections  of  lamplight  from  the  facets  of  a  cut-glass  tumbler 
sweep  round  on  the  table-cloth 
twice  as  fast  as  the  glass  is 
turned.  For  when  the  angle  of 
incidence  is  increased,  the  angle 
of  reflection  increases  equally ; 
and  therefore  both  together, 
which  make  up  the  angle  between 
the  direction  of  the  light  before 
and  after  reflection,  increase  by  twice  as  much. 

Thus  in  Fig.  176,  when  the  little  mirror  M  turns  from  facing  n 
to  facmg  7i',  the  ray  it  reflects  from  lamp-and-slit  L  moves  from  8 
to  s',  and  the  angle  sUs'  is  double  wMn'.  This  is  actually  the  con- 
trivance of  lamp  and  scale  used  for  observing  the  movements  of 
sensitive  galvanometers,  §  764,  when  M  is  a  concave  muror  of  raduis 
ML,  which  is  usually  a  metre. 

387 


388 


LIGHT 


[§483 


§  483.  In  the  Sextant  (Fig.  177)  a  small  telescope  T  looks  through 
the  clear  upper  half,  nearest  you,  of  the  '  horizon  glass  '  H  at  one 
object,  and  also  receives  a  ray  by  reflection  in  the  silvered  lower  half 
of  H,  from  the  '  index  mirror  '  I,  whither  it  comes  from  another 
object.  When  I  and  H  are  parallel  (B  at  extreme  right,  reading 
zero),  these  rays  IH'  and  TH  are  parallel,  and  start  from  the  same 
distant  source,  but  when  the  swinging  index  bar  B,  which  carries  I, 
is  moved  round  the  graduated  frame  F,  to  which  H  and  T  are  fixed, 
the  ray  SI  turns  through  double  the  angle.  The  angle  SIH',  e.g. 
the  altitude  of  a  star,  is  therefore  obtained  by  moving  I  round 
until  the  reflected  star  appears  on  the  horizon,  and,  in  general,  the 
angular  distance  between  two  objects  by  making  one  apparently 
overlap  the  other. 

For  convenience,  the  sextant's  graduations  are  figured  double. 
The  diagram  shows  the  fine  adjustment  screw  M  and  magnifying 


Fig.  178. 


glass  for  the  vernier,  which  reads  to  10  sec.  of  arc,  but  is  being 
superseded  nowadays  by  greatly  improving  M,  giving  it  a  large 
graduated  head,  and  using  it  as  micrometer,  reading  very  plainly. 
The  sun  is  being  observed  through  the  dark  glasses  d  which  can  be 
turned  out  of  the  way,  as  are  d' ,  when  not  required. 

§484.  The  reflected  image  in  a  plane  mirror.  In  a  looking- 
glass  one  sees  the  image  some  distance  behind  the  surface.  What 
distance  ? 

Fig.  134  has  answered  that :  it  is  as  far  behind  as  the  object  is 
in  front ;  but  if  you  are  not  convinced,  here  is  an  alternative  treatment : 

In  Fig.  178,  E  sees  the  object  O  reflected  along  ME  where  angles 
at  M  are  equal. 

E',  the  left  eye,  sees  it  along  E'M'  where  angles  at  M'  are  equal. 
The  actual  waves  are  shown  along  this  track,  centred  first  on  O  and 
then  on  I ;  take  a  pair  of  pencil  compasses  and  continue  every 
arc  both  ways,  and  you  get  the  wave-system  of  which  the  diagram 
is  a  few  bare  sticks. 


§485]  REFLECTION   AND   REFRACTION  389 

Consequently  it  must  appear  to  be  at  I  where  these  du-ections 
cross,  and  you  easily  prove  that  I  is  on  the  perpendicular  OP 
produced  as  far  behind  the  mirror  as  the  object  is  in  front. 

Having  no  real  existence,  it  is  described  as  a  Virtiuil  Image,  and 
its  apparent  distance  is  actually  judged  stereoscopically,  §  604, 
by  the  muscular  convergence  of  the  two  eyes,  for  a  single  line  OME 
gives  no  information  as  to  the  distance  of  I.  These  means  soon  fail. 
You  never  thought  of  the  moon's  reflection  in  the  lake  as  being 
200,000  miles  below  it. 

Treating  similarly  other  points  O'  on  a  solid  object,  the  image 
is  found  to  be  equal  in  size,  but  '  laterally  reversed,'  i.e.  upside 
down  only  or  left  for  right  only,  not  both  together.  We  never 
'  see  oorsels  as  ithers  see  us.' 

Multiple  Reflections  in  parallel  and  inclined  mirrors  often  receive 
notice  of  sorts,  but  the  careful  and  accurate  treatment  I  gave  them 
long  since  was  a  work  of  supererogation,  for  though  occasionally 
amusing  in  tea-rooms,  kaleidoscopes,  etc.,  I  have  never  found 
them  of  any  medical  use,  and  they  are  omitted  hence. 

§  485.  Laws  of  Refraction.     Part  of  the  light  that  falls  on  the 
surface  of  a  transparent  substance,  or  '  medium,'  passes  into  it, 
but  in  so  doing  becomes  suddenly  bent  from  its  course.     The  formal 
Law  I  of  this  Refraction  is  the  same  as 
that  for   Reflection,  so  that  the  whole 
diagram    of    incident,    reflected,    and 
refracted  '  rays,'  and  '  normal,'  lies  in 
the  paper. 

Law  II  was  worked  out  and  stated 
in  §  407  :  The  ratio  of  the  sine  of  the  angle 
of  incidence  to  the  sine  of  the  angle  of 
refraction  into  the  second  medium  is 
constant,  and  is  called  the  Refractive 
index  of  the  second  medium  relative  to 
the  first  (and  is  actually  the  direct  ratio  Fig.  179. 

of    the    velocities,    first    and    second). 

Refer  to  Fig.  140,  and  now,  to  represent  the  same  thing,  draw  a 
new  Fig.  179.  Describe  a  circle  about  the  point  A  where  '  ray  ' 
meets  surface,  draw  normal  NAZ,  and  draw  SH,  EK  parallel  to 

Then  SH/SA  =  sine  i,     EK/EA  =  sine  r 

•  •  Sine  r       SA/  EA       EK       ^' 

For  EA  =  SA,  which  is  the  reason  for  the  circle.     Notice  that  the 
angles  of  incidence  and  refraction,  which  are  the  angles  the  waves 
make  with  the  surface,  are  the  angles  between  '  rays  '  and  •  normal. 
Notice  also  that  Bending  is  towards  the  normal  on  entering  the  more 
refractive  medium. 


\y^ 

\^ 

/        ^\*' 

V 

\ 

v- 

^--Jl. 

L^ 

\. 

390 


LIGHT 


[§485 


Some  Refractive  Indices  relative  to  air  are  : 


Realgar    .      .      . 

2-45 

Rock  salt 

1-54 

Canada  balsam 

1-53 

Diamond . 

2-42 

Fluorite   .      . 

1-43 

Ethyl  benzoate 

1-51 

Phosphorus  . 

2-16 

Ice      .      .      . 

1-307 

Xylol        .      . 

1-50 

Flint  glass  (dense) 

1-72 

Mono-bromo- 

Glycerine 

1-47 

>»        »»         •      • 

1-62 

naphthalene 

1-66 

Alcohol    . 

1-36 

Crown  glass  (com- 

Carbon    disul- 

Water      .      . 

1-33 

mon)     . 

1-52 

phide    . 

1-65 

Media  with  high  indices  are  often  spoken  of  as  '  optically  dense.' 

The  Sine  Law  you  test  in  the  laboratory  as  in  Fig.  180,  which  em- 
bodies an  alternative  construction  for  finding  the  refracted  ray. 
The  rectangle  is  a  block  of  glass  lying  on  a  drawing-board  (or  trough 
of  water  standing  beside  a  board).  0  is  a  scratch.  Mark  on 
the  board  the  refracting  surface  XY,  and  ONZ  perpendicular. 
Sight  O,  and  stick  in  wide-apart  pins  jpjp' ,  qq' ,  etc.,  along  various 
lines  of  sight.  Draw  pp'K,  qq'B,  etc.,  AO,  BO,  etc.,  and  where 
PA  produced  meets  NO  put  letter  M. 

PAMNZ  is  an  angle  of  incidence,  sine  i  =  AN/AM. 

AON  is  corresponding  angle  of  refraction,  sine  r  =  AN/AO. 

AN  /AN      AO  ^, 

/.  [i.  =  -r^  I  T-p^  =  -T^TF ;  measure  them 


V> 


AM/  AO      AM 
and  BO/BM',  etc.,  should  give  the  same  result. 

§486.  Apparent  reduction  of   depth    in    refractive   media.    In 

Fig.  180  two  close  lines  QR  will  cut  at  I,  light  from  O  appears  to 

reach  your  two  eyes  from  I, 
_  which  is  therefore  the  Virtual 

Image  of  0. 

Looking  nearly  vertically 
down  with  both  eyes  (cf .  §  484) 
along  ZN  and  SD,  J  is  the 
position  of  the  image  of  O  ; 
OD  is  always  [i  times  JD, 
therefore  now  ON  practically 
=  (i,  JN,  or  the  Beal  depth  is 
[I  times  the  apparent  depth,  as 
was  shown  by  the  wave  method 
in  §  405,  Fig.  138. 

Thus  a  glass  block  is  half  as 
thick  again  as  it  looks ;  you 
measure  this  by  parallax  be- 
tween a  pin  behind  and  an 
inverted  drawing-pin  on  top  of 
the  block. 

And  water  is  one-third  as  deep  again  ;   stand  in  it  shoulder-deep 
and  look  down  at  yourself. 

A  Stick  slanting  into  water  appears  bent  upwards ;   for  all  parts 
of  it  under  water  lift  up. 


V" 

R  / 

\ 

//^  ^ 

d\    N 

cA/^^^y? 

X      \ 

/iji-^/     Y 

\M 

\\ 
\\ 
\\ 

J\ 

//  ^ 

o\ 

F 

Fig.  180. 


§  488J  REFLECTION   AND   REFRACTION  391 

A  method  of  measuring  y.,  for  a  liquid,  say — under  a  low-power 
microscope  possessing  a  graduated  fine  adjustment — is  this : 
focus  on  a  scratch  inside  a  watch-glass,  pour  in  the  liquid  and  refocus, 
then  refocus  on  dust  on  top  of  the  liquid  : 

(third  —  first)  =  tx(third  —  second). 

Looking  obliquely  with  both  eyes,  as  along  QB  and  RC,  Fig.  180, 
I  is  much  nearer  the  surface.  The  shallow  bottom  of  the  pool 
appears  impassable,  its  pond-weeds  touching  the  surface,  but  it  all 
sinks  down  under  your  boat,  to  reappear  just  as  shallow  a  few  yards 
astern. 

In  appearance,  the  image  of  point  O  moves  along  what  is  called 
a  '  Caustic  curve  by  refraction  '  (cf .  §  584),  to  which  all  the  lines  of 
sight  out  of  water  are  tangent  when  produced  ;  you  see  the  beginning 
of  it  in  Fig.  180,  and  it  is  traced  farther  in  Fig.  192  :  two  close  lines 
meet  on  the  curve,  and  there  lies  the  image  for  that  pair  of  eyes. 

Fig,  192  is  further  referred  to  in  §  491. 

§487.  Successive  parallel  layers  of  different  refractivities.  For 
refraction  from  medium  y.^  to  medium  jxg  one  uses  the  relative  index 

V/V 


^,  for  in  velocities  it  is  -  /  —  =  -i 

(^1  V2  /  Vj         Vg 


And  going  on  to  media  (Xg,  (x^,  etc.,  the  whole  refraction  is  given 
by  [i-Jiiv  since  it  is  y.J[L^  X  [ijiiz  X  [ijv-i  =  vjv^,  provided  the 
surfaces  are  all  parallel. 

Light  therefore  resumes  its  original  direction 
after  passing  through  a  parallel-faced  pane  of 
glass,  and  this  must  always  be  the  case  when 
first  and  last  media  are  the  same. 

But  any  particular  ray,  though  parallel  to 
itself,  is  slightly  'side-stepped,'  Fig.  181,  the 
amount  varying  from  nothing,  when  perpen- 
dicular, to  nearly  the  thickness  of  the  plate, 
when  very  oblique. 

This  is  the  oblique  rise  of  I  mentioned  in  the 
last  paragraph,  and  it  is  the  lateral  shift  which  Fio.  181. 

is  ignored  in  the  Theory  of  Thin  Lenses,  Chapter 
XXXIII,  to  be  taken  into  account  again  in  Thick  Lenses,  Chapter 
XXXVI. 

§488.  Atmospheric  Refraction.  The  clear  atmasphere  can  bo 
regarded  as  a  succession  of  parallel  strata  gradually  increasing  in 
density  and  refractive  power  from  above  downwards,  Fig.  182. 
Starlight  entering  it  obliquely  therefore  gradually  changes  direction 
along  a  curved  path,  and  strikes  the  eartli  more  steeply,  i.e.  the  sUr 
is  seen  slightly  raised  in  the  sky,  as  if  in  the  dotted  direction. 
Taking  advantage  of  the  last  paragraph,  one  need  consider  only 
the  index  of  the  air  next  the  earth,  thus  to  calculate  the  nse  for 


392 


LIGHT 


[§488 


a  star  10°  above  the  horizon  (neglecting  curve  of  earth),    y.   air 
100030 

sine  i  =  sine  80°  =  0-98480  =  1-0003  sine  r 
/.  r  =  79°  54'     .-.  i  —  r  =  rise  =  6' 

an  error  of  6  miles  if  the  navigator  omitted  to  correct  for  it. 

The  refractive  lift  increases  rapidly  near  the  horizon  :    the  sun       j 


Fig.  182. 

and  moon  have  angular  diameters  of  about  J°,  at  5  diam.  high  they 
are  lifted  half  a  diameter,  on  the  horizon  they  are  lifted  their  whole 
diameter ;  Fig.  183,  from  a  tele-photo,  shows  the  sun  setting  over 
a  distant  island  when  without  refraction  he  would  be  just  completely 
set ;  notice  how  reluctant  his  lower  part  is  to  go  down,  so  that  he 
squashes  into  plum -pudding  shape.     There  is  a  real  gain  in  length 

of  day,  of  several  minutes,  at 
both  ends,  made  possible  because 
the  sphere  which  catches  the  sun- 
light is  the  earth  enlarged  by  its 
shell  of  atmosphere. 

Variation  of  temperature  alters 
the  refractivity  of  a  fluid  very 
considerably  :  great  differences  at 
irregular  interfaces  cause  visible 
streaming  of  hot  air  in  cold. 
Through  field-glasses,  on  a  hot  day, 
the  distant  landscape  trembles 
and  jumps  ;  a  larger  telescope  only 
makes  matters  worse,  a  4-in.  is 
the  biggest  that  is  any  use  to  the 
coastguard.  Stars  twinkle  for  this  same  reason  of  air  currents 
of  varied  temperature  ;  through  big  telescopes  they  cease  to  twinkle, 
but  jump  about,  long  exposed  photographs  at  Greenwich  smudge 
little  stars  to  five  times  their  size  ;  the  moon  '  boils  '  and,  just  as 
one  can  read  a  paper  in  the  train  but  cannot  photograph  it,  so  the 
best  photographs  of  the  moon  do  not  show  the  detail  a  patient 
watcher  can  glimpse  bit  by  bit.  This  is  what  sends  astronomers 
to  hill-tops  of  steady  temperature  in  distant  climes. 

§  489.  Mirage     is    due    to    low-down    atmospheric    refraction 
greatly  exaggerated.     It  is  by  no  means  confined  to  the  tropical 


Fig.  183. 


490] 


REFLECTION  AND   REFRACTION 


393 


desert ;  you  have  seen  it  often,  the  shimmer  of  light  which  glosses 
over  the  hot  black  sunlit  road  as  the  car  breasts  a  hill.  Look  out 
to  sea  through  a  pocket -telescope  almost  any  sunny  summer  day, 
and  you  will  seethe  strangest  antics  among  distant  shipping,  especially 
if  you  lie  down  on  the  warm  shingle.  Funnels  and  foc'sles  appear 
weirdly  drawn  out,  and  the  ship  may  even  seem  floating  high  up 
on  its  own  partial  reflection  in  a  shimmering  silvery  sea.  The  hot 
surface  warms  the  air  just  above  it  and  makes  it  less  refractive,  i.e. 
light  travels  faster  in  it.  The  lower  parts  of  light-waves  which  are 
passing  over  therefore  gain  on  the  upper  parts,  and  the  wave-train 
is  warped  upwards,  and  reaches  the  eye  as  if  it  came  from  a  lower 
point.  A'  instead  of  A,  Fig.  184.  Wave-trains  venturing  lower, 
into  hotter  strata  still,  may  become  so  oblique  that  at  last  they 
suffer  total  reflection,  §491,  and  reach  the  eye  as  if  from  B'  :  the 
silvery  shimmering  '  water  '  really  being  totally  reflected  sky. 


Fig.  184. 


Sometimes,  especially  on  fine  warm  days  in  Spring,  while  the  sea 
is  very  cold,  the  opposite  effect  is  seen.  The  light  waves  are  pinched 
together  at  their  lower  ends  in  the  cold  air,  and  the  wave-train  is 
warped  downwards,  shipping  is  squashed  flat,  the  Cork  lightship 
two  miles  off  becomes  a  raft ;  the  curving  train  clings  to  the  curve 
of  the  earth,  the  Sunk  16  miles  away,  which  we  know  only  as  a  mast 
with  knobs  on,  shows  in  plain  view,  and  we  actually  see  shipping 
in  the  sea  beyond  her.  See  the  lower  figure  in  Fig.  184,  where  the 
straight  tangential  line  of  sight  contrasts  with  the  arched  one, 
which  runs  at  right  angles  through  all  the  wave-fronts,  pinched 
together  at  their  feet  by  the  colder,  denser,  slower  air.  You  can 
supply  these  for  yourself  in  the  upper  figure,  there  widened  out  at 
their  feet  in  the  warmer,  lighter,  faster  air. 

§490.  The  Prism.  Looking  through  a  triangular  prism,  such 
as  the  '  lustre  '  from  a  Victorian  vase  or  chandelier,  objects  appear 
lifted  up  towards  the  narrow  end,  the  refracting  angle,  of  the  wedge. 
That  is,  light  has  been  permanently  deviated  '  towards  the  thick 
end,'  very  likely  by  as  much  as  45°.  How  much,  can  always  be 
found  by  applying  Fig.  179  to  both  faces  in  turn,  but  when  the  light 


394  LIGHT  [§  490 

passes  symmetrically  the  deviation  proves  to  be  least — a  minimum 
deviation,  D — and  in  Fig.  185  : — 

Light  travelling  along  OR  has  been  deviated,  to  travel  along  QE, 
by  the  angle  at  P,  =  D. 

In  the  flat  triangle  PQR  either  angle  x  =  JD,  the  minimum 
deviation 

QS  and  SR  are  '  normals  '  to  the  prism  faces,  .'.  angle  at  S  =  A  the 
refracting  angle  of  the  prism  (lay  one  book  on  another,  askew,  and 
look  at  the  angles  made). 

In  the  flat  triangle  SQR  either  angle  o  =  JA,  the  refracting  angle, 
and  this  angle  o  is  the  angle  of  refraction  r  in  the  glass 

Angles  X  -\-  0  =  angle  PQS  =  external  angle  of  incidence  i  in  air 

_  sine  i  _  sine  {x  -\-  o)  _  sine  J(A  +  D) 
' '  ^  ~  sifie  r  ~      sine  (o)  sine  J  A       ' 

a  result  you  will  do  well  to  recollect ;  and  recollect  also  that  the 
half-angles  must  be  worked  out  complete  before  looking  into  the 
table  of  sines,  and  that  it  is  A  below,  and  not  D. 

Recollect,  too,  that  a  prism  is  like  the  toe  of  a  sandbank  ;  waves 
drag  their  inner  ends  in  the  shallows,  and  therefore  curl  round  towards 
the  thick  part. 

Note  on  Minimum  Deviation.  If  this  did  not  occur  in  the  symmetrical 
position,  let  it  be  along  one  track  in  Fig.  186,  then  the  perfectly  similar,  though 
left-handed  track,  would  also  give  minimum  deviation,  therefore  the  two 
settings  would  give  two  minima ;  but  there  is  only  one,  .•.  etc. 

It  is  the  position  you  find  for  yourself  after  a  few  seconds'  handling  a  prism, 
because  it  gives  the  best-defined  view  with  least  distortion.  It  is  a  very 
'  flat  minimum,'  and  the  prisms  can  be  set  quite  accurately  enough  by  hand, 
even  in  the  spectrometer.  If,  in  the  row  of  pins  method  you  learn  in  the 
laboratory,  it  seems  troublesome  to  find,  re-set  the  prism  half  a  dozen  times, 
always  to  give  the  best  (coloured)  view  of  the  two  pins  beyond,  planting  your 
two  near  pins  and  measuring  the  deviation  each  time  :  at  least  two  will  be 
less  than  the  rest,  and  equal.     Recollect  it  is  the  symmetrical  position. 

Warning. — Three-side-polished  glass  prisms  swarm  with  brilliant 
colourless  total  reflections;  be  on  your  guard  against  these,  or  kill  them  by 
plastering  the  unused  side  all  over  with  gummed  paper. 

Fat  prisms.  There  is  a  limit  to  the  possible  angle  for  a  prism,  and 
it  is  reached  in  Fig.  187,  where  light  creeping  up  along  the  face 
plunges  in  at  the  '  critical  angle  '  with  the  normal  (next  §),  and  re- 
verses this  process  on  emergence.  You  can  see  plainly  enough  that 
this  Maximum  Refracting  Angle  is  twice  the  Critical  Angle  c. 

For  glass,  the  Critical  Angle  is  41°  or  less,  and  you  can  not  see 
through  a  square  corner  of  glass  ;  so  take  care  when  you  are  given  a 
'  right -angle  prism  '  for  refracting  purposes. 

For  water,  the  Critical  Angle,  the  sine  of  which  is  1/1-33,  is  48|°, 
and  you  can  see  through  the  corner  of  a  square  water-tank.  This 
shows  very  well  in  the  Brighton  Aquarium,  where  light  entering 
the  moving  surface  of  the  water  emerges  from  the  glass  front  and 
plays  in  broad  rippling  bands  of  colour  (§  552)  on  the  sill. 


§491] 


REFLECTION  AND  REFRACTION 


396 


Thin  prisms.  We  saw  in  §  155  that  there  is  practically  no  diflference 
between  small  angles  and  their  sines.  Consequently  for  prisms  of 
angle  only  a  few  degrees  it  is  permissible  to  write  : 


[I  = 


_i(A  +  D) 


iA 


"^  A  ^''  '^ 


1  = 


D 


.*.  for  thin  prisms  D  =  ((x  —  1)  A, 

as  was  proved  also  in  §  409.    And  now  it  signifies  very  little  whether 
the  light  passes  through  symmetrically  or  not ;  Fig.  188. 

Fig.  185. 


Fig.  186. 


Fig.  187. 


Fig.  188. 


Fig.  189. 


Fig.  189  is  a  full-size  section  of  Pilkington's  prismatic  window- 
glass,  which  is  remarkably  effective  in  throwing  skylight  to  the 
backs  of  rooms.  The  other  way  up  you  will  be  able  to  make  out 
that  it  can  also  be  used  '  by  total  reflection  '  to  reach  even  farther 
back. 

§491.  Total  Reflection.  In  §408,  Fig.  141,  we  saw  that  waves 
travelling  practically  parallel  to  the  surface  of  a  slower  medium 
send  into  it  waves  at  a  sharp  angle.  Fig.  179  now  modifies  into 
Fig.  190,  which  repeats  the  directions  of  travel  of  the  waves  of 
Fig.  141.  The  sun  is  setting,  and  its  last  grazing  *  ray  *  SA  glints 
along  the  surface  and  plunges  in,  in  direction  AE,  the  angle  NAE 
being  the  last  and  largest  angle  of  refraction — call  it  c. 

As  always,  SH  =  \i  EK,  and  now  SH  is  the  full  radius  SA  =  EA 

.-.  EA/EK  =  Y/v  =  ii    or    EK/EA  =  sine  c  ==  1/^. 
Any  light  attempting  to  escape  from  the  lower  *  denser  '  medium 
from  within  this  angle  c  can  do  so,  as  shown  by  the  arrow ;   but  if 


396 


LIGHT 


[§491 


it  travel  by  FA,  fx  X  FG  exceeds  SA,  and  cannot  be  within  the 
quadrant,  the  construction  breaks  down,  which  means  that  there  is 
no  escape  by  refraction  into  the  (faster)  '  lighter  '  medium,  but 
complete  total  reflection  back  into  the  denser  medium,  at  the  equal 
angle  of  reflection  shown. 

The  limiting  angle  c  is  the  Critical  Angle  of  Total  Reflection  in  the 
denser  medium,  and  its  sine  is  l/[x  of  that  denser  medium. 

This  accounts  for  the  brilliance  of  the 
under-side  of  the  water  surface  in  a 
tumbler,  the  glistening  of  air  bubbles  in 
liquids,  or  cracks  in  glass  or  ice  ;  light 
in  the  dense  substance  happens  to  strike 
the  air- crack  or  bubble  too  obliquely,  and 
is  totally  reflected  from  it.  Mercury 
poured  into  a  test-tube  dipping  under 
water  appears  less  brilliant  than  the  air- 
filled  part,  and  a  little  water  on  top  of 
the  mercury  apparently  cuts  the  tube  in 
halves. 

All  that  is  necessary  for  Total  Reflection  is  that  the  light  shall 
strike  the  inside  surface  of  the  dense  medium  at  an  angle  exceeding 
the  Critical  Angle,  and  as  this  is  only  41°  for  glass,  right-angled 
prisms,  where  it  would  naturally  be  45°,  are  largely  employed  as 
permanently   brilliant   reflectors.     In   Fig.    191,    (1)    exhibits   the 


Fig.  190. 


tMbJ^ 


Fig.  191. 


attractive  condition  that  every  student  some  time  or  other  blunders 
into  in  getting  'refraction  '  through  any  three-side-polished  prism, 
(2)  is  turning  a  right  angle,  (3)  is  turning  the  beam  upside  down,  and 
is  used  sometimes  to  erect  a  lantern-picture  on  the  screen,  (4)  is 
an  erecting  prism  from  prismatic  binoculars,  and  (5)  answers 
Question  19,  (6)  is  from  a  '  prismatic  pavement  light '  used  for 
illuminating  basements,  (7)  is  a  section  of  a  '  bulkhead  light '  which 
enables  a  well-protected  lamp  to  light  a  wide  sweep  of  deck,  and 
(8)  shows  the  reflection  of  an  oncoming  car  headlight  from  the  right- 
angle- patterned  back  of  a  bicycle  rear-reflector,  or  roadside  warning 
sign,  of  the  cheapest  description  (the  flat  surface  being  often  'flashed ' 
with  red).  (9)  shows  that  bright  ring  you  see  in  the  front  lens  of 
your  '  sixth  '  micro -objective,  the  light  in  the  middle  passes  through 


§491] 


REFLECTION  AND   REFRACTION 


397 


to  the  back,  the  limiting  critical  rays  are  drawn  ;  and  (10)  shows 
how  a  limited  amount  of  total  reflection  is  obtainable  from  any 
rod  or  bead  of  |x  greater  than  ^J2  {i.e.  never  from  water),  the  bright 
line  in  a  glass  rod  held  against  the  dark,  or  a  fibre  of  wool  or  silk 
viewed  as  an  opaque  object  under  the  microscope,  the  eerie  glim- 
mering reflection  of  the  promenade  lights  that  comes  back  to  you 
from  the  grains  of  the  broad  sands  at  night,  or  the  luminosity  of 
a  parrakeet's  green  plumage  when  you  shine  a  torch  on  him.  The 
more  refractive,  the  sharper  the  dotted  angle  may  be,  i.e.  the  wider 
the  beams  that  get  reflected. 

In  diamond,  (i  =  2-4,  c  is  24^°,  and  facets  cut  on  the  back  at 
any  angle  beyond  this  are  sure  to  sparkle  through  the  flat  front 
of  the  gem. 

Fig.  192,  in  which  the  angles  are  accurate,  shows  that  a  fish  may 
see,  in  a  calm  surface,  the  whole  sky  and  landscape,  in  a  circular 
picture  48 J°  in  radius,  framed  in  reflected  pond-bottom. 


Notice  how,  as  always,  the  amount  of  refraction  increases  with 
angle  of  incidence.  The  last  15°  gets  crushed  into  only  2°  in  the 
fish-eye  view,  i.e.  bush  and  angler  aUke  are  dwarfed  do\*'n  to  the 
shape  of  hassocks,  which  probably  explains  why  it  does  not  matter 
much  whether  you  stand  behind  or  in  front.  But  if  you  move,  then 
obey  the  trout-fisher's  three  commandments  :  stand  back,  stand 
farther  back,  stand  back  more  yet. 

The  figure  shows  this  distortion  also  in  the  shape  of  the  emergent 
wave  curve,  which,  instead  of  being  the  dotted  sphere  truly  centred 
on  the  object,  is  the  mushroom-cap  shown  '  tucked  up  '  at  the 
edges— really  the  involute  traced  by  a  radius  (to  which  it  is  every- 
where perpendicular)  rolling  on  the  '  Caustic  '  evolute,  the  curve 
to  which  you  see  all  the  emergent  ray  directions  are  tangent. 

This  will  be  referred  to  again  in  §  584.  ,       _i 

The  edge  of  the  picture  is  coloured,  for  blue  is  more  refracted 
than  red,  and  therefore  bends  down  at  a  steeper  angle,  i.e.  in  an  inside 
cone. 


398 


LIGHT 


[§492 


§  492.  Two  methods  of  obtaining  Refractive  Index  by  measuring 
Critical  Angle  are  in  vogue,  the  one  in  elementary  laboratories, 
the  other  of  value. 

Fig.  193.  A  fiat  glass  air-cell  is  made  by  sticking  a  cover  over 
a  ring  of  cement  on  one  end  of  a  micro -slide  ;  this  hangs  vertical 
from  a  wooden  bridge  and  dips  in  a  square  glass  jar  of  the  liquid 
under  test,  in  front  of  the  window ;  two  long  pins  PQ  mark  a  sighting 
line. 

The  bridge  is  turned  askew  from  square  position  across  the  light, 
until  at  angle  c  a  black  shadow  invades  the  cell  and  settles  its  edge 
on  the  sighting  line  :  the  window  light  has  been  refused  admission 
into  the  air-film  at  the  critical  angle.  The  edge  is  orange-red, 
for  (i,  depends  to  some  extent  on  colour.  Mark  AB,  and  turn  the 
bridge  to  the  same  effect  the  other  way,  CD.  Join  AB,  CD  ;  make 
OC  =  OA,  bisect  CA ;   then  each  of  these  angles  =  c 

.-.  {z  =  OC/JCA. 


s.     _ 

--/ 

p 

N 

— -^ 

a 

^ 

-N 

L^ 

/ 

/A 

\C 

V 

Fig.  193. 


Fig.  194. 


Fig.  194.  Monochromatic  light  shines  on  the  glass  block  A  which 
is  stuck  to  the  right-angle  prism  B  by  a  drop  of  the  liquid  under 
test.  The  last  limiting  ray  that  can  enter  is  in  the  plane  of  the  face 
of  B  ;  it  plunges  in  at  the  glass/liquid  critical  angle,  the  sine  of  which 
is  [x/[Xi  glass,  it  escapes  from  the  second  face,  lying  against  a  straight- 
edge, and  is  sighted  along  pins  PQ  as  the  sharp  edge  of  a  black 
shadow.     NO  is  perpendicular  to  the  straight-edge. 

A  little  trigonometry  shows  that  (x  =  VTl^i^  ~  ^^^^^  NOQ). 

Old  prism -binocular  prisms  serve  well,  and  their  [x^  is  easily 
measured  on  the  spectrometer  ;  it  is  usually  1-570.  Using  red  glass, 
sodium  light,  or  mercury  green,  any  liquid  up  to  immersion-oil 
can  be  measured  with  much  accuracy,  even  on  this  simple  outfit. 

Refractometers  of  this  type,  provided  with  little  sighting-telescopes 
and  ready-divided  direct-reading  scales,  are  sold  at  high  prices,  and 
are  worth  them,  for  the  examination  of  butter,  margarine,  and  all 
greases  and  oils,  tarry  mixtures,  etc.,  for  measuring  the  concentration 
of  all  sorts  of  solutions,  S3T:ups,  body-fluids,  etc.,  and  for  testing 
glasses,  gem -stones,  and  so  forth. 


§493]  REFLECTION  AND   REFRACTION  399 

§493.  Visibility  and  invisibility.  A  self-luminous  object  is 
visible,  but  most  objects  depend  on  obstructing,  bending,  and  re- 
flecting light  from  without,  and  so  producing  contrasts.  Everyone 
knows  the  protective  invisibility  of  birds  and  animals  in  their  natural 
surroundings,  but  we  mean  more  than  this. 

'  Clear  '  water,  a  sheet  of  '  clear  '  glass,  or  a  good  mirror,  may 
show  the  reflected  images  of  objects  (and  often  leads  from  ex- 
perience to  a  suspicion  of  its  presence),  but  is  itself  invisible.  We 
have  all  blundered  into  such  surfaces. 

Smash  the  glass,  and  the  fragments  are  visible  by  their  varying 
refractions  and  total  reflections,  the  most  visible  part  of  a  chip 
depending  on  the  direction  in  which  it  is  viewed.  '  Grind  '  its  surface, 
or  powder  its  fragments,  and  the  multitude  of  reflections  from 
scratches  or  grains  flings  light  practically  equally  in  all  directions. 
A  cloud  is  a  swarm  of  droplets ;  froth,  of  bubbles ;  snow,  of 
crystals ;  paper  and  fabrics,  of  fibres.  Each  individual,  under 
the  microscope,  is  perfectly  pellucid,  but  light  incident  on  the 
immense  irregular  swarm  suffers  so  many  local  and  differently 
aimed  reflections,  etc.,  that  it  is  scattered,  or  '  irregularly  reflected,* 
equally  in  all  directions,  i.e.  the  object  is  equally  visible  in  all 
directions. 

Similarly  light  gets  through  them,  but  irregularly ;  they  are 
translucent. 

The  face  of  pressed  paper  has  been  so  far  flattened  that  it  shows 
much  nearly  regular  reflection  or  gloss,  especially  very  obliquely. 


-"X/^^*^ 


Fia.  196. 


Reduce  refraction,  and  reflection  is  reduced  also.  Ice  has  nearly 
the  same  refractive  index  as  water,  and  in  water  its  outline  almost 
disappears,  while  its  contained  air  bubbles  remain  extremely  visible. 
A  glass  rod  is  more  refractive,  and  not  so  invisible  as  the  ice,  in 
water,  but  in  xylol  or  oil  of  cedar  it  completely  disappears.  Oiled 
silk  and  oiled  ground  glass  are  nearly  transparent. 

How  this  comes  about  can  be  seen  by  considering  the  analog>'  of 
Waves  encountering  a  Reef.  At  low  tide.  Fig.  195,  left,  they  cannot 
pass  over  it  at  all,  but  are  reflected  completely,  except  for  losses  in 
cracks  in  the  rock.  This  corresponds  to  reflection  from  an  opaque 
metal. 

At  haK-tide  a  fraction  of  the  wave-energy  carries  over  the  reef, 
but  only  a  small  one,  for  there  is  not  much  water  to  carry  it,  and  what 
there  is,  is  shallow,  and  therefore  carries  it  slowly,  §  392.  So  that 
when  Y/v  =  y.  ia  large,  there  must  still  be  a  lot  of  reflection  :  when 
there  is  a  large  change  of  refractivity  at  the  surface  there  will  be 


400  LIGHT  [§  493 

a  good  deal  of  light  reflected  for  us  to  see  it  by,  a  diamond  (x  2-4 
will  always  be  more  conspicuous  than  a  chip  of  ice  (i  1-3.  [A  trans- 
parent substance  reflects  the  fraction  (jx^  _  1)2  of  the  light  incident 
upon  it  normally.] 

At  high  tide  the  change  in  speed  is  less,  Y jv  approximates  to  1, 
and  the  bulk  of  the  energy  carries  on  over  the  reef,  which  now  causes 
but  little  throw-back.  This  is  the  case  of  ice  in  water,  glass  in  oil, 
etc. 

[If  the  top  of  the  reef  now  consisted  of  a  large  area  of  very  rough 
weed-grown  rocks,  the  waves  would  neither  be  thrown  back  nor 
pass  through :  this  is  an  absorbent  surface  like  black  cloth,  necessarily 
nearly  opaque.] 

It  is  difference  in  refractive  index  at  irregular  interfaces  that 
produces  the  well-known  visible  streaming  of  hot  air  in  cold,  of 
petrol  vapour  in  air,  of  whisky  or  sjn'up  in  water,  etc.  ;  see  §  488. 

Opacity  helps  visibility ;  directly,  as  in  threading  a  needle 
against  the  light,  or  in  obscuring  the  reflection  from  white  paper 
(Indian  ink  v.  watery  ink) ;  indirectly  by  letting  less  light  leak 
through,  and  so  maintaining  reflecting  power  (contrast  clearness  of 
printing  on  heavy  opaque  white  paper  and  on  tracing  paper) ; 
also  by  casting  shadows. 

Colour  is  a  selective  opacity,  §  557  ;  its  utility  in  producing 
contrasts  needs  little  comment. 

Uniform  illumination  in  all  directions  destroys  all  contrast,  and 
causes  invisibility,  see  §  969.  An  instance,  striking  although  im- 
perfectly conditioned,  is  this  :  a  sunbeam  straying  through  a  chink 
in  the  blind  into  a  dusty  room  looks  almost  solid  ;  pull  up  the  blind 
and  let  the  wide  light  in,  and  the  dust  is  quite  invisible. 

You  will  flnd  this  paragraph  illustrated  to  perfection  in  the  process 
of  Making  your  Micro-slides,  in  Histology.  The  section  reaches 
you,  permeated  by  paraffin  wax,  as  a  translucent  patch  on  the  slide  ; 
dropping  xylol  on  to  dissolve  away  the  wax,  it  vanishes  as  completely 
as  if  you  had  washed  it  off  the  glass,  but  strong  alcohol  to  remove 
the  xylol  makes  it  reappear  as  a  white  patch,  for  its  refractivity 
is  a  good  deal  more  than  that  of  alcohol ;  then  you  dye  it  in  one  stain 
after  another,  wash  with  alcohol,  '  clear  '  that  away  by  oil  of  cloves, 
but  the  section  remains  plainly  visible  now,  on  account  of  its 
colourings  ;  then  this  is  merged  into  the  mountant  of  equal  refrac- 
tivity, and  unstained  structures  are  left  invisible  (not  so  stray 
drops  of  spirit). 


REFLECTION  AND  REFRACTION  401 


EXAM   QUESTIONS,   CHAPTER   XXXII 

See  how  this  chapter  amplifies  §§  405—409  of  Chapter  XXVI  for  the  special 
case  of  light  waves  :  don't  make  a  double  task  of  it,  but,  between  them, 
imderstand  it.  Draw  the  refraction  diagrams  line  by  line,  so  as  to  see  how 
they  grow ;  recollect  these  lines  of  light  are  broad  streams  of  waves. 

In  your  lab.  experiments  notice  the  much  greater  accuracy  of  prism  and 
total  reflection  methods  than  of  block  methods  :  Figs.  193  and  194  any  easy 
enough  to  rig  up,  else  leave  them.  The  practical  questions  are  always  being 
set. 


1.  Give  the  laws  of  regular  reflection  of  light,  and  show  that  the  image  is 
as  far  behind  a  plane  mirror  as  the  object  is  in  front.  How  would  you  allow 
for  thickness  of  glass  ?     How  do  you  explain  the  general  visibility  of  paper  ? 

2.  Give  reasons  for  the  accepted  view  that  light  is  a  wave  motion  in  a 
transmitting  medium.     State  what  you  know  about  waves  of  light. 

3.  How  is  the  refraction  of  light  explained  by  the  wave  theory  ?  Give 
a  diagram  showing  the  refraction  of  a  plane  wave  when  passing  from  air  into 
water,     (/a  =  4/3) 

4.  How  does  the  velocity  of  light  alter  as  it  passes  from  one  medium  to 
another  ?     Describe  illustrative  experiments. 

5.  State  the  laws  of  refraction  of  light,  and  explain  the  physical  meaning 
of '  refractive  index.' 

Light  is  incident  at  60°  on  the  surface  of  a  liquid  of  in^ex  4/3  floating  on 
a  lower  liquid  of  index  5/3.     Show  its  course  in  a  scale  diagram. 

6.  Show  that  the  depth  of  a  liquid  is  always  greater  than  it  appears  to  be, 
at  least  /*  times.  Draw  a  figure  showing  how  the  discrepancy  increases  with 
obliquity.     (  X  2) 

7.  A  coin  appears  to  be  9-3  cm.  under  water  of  index  4/3 ;  when  the  water 
is  replaced  by  another  liquid  to  the  same  depth,  the  distance  is  apparently 
8-1  cm. ;   calculate  index  of  liquid,  and  index  from  water  to  liquid. 

8.  Explain  the  terms  angle  of  reflection,  angle  of  refraction  and  critical 
angle. 

An  object  is  10  cm.  above  the  surface  of  water.  Determine  its  apparent 
position  to  an  eye  in  the  water  20  cm.  vertically  below  it. 

9.  Explain  Angle  of  Minimum  Deviation,  as  applied  to  a  prism,  and  deduce 
a  relation  connecting  it  with  the  angle  and  refractive  index  of  the  prism. 
How  does  it  depend  on  the  colour  of  the  light  ? 

Two  thin  glass  prisms  of  angles  5°  and  7°  are  made  of  glasses  of  refractive 
indices  of  1-50  and  1-65  respectively.  Calculate  the  deviation  produced  by 
them  when  their  refractions  (a)  assist  and  (6)  oppose  each  other.     (  x  5) 

10.  Define  refractive  index,  and  describe  two  methods  of  determining  that 
of  a  liquid  for  monochromatic  light.  Explain  which  method  is  the  more 
accurate. 

11.  Define  the  refractive  index  and  critical  angle  for  two  media,  and  deduce 
the  relation  between  them.  How  would  you  determine  the  critical  angle 
from  water  to  air  ? 

12.  Define  the  critical  angle,  and  construct  it  from  a  medium  of  index 
6/3  (o)  into  air,  (6)  into  water,  4/3.     (  X  2) 

13.  Prove  that  the  largest  refracting  angle  of  a  prism  which  will  transmit 
a  beam  of  light  is  twice  the  critical  angle,  and  calculate  the  maximum  index 
for  transmission  through  a  90°  prism.     (  X  2) 


402  LIGHT 

14.  Under  what  conditions  does  '  total  reflexion  '  occur,  and  how  is  the 
'  critical  angle  '  related  to  the  refractive  indices  of  the  media  concerned  ? 

Give  a  careful  description  of  a  practical  method  of  measuring  refractive 
index  by  utilizing  total  reflexion.     (  X  6) 

15.  A  beam  of  white  light  is  projected  normally  from  water  into  air.  The 
angle  of  incidence  is  then  gradually  increased  to  90°.  Describe  and  explain 
the  effects  observed.     (  X  2) 

16.  A  small  electric  bulb  is  alight  8  cm.  below  the  surface  of  a  liquid  of 
index  5/3 ;  calculate  the  radius  of  the  circle  bounding  the  area  of  emergence 
of  the  light. 

17.  Show  that  a  diver,  looking  up  through  a  flat  pane  of  glass  in  his  helmet, 
would  not  see  the  distorted  view  of  Fig.  192. 

18.  Explain  the  mode  of  action  of  total  reflection  prisms,  and  give  instances 
of  their  use.  Compare  their  advantages  and  disadvantages  with  those  of 
mirrors; 

19.  Looking  into  the  largest  face  of  a  right-angled  prism,  you  see  an  eye 
in  the  corner.     Whichever  eye  you  shut,  this  remains  open.     Explain  this. 

20.  A  narrow  beam  of  sunlight  slants  down  through  smoky  air  into  slightly 
soapy  water  in  a  darkened  vessel.  Crossing  the  beam  where  it  meets  the 
water  appears  another  identical  beam,  equally  inclined  the  other  way,  the 
two  forming  the  two  sides  of  an  X,  and  being  perfectly  straight  throughout. 
Explain  this,  and  show  how  you  could  quickly  prove  that  refraction  really 
is  taking  place. 

21.  Explain  what  happens  when  a  beam  of  light  travels  some  distance 
through  air  of  varying  density.      Give  instances  of  this. 

22.  Show  in  diagrams  the  possible  paths  of  light  through  air  overlying 
surfaces  which  differ  greatly  in  temperature  from  the  atmosphere. 


PBACTICAL  QUESTIONS 

With  pins  and  paper,  plot  several  rays  through  a  glass  block,  and  deduce 
refractive  index. 

Or,  draw  a  caustic,  deduce  it  from  that,  and  confirm  independently. 
Various  forms  of  the  '  real  and  apparent  depth  method.' 
Trace  rays  through  prism,  find  minimum  deviation  and  calculate  refractive 
index. 

Refractive  index  by  measurement  of  critical  angle,  by  air-cell,  etc. 


CHAPTER  XXXIII 


LENSES 


§501.  A  Lens.  Suppose  a  small  prism  of  narrow  angle  A  at 
height  AL  above  an  axis  LF,  Fig.  196.  At  L  on  the  axis  is  a 
prism  of  angle  zero,  i.e.  a  flat  piece.  Of  the  plane  waves  of  a 
broad  '  parallel '  beam  of  light  arriving  from  the  left,  the  portion 
falling  on  A  will  be  bent  down  (§§  409,  490)  and  overlap  the 
portion  from  L  at  F,  where  the  illumination  will  be  increased  at 
the  expense  of  the  stretch  of  shadow  AP. 

The  slope  of  the  beam  AF  is  the  small  deviation  produced  by  A, 
and  can  be  expressed,  as  always  on  railways,  as  a  Gradient  of 
AL  in  AF  or  AL  in  LF,  since  AL  is  hardly  distinguishable  from 
the  arc  of  a  circle  of  radius  AF  or  LF  (recollect  that  angles  get 
exaggerated  in  making  plain  diagrams). 


Fig.  196. 


Half-way  between  L  and  A  put  a  prism  of  angle  JA  ;  this  inclines 
its  light  ^AL  in  LF  and  again  increases  the  light  at  F,  leaving  P' 
in  shadow.  And  if  all  LA  is  filled  with  prisms  the  angles  of  which 
are  proportional  to  their  distances  from  L,  all  the  light  will  be 
concentrated  near  F. 

With  advantage,  a  curved  piece  of  glass  replaces  separate 
prisms  (mere  thickness  matters  little),  and  the  curve  must  be  such 
that  the  angle  increases  regularly  in  proportion  to  the  distance 
from  the  axis  LF.  Now,  to  walk  in  a  circle  one  must  change  one's 
direction  equally  at  every  step,  and  the  whole  change  is  propor- 
tional to  the  distance  walked.  That  is,  a  circular  arc  will  suit 
our  purpose,  provided  that  it  is  so  slightly  curved  that  it  does 
not  signify  whether  we  measure  along  the  arc  or  along  the  chord 
LA.  We  have  arrived  at  a  piece  of  a  '  plano-convex  *  lens  which 
will  concentrate  all  the  sunlight  falling  on  it  to  a  small  focus 
(hearth)  F  in  the  midst  of  a  cold  shadow  PP'. 

403 


404 


LIGHT 


[§501 


Lenses  are  pieces  of  refracting  substance  bounded  by  surfaces 
which  are  portions  of  spheres  (plane  =  infinite  sphere).  Half  a 
dozen  varieties  are  distinguished  in  Fig.  197,  double  or  bi-convex 


Fig.  197. 


1  and  -concave  2,  plano-convex  3  and  -concave  4,  and  meniscus 
or  periscopic  convex  5  and  concave  6  (or  concavo-convex). 

Convex  lenses  are  thickest  in  the  centre  and  concave  thinnest. 


> 


U^ 


Fig.  198. 


Do  not  be  surprised  if  these  '  optical  middles  '  of  round,  oval,  oblong, 
etc.,  spectacle  lenses  are  not  just  in  the  middle  ;  the  size  and  shape 
of  the  edge  is  a  matter  of  fancy,  it  is  ground  down  after  the  lens 
faces  are  finished. 


§  504]  LENSES  405 

§502.  Consider,  then,  a  Convex  Lens  with  plane  ripples  of 
sunlight  falling  on  it  (burning  glass).  They  leave  it,  not  as  separate 
streams  as  in  the  disjointed  diagram,  but  curved  to  circular 
ripples  which  all  close  in  on  the  burning  focus  F,  ami  then  spread 
out  beyond  it.  Fig.  198  (A).  Translated  into  '  rays  '  (i.e.  lines  of 
travel  of  waves,  see  §471,  parallel  rays  become  convergent  radii, 
which  all  pass  through  F,  and  then  diverge  indefinitely.  Working 
in  a  dusty  or  smoky  room,  the  parallel  sunbeam  is  seen  to  become 
a  cone,  brightening  as  it  approaches  the  vertex  F,  and  then  spreading 
until  it  becomes  too  diffuse  to  follow.  All  round  the  cone  is  the  dark 
shadow-space  robbed  of  its  sunlight. 

Using  light  from  the  moon,  or  a  distant  lamp,  which  is  not  blinding, 
put  the  eye  at  F,  and  look  at  the  lens,  i.e.  get  someone  to  hold  and 
move  it  until  he  sees  the  bright  focus  on  the  pupil  of  your  eye. 
Every  part  is  sending  light  to  your  eye,  and  the  whole  lens  appears 
ablaze.  Move  your  eye  into  the  dark  space,  and  the  lens  becomes 
a  black  disc  hiding  the  moon.  Put  your  eye  in  the  cones,  and  the 
size  of  the  bright  patch  seen  becomes  less  and  less  the  farther  you 
go  from  F,  i.e.  the  less  light  you  are  getting. 

Conversely,  let  F  be  a  little  lamp  emitting  light  on  its  own 
account.  Bulging  spherical  waves  spread  along  all  radii.  Those 
that  fall  on  the  lens  are  retarded  in  the  middle  by  the  greater 
thickness  of  slow-speed  substance  there,  issue  as  plane  waves, 
and  travel  straight  in  the  same  parallel  beam  as  before  (Fig.  198,  B), 
but  backwards.  This  occurs  in  railway  signal-lamps,  brilliant 
only  when  seen  full  in  front ;  a  feeble  light  fills  the  dark  space 
because  all  parts  of  the  lantern-box  (suggested  by  the  dotted  lines) 
are  lit  up,  and  scatter  light  through  the  bull's-eye  window. 

§503.  Now  consider  a  Concave  Lens,  Fig.  198  (C).  All  its 
constituent  *  prisms  '  are  turned  the  other  way  about,  and  plane 
incident  waves  become  spreading  circles — parallel  '  rays  '  become 
divergent — ^just  as  if  they  came  from  a  centre  F'.  Standing 
behind,  your  eyes  receive  light  along  directions  LE,  L'E',  ana 
you  are  convinced  that  the  source  is  at  F',  whence  both  streams 
appear  to  come. 

Conversely,  if  another  (a  convex)  lens  were  concentrating  light 
from  the  right  on  F',  the  lens  would  prevent  it  getting  there, 
sending  it  away  in  a  parallel  beam,  the  thicker  slow-motion 
substance  at  the  outside  retarding  the  ripple-ends  just  enough  to 
make  the  ripples  straight. 

There  is  nothing  at  F'  to  be  caught  on  a  screen,  no  hearth  of 
light  and  heat ;  only  through  the  glass  there  appears  to  be  some- 
thing there  ;  F'  is  a  virtual  focus.  In  practice  it  is  located  as  the 
intersection  of  sight-lines  EL,  E'L',  produced. 

[In  the  lab.  a  weak  real  focus  may  occur  near  F',  due  entirely 
to  light  reflected  from  front  of  bi-concave.] 

§  504.  Optical  centre  of  a  thin  lens.  If  the  lens  is  slanted  a  little. 
where  will  F  be  ?     Experiment,  and  you  find  it  stops  where  it  is. 


406 


LIGHT 


[§504 


Near  the  middle  of  the  lens  a  point  L  can  be  found  such  that  straight 
rays  drawn  through  it  meet  both  faces  of  the  lens  at  'places  where  they 
are  parallel.  These  rays  therefore  pass  without  bending,  suffering 
only  a  trifling  '  side-step,'  Fig.  181,  which  in  a  '  thin  '  lens  is  ignored. 
L  is  the  optical  centre  of  the  lens  in  Fig.  197  ;  it  has  been  found  as  the 
intersection  of  two  rays  (shown),  each  of  them  satisfying  the  above 
condition.  Lens  diagrams  are  started  by  drawing  straight  rays 
through  it.  One  of  them  happens  to  be  perpendicular  to  the  lens, 
but  this  is  hard  to  find  in  practice — single-lens  diagrams  have  no 
fixed  '  centre-line.' 

On  any  of  these  central  rays  are  points,  F  for  convex,  F'  for 
concave,  on  both  sides,  at  the  principal  focal  distance  /,  of  the  lens, 
from  L.  This  is  the  same  on  both  sides,  the  illusory  difference 
with  a  meniscus  '  landscape  '  lens  explains  itself  in  Fig.  197  (V). 

§  505.  Waves  from  miles  away  are  flat  enough,  or  '  rays ' 
'  parallel '  enough,  yet  why  is  the  focus  of  the  sun,  with  a  good  lens, 
a  sharp  round  patch,  and  not  a  point  ?  Bundles  of  parallel  rays 
come  from  different  parts  of  the  sun,  but  the  bundles  are  not  parallel 
to  one  another.  Each  has  its  own  point  focus  ;  all  these  lying  side 
by  side  build  up  the  patch.  Some  bundles  start  from  less  brilUant 
parts,  their  foci  look  dark — sunspots.  An  Image  of  the  distant 
object  has  been  formed  in  the  principal  focal  *  plane  '  of  the  lens. 
Any  point  in  this  is  a  principal  focus,  therefore  don't  draw  a  lens 
with  a  solitary  dot  on  each  side  invidiously  exalted  as  '^principal 
focus.' 

§  506.  Now  take  light  spreading  in  circular  ripples  along  rays 
from  a  point  not  far  away.  Fig.  199.  These,  hindered  so  much 
in  their  middles  by  the  thicker  slow-speed  glass  in  the  middle, 
become  concave,  and  travel  down  radii  to  a  centre  I  which  is  found 
thus  : 


Fig.  199. 

(1)  It  lies  on  the  undeviated  ray  OL  through  the  optical  centre. 

(2)  Ray  OA  is  bent  just  the  same  amount  as  before,  in  Fig.  198, 
A,  since  the  particular  direction  of  incidence  hardly  affects  the  devia- 
tion by  a  thin  prism.  Fig.  188. 

When  some  of  the  radiation  from  one  point  concentrates  at 
another,  the  second  point  is  the  image  of  the  first  object  point, 
and  they  are  at  conjugate  focal  distances  from  the  lens,  or  mirror. 


508]  THIN   LENSES 


407 


O  can  be  a  little  pocket-lamp  bulb ;  the  eye  placed  at  I  will  see 
the  whole  lens  flashing  full  of  light,  as  in  §502.  O  and  1  are 
interchangeable  as  far  as  the  lens  is  concerned. 

A  Concave  Lens  lets  the  middles  of  the  waves  through  fa«ter, 
and  they  bulge  more,  as  if  they  came  from  the  virtual  image  of  O,* 
on  the  same  side  of  the  lens  as  O.     It  is  not  interchangeable  with  o! 

Notice  the  Distinction  between  Real  and  Virtual  Images.  Real 
Images  are  formed  where  rays  come  and  meet,  they  are  to  l>e 
seen  actually  in  the  air  by  an  eye  anywhere  within *the  cone  of 
rays  beyond  them.  I  have  had  a  parrot  industriously  pecking 
at  one,  and  remarking  sotto  voce  on  its  unsatisfying  lack  of  flavour 
and  its  indestructibility.  But  Virtual  images  are  apparitions  seen 
only  '  through  '  the  glass  :  characteristically  they  are  usually  your 
own  individual  property,  and  your  eye  is  close  up  to  the  glass. 


THIN  LENSES 

§507.  Referring  now  more  carefully  to  Fig.  197  and  §  504, 
although  the  light  goes  on  in  the  same  direction,  it  has  suffered 
the  slight  'side-step'  of  §487,  Fig.  181.  This  we  here  iynore, 
as  it  compUcates  matters  badly,  but  to  justify  our  doing  so  wt 
must  keep  our  lenses  thin,  and  our  angles  of  incidence  small,  much 
smaller  than  the  exaggerated  angles  necessary  in  clear  diagrams. 
All  that  follows,  up  to  §  535,  deals  therefore  with  lenses  like  spectacle 
lenses,  of  thickness  very  small  compared  with  the  other  distances 
measured. 

In  §  541  we  will  remove  the  thickness  limitation  ;  and  in  Chapter 
XXXVIII  consider  wider  angles. 

§  508.  Relations  will  now  be  worked  out  to  connect  the  refractive 
index  y.  of  the  material  of  a  lens,  the  curvatures  l/r^  and  l/r,  of  its 
faces,  its  focal  power  1//,  and  conjugate  focal  powers  1/a  and  1/6. 
All  these  quantities  appear  as  reciprocals  because  we  persist  in 
measuring  the  lengths  of  slopes  instead  of  their  steepnesses. 

In  Fig.  200  a  few  waves  are  admitted,  as  a  hint  of  what  is  really 
going  on,  and  then  they  are  faded  out  and  their  lines  of  flight  con- 
sidered.    The  whole  diagram  is  strictly  to  horizontal  scale. 

In  Fig.  200  I  the  angle  A  between  the  faces  of  the  lens  at  its  edge 
is  also  the  angle  between  their  radii  of  curvature  there,  §  152,  for 
each  radius  is  perpendicular  to  its  sphere.  A  is  therefore  the  sum 
of  the  angles  at  Cj  and  C^,  or,  speaking  railway  fashion,  the  sum  of 
the  gradients  AL  in  LCi  and  AL  in  LCj  ;  L  being  the  Optical  Centre 
and  Ci  C2  the  centres  of  the  spheres,  of  radii  fj  fj,  of  which  the  lens 
surfaces  form  parts. 

Ignoring — because  angles  are  small — the  difference  of  length 
between  AC,  and  LC,,  etc., 

.     .       AL  ,   AL 

angle  A  = h  -  -  • 

f,  r^ 


408 


LIGHT 


[§508 


Now,  taking  a  '  ray  '  through  A  parallel  to  the  central  ray  LC 
{i.e.  two  pinhole  streams  of  the  same  wave,  §  471),  it  is  bent  down 
through  an  angle  D  =  (ja  —  1)  A,  §  409,  and  meets  the  central  ray 
at  the  Principal  Focal  Distance  / 


/.  D  =  gradient  AL  in  LF  =  AL// 
/.  D  =  (fji  —  1)  A  becomes 


Now,  we  are  not  compelled,  in  a  diagram,  to  work  to  the  same 
scale  horizontally  and  vertically,  any  more  than  a  profile-map- 


FiG.  200. 


maker  is,  and  we  can  call  AL  whatever  we  like 
'  units  of  vertical  height.'     Choosing  the  simplest 


1  or  40  or  100 


D 


j=i^ 


1) 


i  +  i 

^1       ^f. 


which  says  that  The  focal  power  of  a  lens  is  the  product  of  {the  re- 
fractive index  of  its  material,  less  1)  and  the  aggregate  curvature  of  its 
faces. 

§  509.  Now  let  us  give  an  additional  new  definite  meaning  to  D. 

The  optician  uses,  as  Unit  of  Focal  Power  or  Strength  of  a  Lens, 
the  metric  Dioptre,  which  is  the  strength  of  a  lens  the  sun's  image  of 
which  is  1  Metre  away  from  it,  1  m.  principal  Focal  Length. 

We  are  going  to  follow  him,  and  save  ourselves  a  tangle  of  algebra. 
Hence,  again.  The  strength  of  a  lens  in  Dioptres  is  [l  —  I  times  the 
aggregate  curvature  of  its  faces,  their  radii  being  in  metres. 


D  =  (ix-l)(i  +  i)  dioptres. 


§509] 


THIN   LENSES 


409 


He  uses  a  Lens  Gauge,  Fig.  201,  which  is  a  spherometer  with  two 
fixed  points  only  ;   the  spring  middle  point  works  magnifying  gear 
and  reads  direct  on  the  dial  the  bulge  of  the  lens  above  or  below 
the     straight     chord    between    the    fixed 
points  :    this  is  checked   by   a   flat   glass, 
which  must  read  0  on  the  dial. 

Pressing  on  the  two  sides  of  the  lens  in 
turn,  the  readings  would  therefore  be  l/fj 
and  l/r-g,  but  he  makes  it  even  more  direct 
reading  by  embodying  the  multiplying 
factor  (ji  —  1  in  the  machine.  For  hard 
crown  spectacle-glass  (jl  =  1-515  and 
[1  —  1  =  ^  nearly  enough  for  the  oculist, 
therefore  the  dial  is  graduated  to  read 
straightaway  i{l/r^)  and  iil/r^),  and  D  is 
the  sum  of  the  readings.  Due  heed  must 
of  course  be  paid  to  sign  ;  you  see  on  the 
dial  concave  curves  are  reckoned  —  falling 
short  of  the  flat,  and  convex  bulges  are  + . 

A  convex,  convergent,  burning,  magnifying,  long-sight  lens  is  a  -\-  D. 

If,  instead  of  measuring  in  metres,  the  old  English  custom  of 
measuring  /,  r^  and  rg  in  inches  is  adhered  to,  the  oculist  reckons  a 
metre  as  40  in.  and  writes  : 


Fig.  201. 


-^  =  D  =  (, 


ihd 


in  inches 


Or,  as  we  mostly  work  in  centimetres,  the  form  of  the  equation 
which  should  be  adhered  to  in  the  laboratory  is 


100 
/ 


D 


(,-.||!^+'"' 


(^-f') 


m  cm. 


Evidently  Dioptric  strength  =  reciprocal  of  focal  length  in  metres, 
e.g.  a  2D  lens  is  a  half -metre  convex ;  a  —  4D  is  a  25-cm.  focus 
concave  ;  a  20D  is  a  2-in.  focus  pocket-lens. 

The  Dioptre  is  the  bending  that  the  lens  does,  and  Fig.  202  shows 


SCALE    OF    DIOPTRES 


io   f^jr^^i 


Fio.  202. 


you  a  scale  of  them,  as  well  as  can  be  done.  If  you  will  put  a  pro- 
tractor to  A,  and  measure  the  angles,  you  will  find  that  the  first 
four  are  practically  5-5°  each  ;  after  that  they  fail,  but  simply 
because  to  make  a  diagram  at  all  the  vertical  height  had  to  be 


410  LIGHT  [§  509 

exaggerated  tenfold,  making  distortions  a  hundredfold  :  actually 
the  angle  ascribed  to  4D  is  that  of  40D,  an  inch  focus  strong  pocket - 
lens.  These  angles  were  drawn  by  joining  the  reciprocals  of  the 
metric  lengths  to  A,  so  you  can  appreciate  how  closely  this  dioptre 
scale  of  '  shortnesses  '  represents  equal  increases  in  refraction  in 
the  lens  :  beyond  40D  the  microscope-maker  reverts  to  focal  lengths. 
A  is  where  you  skid,  and  the  dioptric  figure  where  you  hit  the  wall 
measures  the  badness  of  the  skid  :   does  that  make  it  any  plainer  ? 

§  510.  Fig.  200,  II  (on  half  the  scale  of  I  and  III,  to  get  it  in) 
shows  the  condition  of  §  506,  Fig.  199,  when  the  '  object '  source 
of  light  is  near  at  hand  ;  spherical  waves  are  spreading  from  it,  of 
which  OL  and  OA  are  radii,  or  '  rays.' 

At  A  the  same  bending  D  is  suffered  as  before,  and  is  now  the 
aggregate  of  the  two  gradients  AL  in  LO  and  AL  in  LI. 

Call  the  oBject  distance  b  and  the  imAge  distance  a 

—  or  D  =  — \-  r  '  •        in  metres 

/  ah 

100       -r.       100   ,    100 

—ir-  or  D  = — J—    .  .  m  cm. 

/  a     '      b 

or  The  Dioptric  strength  of  a  lens  is  the  aggregate  of  its  conjugate  focal 
powers,  O  and  I  being  conjugate — yoked  together — in  a  way  which 
laboratory  practice  will  make  very  plain.  Incidentally  you  see 
I  have  turned  the  lens  round  just  to  show  that  it  makes  no  difference 
(until  you  come  to  Chapter  XXXVIII). 

§511.  Fig.  200  III  shows  the  third  case,  when  the  Object 
point  has  come  very  close.  D  is  now  not  enough  to  bring  the  wide- 
spreading  light  in,  and  it  continues  to  spread,  though  not  so  fast, 
as  if  from  a  more  distant  Image  point  I,  which  is  plain  enough  to  see 
when  you  look  through  the  lens  towards  it,  but  has  no  real  existence, 
the  long  pecked  line  being  only  the  direction  of  the  emergent  light 
produced  backwards  ;  it  is  a  Virtual  Image,  call  it  I'. 

D  is  now  the  difference  between  the  angles  at  I'  and  O,  1/6  -^  I /a. 

What  business  have  we  to  expect  our  previous  relation  to  hold  ? 
I'  never  handles  the  goods  at  all,  it  is  *  but  a  magic  shadow  show.' 

Well,  the  simplest  way  of  expressing  that  is  to  label  it  with  a 
minus  sign,  and  then  if  you  write 

and  put  in  your  arithmetical  values  (try  the  diagram  itself),  the 
thing  comes  out  right.  Don't  tinker  with  the  signs  on  the  main 
line — they  all  add — until  you  have  an  actual  arithmetical  value 
in  hand  to  subtract. 

Fresh  from  school  algebra,  you  will  look  askance  at  my  way  of 
using  the  minus  :    it  doesn't  mean  that.     Doesn't  it  !     Wait  until 


512] 


THIN   LENSES 


411 


you  have  lent  a  few  friends  (and  a  few  firms)  various  sums  of  money, 
i.e.  you  have  —  received  them,  and  have  entered  them  on  the  cremt 
or  —  side  of  your  account  book.  Now  go  and  get  them  back  ;  and 
you  will  be  supernaturally  lucky  if  you  don't  find  that  '  minus  * 
has  a  far  more  fundamental  meaning  in  real  life  than  algebra  ascribes 
to  it. 

In  II,  object  and  real  image  are  completely  interchangeable, 
but  here  in  III  real  object  and  conjugate  virtual  image  are  as 
Orpheus  and  Eurydice.  Interchange  now  would  mean  this : 
that  if  by  some  converging  lens  or  mirror  you  produce  streams  of 
light  converging  on  I',  the  interposition  of  L  will  cut  off  its  supplies 
and  converge  them  on  O  instead,  making  it  a  real  image  of  what 
may  now  be  called  a  Virtual  Object.  This  is  made  use  of  in  measur- 
ing concave  lenses,  see  Questions  9  and  28,  and  §§  535,  627. 

§512.  Now  look  at  Fig.  203.  The  top  figure  you  recognize 
as  Fig.  199  twice  over,  one  for  the  top  and  one  for  the  bottom  of  the 
luminous  object  on  the  left,  showing  how  their  waves  converge 


Fia.  203. 

on  to  image  points  on  the  right,  and  then  spread  away,  unless 
stopped.  Some  direction  lines  in  which  the  waves  are  travelling 
have  been  sketched  in ;  it  is  what  is  happening,  but  looks  rather 
a  tangle,  although  only  every  4000th  wave  is  drawn. 

To  cope  with  it,  in  the  middle  figure  we  have  emphasized  certain 
convenient  little  short  sections  of  the  wave-fronts,  showmg  in  them 
every  thousandth  wave,  and  slighting  the  rest.  Indeed,  one 
system  has  only  enough  left  to  make  a  backbone,  to  which— just 
as  well  as  not — we  have  kept  a  stream  in  the  other  parallel. 

In  the  lower  figure  I  have  packed  the  waves  1000  times  closer 
along  these  directions,  as  they  really  are,  and  have  left  out  everj' 


412  LIGHT  [§  512 

other  part  of  them.  Students  often  know  quite  a  lot  about  diagrams 
hke  this,  but  haven't  the  faintest  idea  that  they  mean  anything 
in  practice.  After  all,  it  would  take  a  skilful  comparative  anato- 
mist to  reclothe  its  bones,  and  that  is  why  I  have  shown  you  the 
complete  animal  and  its  dissection  :  I  want  things  to  be  more  use 
than  a  feebly  amusing  game  of  cat's-cradle.  But  now  this  final 
skeleton,  or  Standard  Geometrical  Construction,  is  absolutely 
indispensable,  and  you  must  make  yourself  thoroughly  familiar 
with  it. 

Merely  sketched,  it  checks  blunders  in  calculation ;  drawn 
carefully  to  scale,  it  saves  all  calculation  ;  and  be  assured  that  good 
graphical  construction  is  at  least  as  acceptable  to  any  of  your 
examiners  as  is  numerical  calculation. 

To  make  Fig.  203  (lower),  from  both  ends  of  the  object  O,  placed 
parallel  to  the  lens,  draw  axial  '  rays  '  straight  through  the  Optical 
Centre  of  the  lens.  I  call  these  the  '  Scissors  Rays,'  from  the  way 
they  open  and  shut  as  the  object  pushes  between  them  to  or  from 
the  lens.  Necessarily  the  image  also,  wherever  it  is,  has  its  ends 
on  these  two  rays,  because  the  lens  has  to  collect  all  rays  from 
object-point  into  its  image-point,  and  here  are  rays. 

From  one  end  of  the  object  (I  have  used  the  top)  draw  a  third 
ray  parallel  to  that  from  the  other  end.  This  ray,  at  A,  does  all 
the  bending  in  the  picture,  to  cross  over  the  long  unbending  axial 
one  at  F,  at  the  Principal  Focal  Distance/  (marked  by  an  arc)  from 
the  lens  ;  and  then  continues  and  meets  its  brother  ray  in  the  image 
of  the  one  point  from  which  both  sprang. 

Draw  in  the  rest  of  the  Image  parallel  to  the  lens  and  object  : 
here  it  is  evidently  inverted. 

§  513.  Moving  object.  Standard  Construction  shows  readily 
what  happens  to  the  image  when  the  object  moves  to  or  from  the 
lens.  The  two  parallel  rays  can  be  regarded  as  '  Rails  ' ;  of  course, 
one  rail  is  also  one  '  scissors  ray,'  but  keep  it  still  and  it  will  not 
cause  you  any  confusion. 

The  Object  runs  on  the  Rails,  always  keeping  its  ends  on  them. 
They,  and  their  '  crossover  '  at  F,  remain  fixed,  and  only  the  long 
line  (pecked  in  its  second  positions  in  both  figures  of  Fig.  204), 
the  moving  jaw  of  the  '  scissors,'  alters  its  inclination,  as  shown  by 
the  dotted  arcs  in  Fig.  204. 

When  O  is  far  away,  the  swinging  line  is  only  slightly  inclined 
to  OLF,  and  I  is  near  F  ;  as  we  expect,  for  F  is  the  real  image  of 
O  '  at  infinity.'  (Consequently  the  dotted  arcs  start  from  the  line 
OLF.) 

As  O  travels  nearer,  the  line  tilts  more  and  more,  and  I  recedes 
along  AFI  away  from  the  lens,  and  gets  bigger.  (A  little  mark  on 
the  arcs  shows  the  swinging  line's  position  in  Fig.  203.)  Fig.  204 
starts  with  Og  at  twice  the  focal  distance  from  the  lens  :  that  makes 
the  triangles  ALF  and  Fig  equal,  and  I2  is  also  2/,  and  the  same  size 
as  O2  (or  the  length  of  the  hill  O2L  is  twice  the  length  of  hill  AF). 


§514] 


THIN   LENSES 


4)3 


Hitherto  O  has  approached  faster  than  I  receded,  now  object 
and  image  have  come  their  closest,  4/— shown  by  measuring  the 
diagram,  or  calculating  in  1/a  -f  1/6  =  1//  what  a  makes  (a  +  6) 
a  minimum,  or  by  experimental  '  copying  full  size.'  Henceforth 
I  recedes  faster,  rapidly  getting  larger— as  you  see  with  O,  and  I3— 
until  when  at  O4  it  reaches  distance/,  I  has  gone  to  infinity,  OL  having 
become  parallel  to  AF. 


Fio.  204. 


0  moving  nearer  still,  within  the  principal  focal  distance,  O5L  and 
AF  spread  apart,  never  to  form  a  real  image,  but  appearing  to  eyes 
on  the  right  of  the  lens  as  if  they  came  from  a  point  of  I5,  an  enlarged 
erect  Virtual  Image.  This  is  the  important  case  of  a  Magnifying 
Glass,  see  Fig.  266. 

The  construction  presently  fails  in  exactness  because  angles 
become  too  large,  but  it  shows  that  ultimately  object  and  virtual 
image  nearly  coincide  on  the  lens  surface  ;  a  reading-glass  laid  right 
on  the  page  has  practically  no  effect. 

§  514.  The  action  of  a  Concave  Lens,  with  its  virtual  images,  has 
already  been  explained  in  §  503.  In  calculation  its  focal  power 
must  be  written  —  1//,  for  it  has  the  very  reverse  of  combining 


Fio.  205. 


power.  The  Standard  Construction  applies  to  it  as  in  Fig.  205: 
the  same  two  axial  rays  are  drawn  and  the  third  laid  down  parallel. 
But  this  now  bends  up  in  the  direction  found  by  joining  A  and  F', 
which  is  at  the  principal  focal  distance  along  the  parallel  axial  ray 


414  LIGHT  [§  514 

on  the  same  side  as  the  object.  Virtual  I'  is  at  the  point  where  this 
prolongation  AF'  cuts  the  fellow-ray  from  O,  OL ;  for  it  is  seen 
along  LO  and  along  the  deviated  direction  EA.  The  whole  image 
lies  between  the  '  scissors.' 

As  O  runs  along  the  *  rails  '  from  infinity  up  to  the  lens,  I'  runs 
from  F'  up  to  the  lens  ;  two  positions  are  shown.  Image  is  always 
virtual  and  smaller  than  object,  m  increases  from  0,  up  to  I  when 
lens  touches  object. 

§  515.  Magnification.  The  Magnification  is  the  ratio  of  the  length 
of  the  image  to  that  of  the  object,  measured  as  diameters,  i.e.  across. 

Since  both  lie  between  the  '  scissors  '  axial  rays,  their  lengths 
are  evidently  proportional  to  their  distances  from  the  lens. 

,,       .„     ^.  distance  of  image  from  lens      a 

Maemfication,  m  —  3^- .    1  •     ^  j r —  =  t 

°  distance  of  object  from  lens      6 

and  if  they  lie  on  opposite  sides  of  the  lens  the  image  is  inverted. 

Inspection  of  the  diagram  shows  that  w  f or  a  convex  lens  can 
have  any  value  whatever  for  real  images,  but  must  exceed  1  for 
virtual.     For  a  concave  lens  it  is  less  than  1  for  any  real  object. 

§  516.  Magnification  in  depth.  If  the  image  of  a  small  object 
is  magnified  m  times,  its  thickness  along  the  axis  appears  greatly 
out  of  proportion,  being  magnified  m^  times. 

e.g.,  put  a  =  50  cm.  and  6  =  10  cm.,  which  makes  m  =  5 
100/50  +  100/10  =  D  =  12. 

Now  alter  6  by  1  mm.  to  9-9  cm. 

lOO/a'  +  100/9-9  =  12,    i.e.  lOO/a'  +  10-1  =  12 

.*.  100/a'  =  1-9,  a'  =  52-6  cm.,  a  shift  of  26  mm.,  whereas  the 
object  moved  only  I  mm. ;  or  (5)2  times. 

In  practice,  this  has  two  curious  effects  : 

(1)  If  your  Microscope  is  magnifying  500  diam.,  the  m.p.  in  depth 
is  250,000.  That  means  that  only  a  very  small  thickness  of  the 
section  can  possibly  be  in  fair  focus  for  the  eye  at  once,  that  the 
microscope,  unless  mismanaged,  makes  several  '  optical  sections  ' 
out  of  the  very  thinnest  mechanically  cut  one. 

It  also  means  that  a  great  deal  of  comfort  depends  upon  the 
excellence  of  the  fine  adjustment. 

(2)  If  an  ordinary  photograph  is,  say,  l/20th  the  size  of  the 
original,  the  depth  of  the  optical  image  formed  near  the  film  is 
only  1 /400th  the  depth  of  the  original.  That  is,  the  camera  can 
tolerate  some  depth  in  the  object  without  getting  badly  out  of 
focus.  And  if  a  smaller  camera  is  used,  giving  perhaps  1 /100th  the 
size,  the  thickness  of  the  image  reduces  to  1 /10,000th,  and  the  focal 
tolerance  increases,  as  you  can  see  in  any  cinema  picture — almost 
everything  is  in  focus.     Enlarged  up  to  the  same  size  as  before, 


§518]  THIN   LENSES 


415 


the  tolerance  is  actually  five  times  as  much,  a  fact  which  has  put 
big  cameras  out  of  use. 

But  when  attempting  to  reconstruct  Stereoscopic  Solidity  from 
the  squashed  image,  one  gets  a  cut-out-flat  stage-scenery  effect. 

§  517.  Two  or  more  lenses  in  contact.  The  successive  refractions 
D2,  Dj,  etc.,  take  place  practically  at  the  same  point  A,  and  the 
resultant  Dioptric  Strength  is  of  course  the  aggregate  of  the  individual 
strengths  D^  +  Dg  +  D3  +  •  •  •  .  due  regard,  naturally,  being 
paid  to  any  negative  lens. 

For  combinations  of  lenses  not  in  contact  see  §  542. 

§  518.  A  lens  in  water.  Going  back  to  §  409,  y.  is  the  ratio  of 
speeds  outside  and  inside  the  prism.    For  any  medium 

jx  _  V  of  light  in  air 

1       V  of  light  in  the  medium 

similarly  (x'  for  another  medium  =  V/v' 

.*.  ratio  v'/v  of  speeds  of  light  in  any  two  media  =  (x/jx' 

So  all  one  has  to  do  is  to  write  jx/jx'  in  place  of  jx/l,  i.e.  of  a  single 
index. 

.*.   For  prism,  Deviation  =  f-^  —  1  j  A 

For  lens  D  =  (^,  -  l)(l  +  i,). 

So  that  a  glass  lens  which  had  a  strength  (1-5  —  1)  curvature  =0-5 
curvature  in  Air  has  only  a  strength  (1-5/1 -33  —  1)  =  0-125  cur- 
vature in  water,  a  reduction  to  a  quarter ;  and  so  on  for  any  two 
media.  Since  many  oils  possess  refractive  indices  near  to  that 
of  glass,  you  are  quite  likely  to  see  little  lenses-in-water  like  this 
under  the  microscope  ;  clove  oil  drops  in  sections,  perhaps,  or 
butter-fat  in  milk. 

If  the  surrounding  medium  is  of  identical  index,  refraction 
ceases  altogether  (invisible,  §  493,  or  Eye,  §  602). 

If  of  greater  index — e.g.  glass  1-5  in  carbon  disulphide  1*67  ; 
or  water  in  oil  or  balsam,  in  badly-cleared  micro-shdes — 1//  becomes 
proportional  to  (1-5/1-67  —  1)  =  —  0-1,  i.e.  changes  sign,  and 
begins  to  gain  strength  as  a  concave  lens.  Optically  it  is  a  cavitv 
between  two  hollow  refracting  cheeks,  like  an  air-bubble  in  liquid, 
another  favourite  microscopic  object. 


416  LIGHT 


EXAM.   QUESTIONS,   CHAPTER  XXXIII 

The  study  of  thin  lenses  is  commonly  carried  on  under  the  shadow  of  an 
algebraic  convention  as  to  '  sign  '  which  makes  distances  measured  opposite 
ways  from  the  lens  opposite  in  sign. 

That  sounds  simple  and  rational  enough,  and  so  it  is  from  the  point  of  view 
of  geometry,  but  it  has  not  the  slightest  physical  basis.  On  the  contrary, 
it  makes  out  to  you  that  what  is,  is  not;  and  what  does,  does  not.  The 
practical  result  is  that  students,  after  diligent  training,  come  to  their  exam 
and  select  a  sign  at  random,  for  that  gives  them  a  one  in  eight  chance  of  being 
right,  and  that  is  about  the  proportion  of  correct  calculations  that  most  of  us 
find  in  marking  many  thousands. 

The  convention  carries  you  no  further,  neither  into  optics  nor  into  the 
practical  work  of  lenses  which  is  in  the  hands  of  the  optician  and  the  oculist ; 
they  never  torment  themselves  with  what  is  good  enough  for  the  scholar 
because  his  writer  hasn't  looked  out  into  the  technical  world  this  century. 

I  have  given  a  physical  meaning  to  +  and  —  ;  I  have  steered  as  near 
professional  treatment  as  I  can  with  our  wide  range  of  problems  :  if  you 
can't  get  on  with  dioptres,  get  hold  of  some  book  of  your  grandfather's,  and  go 
without.  This  book  will  have  nothing  to  do  with  a  fallacious  bogey  which 
has  been  a  cloud  by  day,  obscxu'ing  the  whole  teaching  of  lenticular  optics, 
and  a  will-o'-the-wisp  by  night,  misleading  generations  of  students  through 
darkness  to  damnation. 

Please  pay  heed  to  the  way  the  line  diagrams  develop  from  the  full  wave 
systems — nobody  wants  you  to  reproduce  these  latter,  they  are  moving,  and 
you  want  a  steady  plan  of  operations,  such  as  you  can  make  out  looking  down 
at  streams  of  ants  at  your  feet — but  the  point  is,  they  are  physical  operations, 
not  cat's  cradles  nor  crochet  patterns. 

The  development  of  the  complete  relation  of  §  509  is  not  asked  for,  but 
several  Questions  hinge  on  your  recollection  of  its  result.  Work  at  the  whole 
Chapter ;  for  look  at  the  mass  of  questions,  and  more  complicated  ones  follow 
in  later  chapters. 

1 .  In  what  unit  is  the  Power  of  a  lens  expressed,  and  how  is  it  related  to 
the  focal  length  ?  What  convenience  arises  from  the  use  of  powers  rather 
than  of  focal  lengths  ?  How  could  you  at  once  distinguish  by  inspection 
(and  without  exploring  the  surfaces  with  the  finger)  a  convex  lens  from  a 
concave  ? 

2.  Give  experiments  which  distinguish  between  shadow,  real  image,  and 
virtual  image. 

3.  What  is  the  convenience  in  speaking  of  virtual  images  in  connection 
with  lenses  and  mirrors  ?  In  what  circumstances  does  a  concave  mirror 
form  real,  or  virtual,  images  ?  In  the  case  of  a  distant  object,  what  would 
be  the  difference  in  the  appearances  cast  on  a  screen  by  (a)  a  long  focus  convex 
lens,  (6)  a  short  focus  convex  lens,  (c)  a  concave  lens,  each  at  its  focal  distance 
from  the  screen  ? 

4.  Show  how  to  construct  image  in  a  thin  biconvex  lens.  Object  being 
at  a  distance  exceeding  /  from  lens,  will  an  increase  in  /  increase  or  diminish 
size  of  image  ? 

5.  Draw  diagrams  showing  formation  by  convex  lens  of  (a)  inverted 
magnified,  (6)  inverted  diminished,  (c)  erect  magnified,  images. 

6.  Draw  a  curve  showing  for  a  convex  lens  the  connection  between  distance 
of  object  from  one  principal  focus  and  of  image  from  the  other. 

7.  If  a  convex  lens  is  held  in  front  of  the  page,  and  moved  to  the  right, 
the  print  moves  to  the  left;  with  a  concave  it  moves  to  the  right.  Explain 
these  by  aid  of  diagrams. 


THIN   LENSES  417 

8.  How  can  you  quite  readily  distinguish  between  weak  lenses  and  plain 

Las?  "^ 


9.  A  lens  intercepts  light  converging  to  a  point  6  in.  beyond,  and  alters 
its  point  of  convergence  to  12  in.     Find  its  focal  length. 

[The  light,  at  the  lens,  possesses  40/6  =  6-66  dioptres  of  convergence;  and 
the  lens  alters  this  to  40/12  =  3-33,  takes  away  3-33,  is  of  —  3-33  D,  a  concave 
of  40/3-33  =  12  in.  focal  length. 

Or,  by  our  formula  1  /a  -f  1  /  6   =  1  //,  using  inches 

40/12  +  40/  (  -  6)  =  40//  =  D 

the  light  really  converges  at  12,  the  6  is  only  a  virtual  point,  because  the 
glass  is  in  the  way,  therefore  6  is  made  — ,  giving  3-33  —  6-66  =  —  3«33  = 
40/(-  12).] 

10.  What  lenses  would  produce  an  image  distcuit  20  in.  of  an  object  distant 
80  in.  ? 

[The  light  arrives  at  the  lens  with  an  uphill  slope,  a  spread  or  divergence 
40/80  =  0-5  dioptre.  The  lens  converts  this  into  a  divergence  from  a  point 
only  20  in.  away,  i.e.  a  divergence  of  40/20  =  2  dioptres,  evidently  by  adding 
1-5  D  divergence;  i.e.  it  is  a  spreading  or  diverging  lens,  a  —  1-5-D  lens,  a 
concave  of  40/1-5  =  26-7  in.  focal  length. 

Or  the  lens  may  converge  this  light  to  a  real  focus  20  in.  away,  i.e.  give  it 
a  real  convergence  of  40/20  =  2  dioptres,  having  first  destroyed  its  divergence. 
This  lens  has  to  be  a  +  2-5  D,  a  concentrating,  convex,  lens  of  40/2-5  =  16  in. 
focal  length. 

Or,  by  our  formula     1  /o  +  1  /6      =  llf,  using  inches 
40/20  +  40/80  =  D  =  40// 

if  the  image  is  left  virtual,  put  —  20 ;  if  made  real,  leave  it  -f ,  so  that  D  = 
either  —  1-5  or  +  2-5.] 

11.  Establish  the  relation  between  the  focal  length  and  the  distances  of 
object  and  image  from  a  lens. 

12.  A  lens  of  30  cm.  focal  length  produces  a  virtual  image  the  linear  dimen- 
sions of  which  are  one-third  those  of  the  object.  What  kind  of  lens  is  this  ? 
Determine  the  positions  of  the  object  and  image. 

[By  §  515  this  means  image  is  at  1/3  distance  of  object,  from  lens. 

Our  formula  l/a  -\-  lib     =  f  becomes,  using  Metres 

1/a  +  l/3a  =  1/0-3  dioptres,  ?  -f  or  — . 

As  the  image  is  to  be  virtual,  unreal,  put  a  —  to  its  a,  giving 

3/(-3a)  +  l/3o  =  -  213a  =  1/0-3 

3a  is  the  distance  of  the  real  object,  and  is  therefore  essentially  -f ,  hence 
the  lens  is  a  concave,  —  3-3  D.     3a  is  plainly  0-6  m.] 

13.  With  a  camera-lens  of  15-cm.  focal  length,  a  photograph  is  taken  of 
a  man  180  cm.  tall  and  4  m.  away ;  find  his  height  in  the  picture. 

[Find  its  distance,   1  /a  -f  1  /6      =  1  //  becomes,  using  Metres 
(Real  image)     1  /a  +  1  /4-0  =1/0-15 

I/O  4-  0-25    =  6-67  dioptres 

.-.  a  =  1/6-42  =  0-156  m. 

.-.length  of  image  =  object  x  o/6  =  180  X  0-156/4  =  7  cm.] 

14.  A  convex  6-25-diopter  lens  projects  an  image  on  a  screen  1  m.  from 
object.     In  what  two  positions  may  the  lens  be  placed  ?  ,    .    • 

[Here  the  smn  of  the  conjugate  focal  powers,  l/o  -f  1/6  =  6-25,  all  bemg 
essentially  -\- 

and  o  +  6  =  1 

.-.together  l/a(l  —  a)  =  6-25  by  recourse  to  algebra 
or  a*  -  a  +  0-16  =  0 

P 


418  LIGHT 

a  quadratic  solvable  by  formula,  or  by  trial  and  error,  giving  a  =  0-2  or 
0-8  m.,  the  two  conjugate  positions,  producing  an  image  four  times  smaller 
on  four  times  magnified.     Make  a  diagram.] 

15.  A  candle  is  6  ft.  from  a  wall.  What  lens  midway  between  would  focus 
its  image  on  the  wall,  and  where  would  a  1-ft.  focal  length  lens  have  to  be 
placed,  and  what  magnification  would  it  produce  ?  Show  that  the  size  of 
the  object  is  the  geometrical  mean  of  the  image  sizes.     (  X  2) 

16.  Show  that  there  are  two  positions  for  a  convex  lens  to  form  a  real 
image  at  a  given  distance  from  the  object;  but  that  at  a  minimum  distance 
(which  find)  these  coalesce.  If  the  image  in  one  position  is  four  times  as 
long  as  in  the  other,  and  object  and  image  are  64  cm.  apart,  find  /  of  lens, 
and  its  distance  moved. 

17.  At  what  two  distances  from  a  10-cm.  convex  lens  may  an  object  be 
placed  to  have  an  image  magnified  nine  times  in  area  ? 

[Here  1/36  +  3/36  =  10  dioptres 
the  first  term  may  be  — ,  magnifying  glass;  or  +>  projection  lens,  giving 
2/36  or  4/36  =^  10,  hence  6  in  metres.] 

18.  What  is  the  largest  object  2-5  m.  away  that  can  be  photographed  on 
a  6  X  9 -cm.  film  with  a  12 -cm.  lens  ?  What  would  be  the  effect  of  a  larger 
diameter  lens  ? 

19.  Calculate  how  far  forward  of  the  distant  focus  position  must  be  the 
2-m.  and  the  1-m.  marks  on  the  focussing  scale  of  a  hand  camera  with  a  12-cm. 
focal  length  lens.     (  X  2) 

20.  A  simple  plano-convex  camera  lens  3  cm.  diam.  and  11-5  cm.  focal 
length  has,  2  cm.  in  front  of  its  flat  face,  a  '  stop  '  pierced  by  an  axial  aperture 
1  cm.  diam.  Sunlight  shining  through  this,  and  just  touching  the  rim  of 
the  lens,  forms  an  image  of  the  sun  on  the  focussing  screen  beyond.  Show 
exactly  where  this  is  and  calculate  its  size  (the  angular  diameter  of  the  sun 
being  0*5° ).     Is  it  exactly  round  and  white  ? 

21.  Show  that,  if  two  thin  lenses  are  in  contact,  the  power  of  the  combina- 
tion is  the  sum  of  the  powers  of  the  separate  lenses. 

22.  Two  thin  convex  lenses,  each  of  6  in.  focal  length,  can  be  put  in  contact, 
and  then  gradually  moved  apart.  An  axial  parallel  beam  falls  on  the  first ; 
where  will  it  be  brought  to  a  focus  when  the  interval  between  them  is  (a) 
zero,  (6)  1  in.,  (c)  6  in.,  {d)  12  in.  ? 

23.  Draw  a  scale  diagram  of  the  formation  of  an  image,  by  a  3-3-D  lens, 
of  an  object  60  cm.  from  it.  Now  put  a  12-cm.  convex  lens  in  contact  with 
the  other  and  find  the  new  image. 

24.  Show  how  a  concave  lens  acts  as  a  '  view-finder,'  e.7.  in  the  back  window 
of  a  car. 

25.  Show  how  to  find  the  focal  length  of  a  concave  lens.  What  difficulties 
have  you  experienced  in  obtaining  an  accurate  result  ?  Why  are  such  lenses 
used  in  the  case  of  short  sight  ? 

26.  Construct  the  path  of  rays  from  an  object  20  cm.  distant  from  a  concave 
lens  of  15  cm.  focal  length,  to  an  eye  15  cm.  beyond  the  lens.  Measure,  or 
calculate,  the  position  of  the  image. 

27.  Define  dioptre,  focal  length,  optical  centre.  Show  in  a  diagram  a  lens  pro- 
ducing at  25  cm.  distance  a  real  image  of  an  object  also  at  25  cm. ;  then  put  a 
40-cm.  concave  lens  in  contact  with  the  other  and  find  the  new  image.    (  X  2) 

28.  Explain  how  a  concave  lens  can  be  made  to  yield  a  real  image.  A 
convex  lens  produces  a  real  image  24  cm.  away,  a  concave  is  now  put  18  cm. 
from  the  convex  and  the  image  moves  6  cm.  farther  on,  and  becomes  larger. 
Draw  a  diagram,  find  by  it,  or  by  calculation,  the  strength  of  the  concave 
lens,  and  the  magnification  of  the  new  image. 

29.  A  convex  lens  of  focal  length  9  in.  lies  over  the  mouth  of  a  gas-jar 
12  in.  deep.  Where  must  a  match  be  held  so  that  its  image  is  formed  on 
the  bottom  of  the  jar  (a)  if  empty,  (6)  if  half  full  of  water  ?   Give  a  diagram. 


THIN   LENSES  419 

30.  Calculate  the  curvature  necessary  for  the  faces  of  an  equi-convex  lenn 
of  6  in.  focal  length  made  of  glass  /x  1  55, 

31.  Show  that  the  focal  length  of  an  ordinary  glass  lens  in  water  is  four 
times  that  in  air. 


PRACTICAL  QUESTIONS 

Plot  the  object  and  image  distances  for  a  convex  lens,  and  deduce  the  focal 
power  from  the  conjugate  focal  powers. 

Find  the  focal  length  of  a  convex  lens  by  throe  methods. 

Measure  the  curvature  of  a  lens  faces  by  the  spherometer,  and  calculate 
its  refractive  index. 

Measm*e  the  radii  of  a  concave  lens,  and  its  focal  length  by  a  parallax 
method. 

Measure  the  focal  length  of  a  concave  lens  by  two  methods. 

Measure  the  focal  lengths  of  two  lenses  (a  stronger  convex  and  a  weaker 
concave). 

By  plano-convex  lens,  and  mirror,  compare  the  refractive  indicee  of  two 
liquids ;  or  find  that  of  one  liquid. 

Lamp  and  screen  are  not  provided,  so  that  aerial  images  have  to  be  looked 
for. 

Convex  and  concave  lenses  may  have  to  be  stuck  together,  and  the  focal 
power  of  the  close  convergent  combination  measured,  and  that  of  the  convex 
deducted  from  it.  Recollect  this  combination  may  be  of  long  focus,  so  keep 
well  back,  and  use  plenty  of  room.  If  a  convergent  lens  is  of  too  long  focus — 
more  than  a  quarter  the  space  available — it  can  be  examined  as  a  magnifying- 
glass,  both  pointers  beyond  it,  the  farther  one,  seen  over  the  top,  being  ma<le 
to  coincide  with  virtual  image  distance. 


CHAPTER   XXXIV 


SPHERICAL  MIRRORS 


§521.  Returning  to  reflection,  let  us  consider  the  image- 
producing  properties  of  mirrors  which  instead  of  being  plane  are 
hollowed  (concave)  or  bulged  out  (convex)  into  portions  of  a 
spherical  surface. 

An  approximation  to  the  continuously  curved  surface  may  be 
built  up  of  many  little  flat  facets.  If  hollow,  all  face  inwards 
and  reflect  the  light  more  or  less  exactly  to  one  place  ;  if  convex, 
they  face  outwards  and  scatter  it  as  if  it  originated  at  one  place 
behind  them.  Here  are  concave  mirror  with  real  focus  like  convex 
lens,  and  convex  mirror  with  virtual  focus  like  concave  lens. 

The  study  of  mirrors  therefore  resembles  that  of  lenses,  but 
is  more  simple,  for  there  are  no  refractive  indices  coming  into 
account. 

It  is  related  that  Archimedes  destroyed  the  Roman  galleys  at 
Syracuse  by  setting  them  afire  by  sunshine  reflected  from  a  concave 
mirror  ;  and  he  stands  in  marble  in  the  Arethusa  Fountain,  by  the 
water-side,  carrying  a  sort  of  shallow  dish-cover  in  commemoration 
of  the  legend.  It  sounds  ridiculous  ;  yet,  as  the  vessels  were  quite 
likely  pulled  up  on  shore  barely  fifty  paces  away,  a  structure  carrying 
a  couple  of  score  polished  Greek  shields,  each  inclined  so  as  to  reflect 
its  flash  of  the  furious  Trinacrian  sun  on  to  one  and  the  same  already 
sweltering  patch  of  pitch-caulked  ship's  side,  might  quite  conceivably 
have  started  a  blaze. 

§  522.  Reflection  in  a  spherical  mirror :  relation  between 
Curvature,  principal  Focal  Power,  and  conjugate  focal  powers. 

The  Concave  Mirror,  Fig.  206,  is  described  from  centre  of  cur- 


FiG.  206. 


vature  C.  The  upper  figures  show  what  is  happening  to  the  light 
waves  :  in  the  lower  figures  all  is  blotted  out  except  the  narrow 
streams  of  these  waves  which  form  rays  conveniently  placed  for  us. 

420 


§523]  CURVED   MIRRORS  421 

One  passes  through  C  and  strikes  the  mirror  as  a  radius,  i.e.  per- 
pendicularly, and  returns  straight  back  on  itself.  Another  strikes 
at  A,  making  angle  OAC  with  the  radius  AC,  the  *  normal.'  It  is 
therefore  reflected  at  an  eqtuil  angle  CAI  on  the  other  side  of  AC, 
and  meets  its  returning  fellow-ray  in  I,  which  is  therefore  the  image 
point  of  0. 

As  with  lenses,  all  angles  must  be  small,  §§  501,  507. 
Gradient  of  AI  is  greater  than  that  of  AC  by  Z.  e  lAC. 
AO  is  less        „  „  „        Z  e  OAC. 

Adding  up,  these  equal  angles  cancel  and 
Gradient  of  AI  +  of  AO  =  twice  gradient  of  AC. 
Putting  AO  or  MO  =  b,  AI  or  MI  =  a,  radius  AC  =  r 

AM      AM  _  2AM 
a    '^    b    ~~V' 

Now,  we  can  put  AM  =  1  or  40  or  100,  as  we  like,  just  as  in  §  507, 
and  for  the  same  reasons.     Taking  the  simplest 

1  +  1=.? 
a^  b       r 

Now  put  O  very  far  away,  OA  becomes  parallel  to  OM,  the  gradient 
l/b  =  0,  and  MI  becomes  the  distance/  of  the  real  principal  focus. 

1211 

••     f-f-a^b' 

Which  says  that  The  Focal  Power  of  a  Mirror  is  eqtml  to  the  aggregate 
of  any  pair  of  conjugate  focal  powers,  and  is  double  its  Curvature. 

It  is  measured  in  Dioptres,  as  for  lenses,  and  working  with 
centimetres  in  the  laboratory  it  is  as  well  to  keep  100  in  the  formula 
instead  of  1. 

The  Principal  Focal  Distance  is  evidently  half  the  Radius  of 
Curvature. 

[The  Convex  mirror,  in  this  method,  leads  to  a  confusion  of  angles, 
so  see  the  next  paragraph.] 

Notice,  again,  there  is  no  single  *  principal  focus  ' ;  it  is  any  point 
in  the  principal  focal  surface  (which  is  now  a  sphere  of  half  the 
radius  of  the  mirror). 

§523.  Standard  geometrical  construction  for  mirrors,  Fig.  207. 

Concave.  From  the  ends  of  an  object  draw  '  scissors  '  rays 
through  centre  of  curvature  C  (which  now  replaces  optical  centre 
of  a  lens).  Both  strike  mirror  radially  (perpendicularly),  and 
return  back  on  themselves. 

From  one  end  draw  another  ray  parallel  to  that  from  the 
other  end,  to  form  the  '  rails.'  This  is  reflected  back  and  crosses 
over  the  other  axial  '  rail '  at  the  principal  focal  distance,  half-way 
between  mirror  and  centre.  Continuing,  it  meets  its  fellow  ray 
in  I,  the  Image  of  the  point  from  which  both  sprang.     Draw  in 


422 


LIGHT 


[§  523 


the  rest   of  image  parallel  to   object   and  mirror,   and   between 
*  scissors  '  rays.     Evidently  it  is  Real  and  inverted. 

Convex.  '  Scissors  '  rays  return  on  themselves  before  reaching 
centre.  The  parallel  ray  is  reflected  directly  away  from  F',  at  the 
virtual  principal  focal  distance,  half-way  to  C,  and  I'  is  where  its 
direction  crosses  the  direction  of  its  fellow  ray.  Fill  in  image; 
evidently  it  is  a  small  erect  Virtual  Image,  since  no  light  ever  reaches 


Fig.  207. 


it  at  all,  the  familiar  little  picture  inside  the  reflecting  globe,  flask, 
teapot,  etc. 

The  Sun  is  in  the  picture,  at  F',  Behind  the  Looking-Glass  where 
nor  heat  nor  light  can  reach ;  it  only  looks  as  if  it  were  there,  it  is 
merely  a  Virtual  principal  focus. 

In  the  formal  relation  this  is  expressed  by  condemning  principal- 
focal  power,  and  image,  and  real-space-destroying  curvature  of 
the  surface,  all  to  the  —  sign. 

§  524.  Motion  of  image.  As  the  object  runs  along  the  '  rails  ' 
of  the  standard  construction,  all  that  happens  is  that  the  slanting 
'  scissors  '  ray  OCI  alters  its  inclination  and  cuts  the  fixed  line 
AF   (produced)   at  different  conjugate  distances,   as  in  Fig.  208 

(and  in  Fig.  207  little  arcs  have 
been  dotted  in  to  show  the  be- 
ginning of  the  process).  There 
is  no  limit  to  its  inclination, 
but  those  who  wish  to  rely 
upon  actual  '  rays '  all  the 
time  can  use  instead  of  it  the 
rays  MO,  Mo  (produced  if 
necessary)  of  the  next  para- 
graph. 
The  virtual  image  in  the  convex  mirror  starts  at  Jr  beneath  the 
surface  for  distant  objects,  and  slowly  comes  forward  until  image 
and  object  touch  on  the  surface ;  increasing  from  0  to  full  size. 
Trace  this  for  yourself  ;  it  is  what  you  see  in  your  driving-mirror 
when  the  other  fellow  overtakes  you. 

You  put  up  with  the  convex  mirror's  exaggerated  perspective 
because,  bending  back  to  face  both  sides,  as  it  does,  it  gives  you  a 
wider  view  (on  a  smaller  scale). 

With    the    concave    mirror    much    more    happens.     The    real 


Fig.  208. 


§526] 


CURVED   MIRRORS 


423 


image  starts  at  F,  Jr  out  in  front,  and  comes  forward  to  meet  the 
object  until  they  meet  at  the  centre  of  curvature,  image  being  in 
verted  and  same  size  as  object.  The  scissors  ray  now  slants  the 
other  way,  and  carries  I  rapidly  out  along  AFI :  the  mirror  is  pro- 
ducing a  larger  distant  aerial  image  I^  of  a  small  object  0,.  When 
object  reaches  F,  OC  and  AF  are  parallel,  I  has  '  gone  off  to 
infinity.  '^ 

When  the  object  is  within  its  principal  focal  distance  of  the 
mirror,  O3C  slants  less  than  AF  and  can  never  meet  it  but  both 
appear  to  come  from  a  point  I3  behind  the  mirror,  on  an  enlarged 
upright  virtual  image.  Fig.  208,  which  comes  forward,  diminishing 
until  image  and  object  touch  on  the  surface.  This  is  the  use  of  a 
concave  mirror  as  a  Magnifying  Mirror,  for  shaving,  etc.  See  also 
Fig.  210,  3. 

§  525.  Magnification,  m.  Since  both  lie  between  the  *  scissors  ' 
rays,  evidently  the  Ratio  of  diameters  of  image  and  object,  which  is 
m  =  ratio  of  their  distances  from  the  centre  of  curvature,  when  the 
scissors  cross. 

A  more  practically  convenient  relation  can,  however,  be  deduced 
as  in  Fig.  209.  From  mid-point  X  of 
object  draw  through  C  to  M,  join  OM,  ^_ 
oM.  Ends  of  image  lie  on  these  rays ; 
for  if  not  let  I'  be  end.  By  symmetry 
Z  e  OMC  =  Z.  e  CMI,  by  law  of  reflection 
Z  e  OMC  =  A  e  CMI'.  /.  I  and  I' 
coincide,  and  similarly  i  is  on  OM. 
Hence  again,  as  for  lenses 

m  =  a/b. 


Fig.  209. 


§  526.  Fig.  210  shows  the  Standard  Geometrical  Construction 
at  work  on  the  Problem,  '  Show  the  actual  cone  of  light  by  which 
the  eye  sees  a  point  of  an  object  in  lens  or  mirror.' 

In  all  five  you  recognize  the  standard  construction,  drawn 
to  begin  with ;  in  4  it  is  Fig.  203  ;  in  1,  Fig.  204  virtual  image  case ; 
5  and  2  are  Fig.  207  ;  3  the  virtual  image  case  of  Fig.  208. 

Then  put  in  the  Eye,  and  make  it  a  good  big  one. 

Select  your  point  on  the  Object,  and  now  you  must  transfer  it 
to  the  Image.  For  the  Image  is  what  the  eye  looks  at,  and  the  whole 
business  of  the  lens  or  mirror  is  to  transfer  every  point  of  the  object 
into  corresponding  image  points,  having  done  which  the  actual 
glass  etc.  '  is  as  if  it  were  not.' 

You  do  this  perfectly  easily  by  running  the  pecked  line  straight 
through  selected  point  of  object  and  appropriate  Centre  of  System 
— Optical  Centre  of  a  Lens,  Centre  of  Curvature  of  a  Mirror — until 
it  hits  the  image,  and  from  that  point  of  the  image  draw  the  cone 
to  fill  the  eye. 

Whence  did  that  point  get  its  light  to  send  to  the  eye  ?  From 
the  corresponding  point  of  the  object :    therefore,  wherever  your 


424 


LIGHT 


[§526 


cone  meets  the  glass,  break  it  and  taper  it  down  to  the  selected 
object  point. 

In  Fig.  210,  1,  you  are  looking  through  a  pocket-lens,  near  the 
edge  :  the  bending  of  the  ex-centric  beam  probably  results  in  some 
distortion,  and  colour ;  test  this  on  your  own  straightaway. 
No.  2  is  a  driving  mirror ;  in  3  you  are  using  your  latest  birthday 
present  to  scrutinize,  under  high  magnification,  that  dusky  growth 
on  your  chin. 

Nos.  4  and  5  are  Real  Image  cases.  In  the  absence  of  any 
diffusing  screen,  the  aerial  image  can  only  pass  its  light  straight  on. 
Carry  on  your  cone,  now  expanding,  to  meet  the  glass  :  that  area, 
fed  from  the  object  point,  feeds  to  the  image-point  the  light  which 


Fig.  210. 


the  latter  then  passes  on  into  your  eye.  In  5  the  point  is  beginning 
to  slip  off  the  edge,  part  of  the  pupil  is  no  longer  filled.  These  are 
the  Conjugate  Image  Methods,  using  pins  in  daylight,  that  you 
employ  on  the  optical  bench  in  the  laboratory. 

Notice  in  these  figures,  that  from  the  point  of  the  straight  cone 
to  your  eye  must  be  at  least  10  in.,  §  606. 

Notice  also  that  in  all  virtual  image  cases  the  patch  of  glass  actually 
in  use  in  inspecting  any  one  point  is  smaller  than  the  pupil  of  your 
eye.  So  that  if  over  that  eighth  of  an  inch  the  surface  is  as  truly 
curved  as  your  eye,  you  will  see  a  point  perfectly.  If  the  rest  of 
the  surface  is  inexact,  adjoining  points  will  pack  wrongly ;  the 
picture  as  a  whole  will  be  distorted,  but  each  little  detail  of  it  looks 
sharp  and  clear. 

Consequently  spectacle -lenses  and  magnifying-glasses  need  only 
be  fairly  true  to  curve,  and  are  much  cheaper  in  consequence, 
while  driving-  and  shaving-mirrors  are  merely  cut  from  balloons 
of  blown  glass  and  silvered,  and  serve  their  purpose  quite  well. 


§  526]  CURVED   MIRRORS  426 

Very  different  is  Real  Image  Formation  :  in  4  half  the  lens,  and 
in  5  the  whole  mirror,  has  to  combine  to  illuminate  even  a  single 
point,  and  departure  from  true  curve  by  even  the  thickness  of  a 
bacillus  sadly  impairs  the  working  of  Camera  or  telescope  lens. 
Their  lenses  and  mirrors  cannot  be  cheap. 

I  have  filled  in  the  waves,  with  compass  planted  in  the  vertex  : 
at  any  rate,  they  suggest  what  is  really  happening.  And  if  you 
recollect  Figs.  131,  132,  you  will  not  be  surprised  to  hear,  later  on, 
that  something  of  much  physical  importance  is  going  on  all  along 
these  cut-off  edges  of  the  waves. 


EXAM   QUESTIONS,  CHAPTER  XXXIV 

Your  microscope  possesses  a  concave  mirror,  and  any  car  a  convex ;  or  at 
cheap  fancy-stores  you  can  probably  get  either  in  small  size  for  sixpence. 
All  are  of  the  poorest  optical  quality,  but  they  are  much  more  like  spherical 
mirrors  than  cat's-cradles  on  paper  :   make  all  the  use  of  them  you  can. 

Questions  22,  28,  29,  30  below  strike  me  as  useless  instances  of  perverted 
ingenuity,  belonging  to  no  known  instrument  :  probably  nobody  answered 
them. 


1 .  What  are  the  characteristics  of  convex  and  concave  mirrors  ?  Give 
diagrams. 

2.  Define  the  focus  and  the  focal  length  of  a  spherical  mirror.  Prove  that 
the  focal  length  of  a  convex  mirror  is  half  the  radius  of  curvature. 

3.  Two  reflections  of  the  landscape  are  seen  in  a  hollow  glass  sphere  {t,g. 
a  lamp  bulb).  Where  are  they  inside  the  sphere,  and  what  is  the  difiference 
between  them  ? 

4.  In  3  how  do  they  change  as  the  object  approaches  ?  Can  they  coincide 
as  regards  distance  ? 

5.  How  does  a  concave  mirror  concentrate  light  ?  Give  instimcee  of  its 
use  in  various  instruments. 

6.  Show  how  the  focal  length  of  a  mirror  is  related  to  its  curvature. 

A  lamp  filament  2  cm.  long  is  80  cm.  from  a  concave  mirror  of  radius  of 
curvature  1  m. ;  how  long  is  the  image,  and  how  far  will  it  move  if  the  mirror 
is  rotated  through  one  degree  ?     (  X  4) 

7.  Draw  diagrams  to  show  (a)  a  real,  (6)  a  virtual,  image,  in  a  concave 
mirror  (the  object  being  finite  and  not  a  point). 

A  mirror  is  set  up,  and  a  source  4  cm.  high  is  placed  20  cm.  in  front;  its 
real  image  is  24  cm.  high.  What  is  the  focal  length  of  the  mirror,  and  what 
kind  ?     (  X  2) 

[Magnification  6  .-.  a  =  6  X  20,  then  100/20  +  100/120  =  strength; 
5-83  D,  =  17-2  cm.  focus  concave.] 

8.  The  image  formed  by  a  concave  mirror  is  reduced  five  times,  and  its 
distance  from  the  mirror  is  30  cm.  Draw  a  diagram,  and  find  from  it,  or 
otherwise,  the  focal  length. 

9.  An  object  is  20  cm.  from  a  concave  mirror,  and  its  erect  image  is  X  3 
times ;   obtain/  by  diagram  or  calculation. 

[The  image  distance  is  evidently  3  x  20,  and  it  is  virtual,  so  that  100/20  -f- 
100/(—  60)  =  dioptric  strength  of  mirror  =  3-33,  or/  =  30  cm.] 


426  LIGHT 

10.  Show  that  there  are  two  possible  positions  for  an  object  in  front  of  a 
3-ft.  focus  concave  mirror  to  yield  a  X  3  times  image,  and  calculate  their 
distance  apart. 

11.  Describe  the  changes  in  the  image  produced  by  a  concave  mirror  as 
the  object  moves  up  from  infinity. 

A  pin,  5  cm.  high,  is  placed  16  cm.  from  a  convex  mirror  of  focal  length 
10  cm.  Find  the  position  and  size  of  the  image.  Give  a  diagram  of  its 
formation. 

12.  A  pin  3  cm.  long  is  48  cm.  in  front  of  a  concave  mirror,  the  real  image 
is  formed  at  16  cm.  The  pin  is  moved  24  cm.  towards  the  mirror;  draw  a 
diagram  and  find  the  changes  in  the  image. 

13.  A  line-object  of  length  1  cm.  is  placed  along  the  axis  at  a  distance  of 
40  cm.  from  a  concave  spherical  mirror  of  radius  of  curvature  50  cm.  Find 
the  length  of  the  image. 

14.  Describe  two  optical  methods  for  finding  the  focal  length  of  a  convex 
mirror.  Discuss  the  superiority  of  the  spherical  over  the  plane  driving 
mirror,  and  draw  a  diagram  showing  positions  of  eye  and  object  seen.     (  X  2) 

15.  Show  in  a  diagram  the  cone  of  rays  by  which  an  eye  looking  into  a 
concave  mirror  sees  one  point  of  image  of  an  object  close  in  front. 

16.  A  brilliant  source  2  cm.  diam.  is  at  the  principal  focus  of  a  concave 
searchlight  mirror  of  focal  length  44  cm.  What  will  be  the  diameter  of  the 
patch  of  light  reflected  on  to  a  cloud  2  km.  away  ?  What  effects  on  the  size  and 
brightness  of  this  patch  will  result  from  increasing  the  diameter  of  the  mirror  ? 

17.  A  concave  lens  is  30  cm.  in  front  of  a  concave  mirror  of  radius  40  cm. 
A  pin  is  then  placed  15  cm.  in  front  of  the  lens,  and  coincides  with  its  own 
inverted  image  formed  by  the  lens  and  mirror.     Find  the  lens. 

18.  Convex  lens  produces  real  image  of  flame  50  cm.  from  itself.  Concave 
mirror  100  cm.  from  lens  reflects  the  light  back  through  lens  to  form  an  image 
close  to  flame ;   what  is  /  of  miiTor  ? 

19.  Show  that  if  a  horizontal  concave  mirror  is  filled  with  a  liquid,  its 
apparent  radius  of  ciu'vatiu'e  is  diminished  in  the  ratio  of  /^  of  liquid. 

20.  The  plane  side  of  a  plano-convex  lens  is  silvered,  and  the  lens  then 
acts  like  a  concave  mirror  30  cm.  focal  length.  /^  =  1-5;  calculate  radius  of 
convex  surface. 

21.  A  plano-convex  lens  silvered  on  its  plane  side  acts  like  a  concave  mirror 
of  20  cm.  focal  length.  When  the  convex  side  is  silvered  it  acts  like  a  concave 
mirror  of  7  cm.     Calculate  yL, 

22.  A  pin  stands  midway  between  a  convex  and  a  concave  mirror,  facing 
each  other  40  cm.  apart  and  each  of  radius  of  curvature  25  cm.  Calculate 
the  apparent  distance  apart  of  the  first  two  images  of  the  pin  visible  in  the 
convex  mirror. 

23.  Two  luminous  objects  are  arranged  0-5  m.  apart  crossways,  2  m.  in 
front  of  a  convex  reflecting  surface,  in  which  their  images  appear  10  cm. 
apart. 

Show  how  these  images  are  produced  and  find  the  radius  of  curvatiu'e. 

24.  Describe  a  combination  of  lens  and  concave  mirror  capable  of  always 
reflecting  back  a  bright  beam  of  light  to  a  distant  source  which  moves  about 
anywhere  within  10°  of  the  axis  of  the  arrangement. 

25.  Distinguish  between  real  and  virtual  images,  and  between  vertically 
inverted  and  laterally  inverted  images.  With  a  convex  lens  and  a  plane 
mirror,  you  can  get  two  return  images  of  the  cross  on  the  screen  beside  it  at 
different  lens  distances.     What  can  you  learn  from  these  ? 

26.  Prove  that  a  lens  with  a  plane  mirror  behind  it  behaves  like  a  spherical 
mirror  the  radius  of  ciu-vature  of  which  is  equal  to  the  focal  length  of  the  lens 

27.  A  20-cm.  convex  lens  is  8-1  cm.  in  front  of  a  convex  mirror,  and  a  pin 
30-5  cm.  in  front  of  the  lens  coincides  with  its  own  inverted  image  formed  by 
lens  and  mirjoj-.     Calculate  /  of  mirror. 


CURVED  MIRRORS  427 

28.  An  object  3  cm.  tall  is  set  up  33  cm.  from  a  concave  mirror  of  radius 
40  cm.     Find  position,  size,  and  character  of  image. 

A  12-5-cm.  convex  lens  is  now  placed  58  cm.  from  mirror,  same  side  as 
object ;  describe  the  images  seen. 

29.  An  object  is  32  cm.  from  a  10-cm.  lens,  and  a  real  image  is  produced  ; 
find  its  position.  If  a  concave  mirror,  radius  16  cm.,  is  placed  20  cm.  beyond 
the  image,  facing  it,  where  is  the  final  image  ? 

30.  An  object  is  50  cm.  from  a  concave  mirror  of  radius  20  cm.,  and  a 
5-cm.  convex  lens  is  placed  between  object  and  mirror,  30  cm.  from  mirror. 
Where  will  the  image  produced  by  the  mirror  be  formed  ?     Give  a  diagram. 

31.  Two  convex  lenses  of/  15  cm.  are  mounted  at  a  distance  apart  of  30  cm. 
Calculate  the  distance  apart  of  an  object  and  its  final  image  as  projected  by 
the  system  if  the  object  is  20  cm.  in  front  of  the  first  lens. 


PRACTICAL  QUESTIONS 

Measiu^  the  curvature,  and  the  focal  length,  of  a  concave  mirror,  by  all 
methods. 

Measure  the  focal  length  of  a  convex  mirror  by  pin  and  plotting  method. 

Measure  the  refractive  index  of  a  liquid  by  method  of  Question  19. 

Plot  magnification  against  image  distance  for  a  concave  mirror  and  deduce 
its  focal  length. 


CHAPTER  XXXV 
PRACTICAL  METHODS  FOR  MIRRORS  AND  THIN  LENSES 


§  531.  Out  of  a  host  of  practical  methods  the  following  few  are 
simple  and  sufficient,  and  sounder  than  most. 

For  supporting  things  in  position  the  *  optical  bench  '  of  Fig.  211 
is  satisfactory.  Along  a  stout  plank  metre  scales  are  screwed, 
against  these  slide  wooden  uprights  of  the  plain  shape  shown, 
and  over  the  IJ-in.  holes  in  these  are  strapped,  by  elastic  bands, 
the  lenses,  mirror,  card  screens,  etc.  On  stumpy  blocks  pins  are 
held  by  plasticine. 

Perfectly  good  plano-convex  lenses  up  to  5D  cost  threepence 
each  at  the  cheap  stores.  Larger  lenses  are  undesirable,  being 
thicker.  Rough-edge  concaves  a  friendly  oculist  will  let  you  have  ; 
spheres,  not  cylinders. 

Two-inch  mirrors  cost  ten  times  as  much  at  the  scientific  apparatus 
dealer's.  Concave  shaving-,  and  convex  driving-mirrors  are  merely 
blown  glass,  of  very  variable  curvature,  and  can  only  be  explored 
piecemeal  by  virtual  image  methods  which  are  too  doubtful  to  in- 
clude here.  But  in  a  Dark  Room,  and  if  you  avail  yourself  of  the 
filament  of  the  exposed  bulb  of  a  pocket-lamp  as  '  object,'  the  un- 
silvered  faces  of  lenses  will  give  you  enough  reflection  to  serve 
instead.  Otherwise  the  luminous  object  is  made  by  cutting  a  small 
cross  in  a  card,  on  the  first  upright,  and  lighting  it  from  behind 
with  a  broad  flame  or  lamp  with  diffusing  glass. 

In  Daylight,  a  bright  pin  may  serve  as  object ;  it  must  be  well 
illuminated  on  the  side  facing  the  mirror  or  lens,  or  you  will  see  nothing 
(must  be  lit  from  below  when  working  vertically).  Look  for  its 
image  in  the  air,  in  line  of  pin  and  glass,  keeping  your  head  well 
back,  for  the  image  may  be  anywhere  along,  and  you  can't  catch 
sight  of  anything  less  than  10  in.  from  your  eye,  so  give  it  room. 
By  moving  the  pin  up  or  down  or  sideways  on  its  plasticine,  you  then 
bring  its  image  down  towards  the  centre  line  of  the  bench,  and 
endeavour  to  touch  it  with  an  exploring  pin  on  a  second  upright, 
similarly  adjustable.  If  they  really  are  close  together,  they  keep  so 
as  long  as  you  can  see  them  both,  as  you  move  your  head  sideways  ; 
if  not,  the  image  moves  off,  and  you  must  shuffle  them  along  ;  until 
in  the  end  you  can  get  both  in  focus  at  once  under  your  pocket-lens. 
This  is  more  trouble  than  lamp  and  screen,  even  as  the  screen  is 
bigger  than  your  eye  ;  but  you  must  learn  how  to  manage  it  in 
the  laboratory.  The  best  way  is  to  get  a  cross  and  screen  in 
correct  position,  and  then  to  replace  them  exactly  by  the  pins, 
well  lit ;  so  now  you  start  right,  and  see  what  to  look  for,  then 

428 


532] 


PRACTICAL  METHODS 


429 


displace,  replace,  etc.  This  daylight  outfit  is  all  that  is  provided 
in  the  exam  room  ;  and,  after  all,  it  is  no  bad  principle  to  test  what 
you  can  do  with  the  minimum  of  adventitious  aids. 


, 

VIII 

1        1 

,  -A 

F 

IG.  211. 

§  532.  Distant  object  method.  A  broken  skyline  at  least  100  ft. 
away  in  sight  through  the  open  window  will  serve  for  an  indefinitely 
distant  object ;  the  sun  itself  is  too  dazzling. 

I.  Convex  Lens.  Catch  sharp  image  of  the  distant  chimneys, 
trees,  etc.,  on  screen  behind  lens.  Lens  to  screen  =  /,  100//  cm.  = 
D      Fig.  211,  I. 


430  LIGHT  [§  532 

II.  Concave  mirror.  Ditto,  on  screen  half  shadowing  mirror  ; 
the  image  at  this  edge  of  the  card  only  will  be  free  from  streakiness  ; 
use  this.     Mirror  to  screen  =/,  Fig.  211,  II. 

Especially  with  the  lens,  keep  as  far  back  in  dull  light  in  the  room 
as  you  can  without  losing  the  distant  scene. 

If  the  object  is  not  very  distant,  but  only  twenty  or  thirty  times 
the  image  distance  found,  reduce  this  distance  by  l/20th  or  l/30th 
part  of  itself,  to  give  /;  see  Chap.  XXXIII,  Question  6.  Don't 
neglect  this  ;  very  distant  doesn't  mean  arm's  length. 

Examinees  seem  scared  of  using  these  methods  :  they  are  the 
very  first  thing  you  should  try ;  they  are  the  actual  '  definition  ' 
methods,  and  they  put  you  on  the  right  track  for  all  others. 

§  533.  Methods  by  return  image  of  object  on  bench. 

III.  Convex  lens.  Behind  the  lens  support  a  plane  mirror 
(bit  of  good  thick  looking-glass),  and  move  lens  until  image  of 
the  illuminated  cross  appears  in  sharp  focus  on  the  screen  close 
beside  it,  Fig.  Ill  (or  pin's  own  image  touches  pin).  Lens  to  screen  — 
/,  for  light  that  retraces  its  path  exactly  after  reflection  at  a  plane 
must  necessarily  be  a  plane  wave,  or  '  parallel.' 

IV.  Concave  mirror.  Returns  sharp  image  at  Centre  of  Cur- 
vature, 2/  from  mirror,  for  onlv  radii  can  be  reflected  directly  back. 
Fig.  IV. 

§  534.  Ordinary  spreading  light  is  wasted  by  convex  mirrors 
and  concave  lenses ;  so  for  them  let  us  adopt 

Methods  by  convergent  light. 

With  your  5D  or  6D  lens,  or,  better,  a  camera  or  lantern  lens, 
if  you  can  lay  hands  on  one,  form  a  real  image  I  on  a  screen  down  the 
bench,  and  don't  let  this  lens  he  moved  :  here  is  your  convergent 
light. 

V.  Convex  Mirror.  Insert  as  in  Fig.  V,  and  move  until  a  sharp 
return  image  is  seen  beside  the  cross.  Then  IM  =  r  =  2f,  for  the 
directly  returned  '  rays  '  must  be  radii  {or  the  wave  converging  to 
centre  I  exactly  fits  the  mirror). 

VI.  Concave  lens.  Set  up  the  plane  mirror  and  strap  the  lens 
on  it  (or  keep  separate  as  in  diagram)  and  move  until  a  sharp 
return  image  forms  beside  the  cross. 

Then  IL  =  /,  Fig.  VI,  for  the  light  to  and  from  the  plane  mirror 
is  then  '  parallel.' 

Those  are  the  Six  Best  Methods  for  Mirrors  and  Thin  Lenses, 
and,  as  you  see,  they  involve  no  calculation  at  all. 


§  536]  PRACTICAL   METHODS  431 

You  should  try  a  lens  facing  both  ways  (unless  it  is  Hymmetrically 
biconvex  or  biconcave),  measuring  to  the  same  edge  each  time. 
The  half  sum  is  /,  the  half  difference  has  found  for  you  how  mucli 
the  Optical  Centre  lies  out  of  the  plane  of  this  edge. 

§  535.  Methods  by  conjugate  foci  (much  favoured  in  exams). 

VII.  Convex  lens.  Involves  a  long  bench,  for  conjugate  foci 
cannot  he  less  than  4/  apart.  As  Fig.  VII,  and  as  described  above 
with  pins.  Distance  from  lens  a  and  b  cms.,  conjugate  focal  powers 
100 /a  and  100/6,  divide  out,  and  add  them  to  get  the  Dioptric  strength 
D  =  100//  =  100 /a  +  100/6. 

VIII.  Concave  mirror.  Fig.  VIII  shows  the  mirror  of  II  and  IV, 
with  conjugate  pins  within  and  without  its  centre  (dot).  Same 
argument  as  VII. 

IX.  Concave  lens.  In  VI,  instead  of  using  the  plane  mirror,  put 
in  L  6  cm.  in  front  of  the  screen  I.  This  image,  towards  which  the 
light  was  focussed,  now  serves  as  '  virtual  object '  for  L  (and  there- 
fore gets  a  —  sign)  ;  it  blurs  out,  and  is  rediscovered  in  focus 
farther  along  (and  much  larger)  a  cm.  from  L  (same  lens,  to  scale  ; 
shows  now  plainly  the  less  refraction  of  the  middle  part). 

Then  100/a  +  100/(-  6)  =  100//=  Dioptric  strength  of  L. 

This  serves  for  any  lens  whatever,  provided  you  find  by  trial  a 
suitable  place  for  it.  It  is  valuable  for  weak  lensbs.  A  weak 
convex,  for  instance,  would  give  you  a  less  than  6,  and  would  work 
out  -f  after  all. 

§  536.    Here,  for  reference,  are  Methods  for  lens  face  curvature. 
Concave  lens  faces  act  as  unsilvered  concave  mirrors,  II  and  IV. 

X.  Convex  lens.  Having  found/,  as  in  III,  move  lens  closer  to  the 
cross  (this  is  the  same  lens  as  I,  III  and  VII)  until  at  a  a  sharp 
image  returns.  Fig.  X,  from  the  back  surface,  on  which  light  in  the 
glass  must  therefore  be  falling  radially  (most  of  it  passing  through) 
as  if  it  came  from  its  centre  of  curvature. 

Hence  b  =  r2  and  gets  a  —  sign,  being  virtual, 
lOO/a  +  I00/(-  r^)  =  D. 

Turn  over  for  the  other  face. 

Then  1//  =  ((x  -  l)(l/ri  +  l/r^)  gives  (i  of  glass.  And  you  see 
that  roughly,  for  common  glass,  [i  1-52,  the  radii  of  biconcave  or 
biconvex  lenses  =  their  focal  lengths. 


Exam  Questions  have  appeared  already.  This  is  a  purely  practical 
chapter,  and  the  next  is  largely  so,  and  the  two  of  thoni  should  bo  an 
antidote  to  the  delusion  that  what  doesn't  involve  calculation  (with  all  it* 

risks)  can't  be  any  good. 


CHAPTER  XXXVI 


THICK  LENSES 


§541.  Having,  in  a  sudden  frenzy  of  virtue,  looked  with 
scorn  on  chipped  spectacle-lenses,  and  bagged  the  finest  he-man's 
lens  you  could  find  in  the  drawer,  and  measured  it  with  all  care, 
you  are  vexed  to  find  the  results  by  no  means  so  accurate  as  you 
intended,  and  on  inquiry,  you  are  told  that  you  have  been  using 
too  thick  a  lens,  your  theory  having  stipulated  a  thin  lens,  for  the 
reason  of  §  507. 

Have  you  therefore  been  toiling  at  a  theory  useful  only  for  spectacle 
lenses,  and  exam  questions,  and  such-like  matters  of  minimal 
interest ;    inapplicable  to,  or  inaccurate  for,  the  much  heftier  and 

better  magnifiers,  eyepieces, 
camera  and  lantern  lenses 
which,  you  learn,  owe  their 
complexity  and  cost  to  their 
increased  accuracy  of  action? 

Well,  that  is  the  foot  of  the 
wall  at  which  your  exam 
syllabus  leaves  you  :  stay 
there  if  you  like,  or  let  us 
give  you  a  hand  up  to  an 
altogether  higher  level : 


Fig.  212. 


Is  bull's-eye  lens  NL,  Fig.  212,  thick  enough  to  satisfy  you  ? 
Split  off  a  flake  by  the  plane  SOS  :  this  flake  has  the  curve,  and  there- 
fore contains  the  focal  strength,  of  the  lens,  §  508. 

NO  is  a  deep  pool  of  glass,  §  486. 

Distant  light  arriving  from  the  left  is  focussed  by  L  at  F,  at  the 
principal  focal  distance  LF. 

Looking,  however,  into  N,  for  light  coming  from  distance  on  the 
right,  you  see  the  '  bottom  '  L  nearer  to  the  surface,  at  L',  where 
NL7NL=l/ti.,  §486. 

For  the  glass  of  which  most  convex  lenses  are  made  [i  is  about 
1-5,  therefore  NL'  is  2/3  NL,  and  it  looks  as  if  the  lens  L  has  jumped 
1/3  the  thickness  of  the  lens  to  L'. 

Good-bye  to  the  pool  of  glass,  the  '  jump  '  takes  its  place  ;  from 
its  Virtual  Position  L'  the  thin  lens  focusses  light  going  towards 
the  left  at  F',  at  the  natural  focal  distance  L'F'  =  LF. 


THAT  IS  ALL  :  instead  of  the  thick  glass  lens  you  have  a  thin 
lens  which  jumps  by  one-third  the  thickness. 

432 


§542] 


THICK   LENSES 


433 


Plainly,  for  the  plano-convex  lens,  we  have  chosen  L  and  L' 
in  the  right  places ;  L  works  unhindered  across  the  air  LF ;  and 
§  486  has  told  us  how  to  locate  L'. 

Symmetry  assures  us  that  in  an  equi-convex  lens  the  space- 
annihilating  jump  is  across  the  middle  third  of  the  thickness. 

Similarly  for  plano-concave  and  bi-concave  lenses ;  but  often 
these  are  of  more  refractive  glass,  for  which  ((x  —  l)/(x  exceeds 
1/3  ;  yet  as  they  are  thin  in  the  middle  1/3  will  do  for  us. 

These  virtual  positions  of  the  Lens  are  called  its  Principal  Planes  ; 
each  lies  at  the  principal  focal  distance  from  the  Principal  Focal 
Plane  on  its  side ;   between  them  Space  has  ceased  to  exist.     You 


Fig.  213. 

take  your  ordinary  lens  diagram,  whatever  it  may  be.  Fig.  203, 
in  variety,  and  cut  it  clean  down  the  middle  of  the  lens,  and  apply  the 
cut  edges  to  the  two  principal  planes  :  the  lens  receiving  the  light 
in  the  first  p.p.  jumps  with  it  quite  unaltered  to  the  second  p.p., 
and  thence  discharges  it  as  usual.  The  diagram  is  now  as  accurate 
as  ever,  Fig.  203  has  become  Fig.  213. 

§  542.  Turn  now  to  the  common  lens-combination  of  two  thin 
lenses  set  in  a  tube  some  distance  apart.  Fig.  214,  Lj  of  principal 
focal  distance  LjF^,  and  Lg  p.f.d.  LgFg. 

Of  two  parallel  beams  of  a  plane  wave  meeting  AjLj,  the  upper 
bends  to  meet  the  undeviated  axial  one  at  F^,  but  strikes  the  second 
lens  at  E. 


Fia.  214. 


Parallel  to  A^E  draw  the  axial  line  L-Fj  ;   then  AjE  and  this  are 
two  streams  of  a  plane  wave  meeting  the  lens,  the  axial  ray  going 


434 


LIGHT 


[§542 


straight  through,  and  the  other  bending  down  towards  meeting  it 
at  principal  focal  distance  LgFg. 

The  two  sample  streams  of  the  original  plane  wave  therefore 
meet  at  F.  Now,  what  equivalent  thin  lens  would  take  those  two 
streams  on  the  left  and  converge  them  at  F,  along  L^LaF  and  down 
along  EF  ?  Evidently,  produce  dA^  to  the  right,  and  FE  to  the  left, 
to  meet  it  at  A — AL  is  the  Equivalent  Thin  Lens  that  would  do  the 
job,  and  LF  is  its  Principal  Focal  Length. 

Now  turn  the  diagram  upside  down,  and  carry  out  the  same 
construction  (do  it,  it  is  simple  enough)  for  yourself,  meeting  now 
the  stronger  lens  first,  and  the  places  are  marked  gnj^  where  you  will 
find  the  other  principal  focus  and  a  fresh  position  for  the  equivalent 
lens  ;  and  they  come  LF  apart,  just  as  before,  and  L'^  is  the  space- 
annihilating  jump  that  the  equivalent  thin  lens  takes,  from  Principal 
Plane  to  Principal  Plane.  Only  notice,  it  is  now  a  jump  backwards, 
the  principal  planes  are  '  crossed.' 

Draw  any  required  thin  convex  lens  diagram  whatever,  for  focal 
length  LF  ;  cut  it  down  the  lens,  put  one  cut  edge  down  AL,  and  lap 
the  other  over  it  down  along  '^y,  and  your  diagram  represents 
accurately  the  action  of  the  lens  system. 

The  Principal  Planes  are  at  quite  different  distances  from  the 
two  glass  ends  :  that's  nothing. 

In  Fig.  214,  I  is  the  diagram  for  two  lenses  in  contact,  cf.  §  517, 
and  III  you  will  get  by  following  the  foregoing  instructions  using 
a  strong  concave  as  second  lens  :  the  equivalent  lens  is  far  out  in 
front,  §627. 

§  543.  Yes,  but  what  if  we  have  a  fat  lens  bulgy  both  sides, 
but  unequally  ;  or  we  can't  get  the  two  thin  lenses  apart  to  measure 
their  strengths  individually  ;   or  they  prove  to  be  two  thick  lenses  ? 

Do  the  job  practically  ;  do  it  at  home  after  supper. 

Get  a  pencil  and  squared  paper,  a  box  of  matches  and  your 
pocket-lens,  and,  best  of  all  possible  lens -combinations  for  the  pur- 
pose, your  microscope  eyepiece. 


d 

A 

^PF 

d' 

L 

L'    f- 

\i^ 

f 

S                i 

-uu 

rual 

a  1 
LIGHT. 

•m=b\^ 

m 

B   1 

m 
tri-irerred 

=  a              ms3 

erecr 

Image. 

tmage. 

CL^ 

\ 

Fig. 

215. 

In  Fig.  215,  dK,  dh,  are  the  two  streams  called  '  parallel  rails  ' 
in  Fig.  204,  along  which  the  object  approaches,  and  AL  is  the  length 
of  the  object.  Arrived  at  the  first  p.p.  AL,  we  jump  unchanged 
to  the  second  p.p.  A'L',  the  axial  line  goes  on,  the  other  bends 


§543] 


THICK   LENSES 


435 


down  to  cross  it  at  F,  L'F  being  its  principal  focal  distance  A  and 
continues  on.  "^ 

1  ^xu^'FiP"^'^'  *^^  Magnification  =  1,  for  A'L'  is  the  unchanged 
length  of  the  object.  (If  you  stuck  a  postage  stamp  on  L  Fig  212 
and  looked  through  N,  the  stamp  appeared  at  L',  unchanged  iii 
size,  §  486.)  * 

At  F  put  m  =  0,  for  here  the  light  of  a  star,  however  vast,  is 
reduced  to  a  point.  Measure  /  along  from  F  and  again  write 
m  =  1,  for,  by  similar  triangles,  a  is  now  as  far  off  the  axis  as  A'. 

At  2/  from  F  put  m  =  2,  at  3/,  m  =  3,  and  so  on. 

Now,  wherever  the  image  of  the  object  may  happen  to  be  formed, 
Its  ends  he  on  these  two  lines,  for  this  is  Fig.  203  extended  to  the 
right,  but  with  the  slanting  cross-line  left  out.  Therefore  you  see 
that  for  an  increase  of  I  in  the  Magnification  the  Image  moves  f 
farther  away  from  the  lens. 


"^ 

<< 

^ 

f*S 

J«*^;«<n.. 

^'S 

s^ 

L 

\^ 

'b 

NTa 

\ 
\ 

m-H 

H 

[^ 

t 

n 

jr^ 

^ 

!  n 

J> 

^ 

^ 

^ 

:?' 

i 

1 

J      IK 

1 

1 
J 

f 

>. 

^ 

. 

1 

, 

, 

/ 

1 

1 
1 
.J_.. 

^    r-^ 

B 
t 

^ 

M^ 

*■    I 

*•    4 

% 

* 

^ 

►^"^ 

1^ 

\ 

Fia.  216. 


Take  two  long  strips  of  your  squared  paper,  fold  sharply  along 
a  line,  and  tear  one  across  ;  this  gives  you  three  stiff  iScales.  I^y 
your  eyepiece  on  the  match-box,  and  adjust  the  short  scale  wiges 
to  the  height  of  its  axis,  Fig.  216.  Using  plenty  of  table-room  and 
a  good  light,  put  the  far  cross-scale  a  goo<l  distance  away,  hunt 
with  your  pocket-lens  for  the  aerial  image  of  it,  and  lay  the  near 
scale  along  it,  adjusting  so  that  when  you  waggle  your  head  side- 
ways scale  and  image  stick  together. 

Measure  the  magnification  ;  5  divisions  of  the  image  occupied 
only  1  of  the  scale,  which  was  half  an  inch  away  from  the  flat  top 
of  the  eyepiece,  which  you  picture  full  size  on  the  rest  of  the  squared 
paper  (reduced  to  a  quarter  in  Fig.  216).     At  this  set  up  m  =  0*2. 


436 


LIGHT 


t§543 


m  =  125  Here. 

iiilinliiiilinil 


F  ( ini  Lcc  efc&tlale.; 


Bringing  the  distant  scale  nearer,  the  image  was  found  2-1  in. 
from  the  flat  end,  and  5  image  divisions  occupied  4-5  of  the  scale, 
so  at  2-1  in.  set  up  m=^  0-9.  ,    ,  o  r,  • 

Object  nearer  still,  gave  at  3-0  m.  m  =  1-4,  and  at  3-7  m.  m  =  1-7. 
Running  the  best  straight  line  possible  through  these  pomts, 
it  cut  the  axial  line  at  m  =  0,  which  is  F,  the  Principal  Focus. 
This  was  confirmed  by  pointing  the  eyepiece  to  a  far-away  lamp 
and  finding  its  image  in  focus  on  a  piece  of  tissue  paper  actually 
touching  the  top  of  the  old  eyepiece. 

Now,  turning  the  eyepiece  end  for  end,  the  three  image  positions 

and  magnifications  marked  on  the  right 
were  found,  and  the  line  through  them 
gives  the  other  principal  focus  well 
inside  the  eyepiece  (which  was  how 
these  got  called  negative  eyepieces). 

Measure  by  the  squares  the  axial 
distances  apart  of  the  places  where 
these  lines  cut  the  m  =  0,  1,  2,  3  lines ; 
these  are  /,  it  was  2-3  in.  and  2*4  in. 
on  the  two  lines  ;   mean  2 '35  in. 

Go  beyond  each  Focus  this  distance 
/,  and  you  find  the  Principal  Planes, 
where  m  =  1,  and  you  see  how  strongly 
*  crossed  '  they  are.  Light  coming  up 
your  microscope  aimed  at  the  one  just 
about  the  very  top  of  the  tube,  jumps 
to  your  eye  as  if  from  a  plane  an  inch  or 
two  lower  down. 
Fig.  217  shows  the  result  of  a  measurement  made  like  this  on  a 
2/3-in.  micro,  o.g. ;  you  see  that  the  '  working  distance '  from  the 
object,  0-16  in.,  has  no  sort  of  relation  to  the  focal  distance,  0-69  in., 
which  measures  to  the  first  p.p.  far  up  inside  the  brasswork.  Your 
1/6  and  1/12  o.gg.  are  measured  in  the  same  way,  only,  naturally, 
with  appliances  more  of  '  watchmakers'  size.' 

For  an  actual  Thick  Lens,  use  one  or  other,  or  both,  of  the  lenses 
of  your  Abbe  condenser. 

In  case  of  poor  definition,  stop  down  apertures  to  a  smaller  hole, 
for  in  the  best  lens- systems  the  principal  planes  are  really  spherically 
curved,  and  you  can  measure  only  with  patches  on  the  axis  small 
enough  to  be  sensibly  flat.  Complex  lens  systems  are  built  for 
more  serious  purposes  than  having  their  focal  lengths  measured. 

So,  without  a  figure  of  calculation,  you  have  diagnosis  and  treatment 
for  any  system  of  lenses,  from  the  simplest  to  the  most  complex  and 
costly.     That  Wall  was  only  a  dry-stone  dyke. 

Go  and  do  likewise  with  your  own  blank  walls  of  difficulty  ; 
seek  hand-  and  foot -hold  and  get  on  top  of  them.  Don't,  when 
you  get  into  practice,  ever  leave  it  to  the  hospital  people  to  get 
the  laugh  of  you. 


Fig.  217, 


§544]  THICK   LENSES  437 

§  544.  How  lenses  and  mirrors  are  made.  Glass  can  be  turned 
by  a  diamond  tool,  but  nearly  all  the  work  on  it  is  done  by  grinding 
with  emery,  or  carborundum,  of  various  grades  of  fineness,  and 
water. 

Discs  of  flawless  glass,  sometimes  fire-moulded  to  roughly  the 
right  shape,  are  stuck  with  hard  pitch  on  to  an  iron  mushroom - 
head,  covering  it  like  a  crude  mosaic.  This  is  kept  in  rotation,  and 
a  corresponding  cast-iron  concave,  turned  to  the  required  spherical 
curve  in  the  lathe,  is  worked  about  all  over  and  over  it,  with 
a  '  pestle-and-mortar '  motion,  wet  emery  mud  being  supplied 
copiously.  The  actual  movement  is  left  as  much  as  can  be  to  chance, 
and  the  result  is  a  uniform  spherical  curve.  This  (or  the  straight 
cylinder)  is  the  only  curve  that  can  be  groimd  with  any  real  approach 
to  accuracy,  all  others  develop  ineradicable  scratches.  The  curve 
is  checked  by  the  spherometer,  which  finds  here  its  real  use.  The 
mud  is  washed  off,  finer  and  finer  grades  of  abrasive  are  supplied 
in  turn,  then  all  is  washed  off  and  prepared  for  polishing. 

A  wooden  polisher  is  coated  with  pitch  and  while  still  soft 
pressed  on  the  block  of  lenses,  lifted,  scored  with  a  hot  knife,  painted 
with  rouge  and  water,  and  the  pestle-and-mortar  motion  resumed 
until  polish  is  attained.  Rouge  is  red  ferric  oxide,  obtained  by 
calcining  green  ferrous  sulphate  :  although  an  impalpable  powder, 
it  consists  of  minute  rounded  granules  just  harder  than  glass ;  it 
is  used  also  in  stropping  razors,  and  the  physics  of  the  two  processes 
is  the  same.  We  have  finished  ploughing,  and  now  we  harrow,  on 
an  ultra-microscopic  scale,  and  the  solid  surface  of  the  glass  or  the 
steel  actually  flows  before  the  pressure  of  the  polishing  granule,  and 
the  ridges  push  down  to  fill  the  furrows.  For  at  a  solid  surface  the 
cohesive  molecular  attraction  is  downwards  to  the  main  mass, 
superficial  molecules  can  be  jogged  along  sideways,  bit  by  bit, 
without  ever  escaping  from  the  cohesion,  and  in  the  end  the  surface 
packs  smooth  and  hard  and  solid  as  ever,  §  145. 

Even  ordinary  looking-glasses  must  be  made  of  glass  ground  and 
polished  in  this  way,  the  finest  '  plate  glass,'  or  else  they  distort 
dreadfully. 

The  minutest  lenses  of  a  microscope  object-glass  are  made  in 
the  same  way,  one  at  a  time.  So  are  the  greatest  telescope  lenses 
and  mirrors. 

The  final  test  of  accuracy  is  to  press  on  the  cleaned  surface  a 
corresponding  concave  polished  *  mould ' ;  the  film  of  air  between 
shows  '  soap-bubble  '  colours,  §  564  ;  it  must  go  a  full  blue  all  over 
— the  least  perceptible  difference  of  tint  means  about  a  millionth 
of  an  inch  out  of  parallelism.  Closer  than  that  there  is  no  means  of 
going,  and  no  man  can  guarantee  perfection  in  his  standard  moulds, 
he  has  to  make  them  by  the  same  process,  and  choose  the  one  he 
considers  best.  You  see,  therefore,  that  no  lens  was  ever  perfect, 
and  that  the  greatest  eye  of  the  telescope  and  the  finest  glass  of  the 
microscope  look  out  with  just  the  patient  skill  and  experience 
of  the  man  who  made  them  ;  they  share  his  humanity,  just  as  they 
share  yours  who  use  them  :   '  it  is  the  man  behind  the  gun.' 


438  LIGHT  [§  544 

Mirrors  are  silvered  chemically  by  the  reduction  of  ammonio- 
nitrate  of  silver  by  various  reducing  agents  in  alkaline  solution ; 
Rochelle  salt  for  ordinary  back-surface  mirrors,  formaldehyde  for 
vacuum  flasks,  and  sugar  for  front -surface  telescope  mirrors,  which 
have  to  be  polished,  and  require  resilvering  every  six  months. 

Quite  recently,  however,  an  electrical  distillation  in  vacuo  process 
has  been  perfected  for  depositing  an  aluminium,  or  Al  Mg  alloy, 
coating,  which  is  almost  incorrodible,  and  has  the  further  advantage 
of  reflecting  the  photographic  ultra-violet  a  great  deal  better  than 
silver  does. 


CHAPTER   XXXVII 
COLOUR 


§551.  Rummage  in  the  family  lumber-room,  and  find  up 
an  old  '  lustre,'  a  triangular  prism  of  glass  from  some  early  Victorian 
vase  or  chandelier  ;  or  go  to  the  second-hand  shop,  they  probably 
have  a  barrel-full ;  get  a  nicely  polished  one. 

Light  a  bunsen  in  front  of  a  dark  background,  and  shut  its  air- 
holes. Look  at  it  from  the  far  side  of  the  room  through  your 
prism,  holding  it  upright,  standing  on  its  triangular  end  :  you  have 
to  look  in  a  direction  nearly  45°  away,  and  you  see  the  luminous 
flame  drawn  out  into  a  broad  rainbow  band,  a  continuous  spectrum, 
the  blue  farther  away  from  the  real  position  of  the  flame  than  the 
red. 

Get  a  friend  in  to  help  you.  Open  the  air-holes,  to  non-luminous, 
and  you  see  only  one  distinct  deep-yellow  flame.  Let  him  knoc*k 
the  burner  on  the  bench,  and  it  brightens:  it  is  the  omnipresent 
saline  dust,  the  flame  of  Sodium  ;  a  little  salt  or  soda  on  an  iron 
wire  gives  it  intensely. 

Then  put  a  little  saltpetre  on  the  wire,  and  as  it  flares  up  you  sec, 
not  one  mauve  flame,  but  three  flames  ;  in  the  middle  your  sodium, 
to  its  left  a  fine  deep  red  flame,  on  its  right  a  dim  violet ;  these  two 
are  the  *  flame-spectrum  '  of  Potassium. 

Try  a  little  calcium  chloride,  and  see  what  you  can  of  Ca,  Fig.  223. 

Open  the  air-holes  wide  to  get  the  bright  roaring  greenish  cone, 
and  you  will  see  in  a  row  a  '  citron-green,'  a  green,  and  a  deep  blue 
cone — the  Carbon  spectrum. 

If  bothered  by  overlapping,  go  farther  away,  or  put  before  the 
flame  a  piece  of  tin  with  a  narrow  vertical  slit  cut  in  it,  and  you  will 
see  distinct  narrow  coloured  slits. 

Take  your  prism  down  the  main  street  at  night,  and  examine  by 
its  aid  the  glowing  lines  of  light  of  the  advertising  signs,  holding 
its  long  edges  parallel  to  them,  and  turning  it  to  give  the  clearest 
view.  Red  Neon  tubes  become  triple,  red,  orange,  yellow  (Fig. 
223  Ne  through  part-closed  eyes)  ;  blue  Mercury  tubes  a  wider 
triple,  faint  yellow,  intense  green,  deep  blue  ;  green  tubes  the  same 
without  the  blue  ;  yellow  tubes  remain  single,  Sodium. 

If  the  colours  overlap,  stand  farther  back,  so  that  the  narrow 
source  of  light  looks  narrower,  until  they  are  clear  of  one  another, 
and  the  spectrum  is  '  pure,'  pure  as  you  can  fairly  expect  from  a 
merely  ornamental  wedge  of  glass  :  you  don't  ask  for  distilled  water 
to  swim  in. 

Stick  up  a  new  pin  before  a  dark  background  in  the  sunshine, 
so  that  it  glows  with  sunlight  ;    it  will  give  you  the  whole  solar 

439 


440 


LIGHT 


[§561 


spectrum  frbm  red  to  violet,  but  not  '  pure  '  enough  to  show  the 
Hnes  that  Fraunhofer  first  glimpsed  in  1814. 

DO  THESE  THINGS. 

§  552.  The  separation  of  colours  has  been  brought  about  in  con- 
sequence of  their  different  speeds  in  Glass,  violet  slowest ;  i.e.  [i  has 
increased  between  red  and  violet,  violet  is  therefore  more  bent  round 
than  red,  and  the  colours  are  seen  as  in  Fig.  218,  in  the  directions 

which  have  been  bent  the  appro- 
w^  priate  amounts. 

^  \  To  obviate  having  to  look  so  far 

p.        ,^"-^    A  round  the  corner  for   everything, 

you  can  use  the  direct-vision  prism 
of  Fig.  219  and  §592,  where  two 
crown  prisms  correct  the  general 
deviation  of  the  dispersing  flint. 
Fig.  218.  And,  as  most  sources  of  Hght  are 

wider  than  distant  gas-tubes,  a 
narrow  adjustable  slit  is  provided  to  put  before  them,  and  the 
narrower  you  close  the  slit  the  less  the  different -coloured  views  of 
it  can  overlap,  i.e.  the  purer  your  spectrum.  One  further  acces- 
sory :  you  can't  possibly  see  the  slit  clearly  if  it  is  nearer  than 
10  in.  from  your  eye,  but  that  makes  an  awkward  long  thing  for 
the  pocket,  so  a  little  lens  is  stuck  in — a  2-in.  convex — and  you 
slide  the  tube,  focus  it  on  the  slit,  and  so  see  it  all  as  if  at  a  com- 
fortable distance  by  practically  parallel  light. 


^ 


Fig.  219. 


That  is  the  Direct-Vision  Spectroscope,  Fig.  219 ;  and  you  can 
wrap  it  round  with  paper  and  stuff  it  into  your  microscope,  in  place 
of  the  eyepiece,  if  you  suspect  micro-blood-stains. 

§533.  For  photographing  the  spectra  of  stars,  the  lens  and 
retina  of  the  eye  can  be  replaced  by  lens  and  sensitive  plate  at 


Fig.  220. 


its  principal  focal  distance,  the  whole  forming  a  long  camera  with 
a  prism  close  in  front.  Fig.  220.  The  star's  light,  coming  in  one 
parallel  beam,  is  refracted  into  different  directions  by  the  prism, 


§554] 


COLOUR 


441 


and  concentrated  into  a  string  of  images  of  the  star  on  the  plate, 
the  latter  moving  slowly  sideways  to  broaden  out  the  beaded  line 
into  a  band. 

Any  attempt  to  produce  a  spectrum  on  a  screen,  without  a  lens 
to  form  clear  images  of  the  narrow  source,  can  only  result  in  a  coloured 
blur. 

It  is  always  best  that  the  light  should  fall  on  the  prism  in  one 
direction  only.  To  secure  this  in  ordinary  larger  spectroscopes 
another  lens  L'  first  catches  the  light,  from  the  illuminated  slit  at 
its  principal  focus,  and  '  collimates  '  it  (brings  it  into  line).  And 
since  the  spectrum  is  small,  an  eyepiece  E  is  used  to  magnify  it. 
Then  for  measuring  purposes  a  scale  of  some  sort  is  provided  in  the 


Fig.  221. 

eyepiece,  or  else  the  telescope  LE  turns  on  a  graduated  scale ; 
and  the  spectroscope  becomes  a  Spectrometer.  Fig.  221. 

The  prism  usually  stands  in  its  position  of  minimum  deviation, 
easily  found  and  excellent  in  definition. 

Large  spectroscopes  may  have  two  or  more  prisms  in  succession. 

Fitted  for  photography,  as  is  usual  nowadays,  they  are  Spectro- 
graphs. 

§554.  The  fact  that  a  Diffraction  Grating,  §  402,  will  do 
instead  of  the  Prism,  shows  that  different  colours  correspond  to 
different  wave-lengths  of  light  or  (reciprocally, 
different  vibration  frequencies),  the  red  to  the 
longer  waves  (slower  vibrations).  The  grating 
enables  spectrometer  readings  to  be  translated 
into  wave-lengths. 

For  in  Fig.  222,  if  a  wire-gauze  with  spacing 
g  be  lit  from  the  left  by  plane  waves  AB,  by 
comparing  with  Fig.  133  you  see  that  a  succession 
of  waves  separated  by  wave-length  g  sin  a  will  be 
thrown  off  at  angle  a,  shorter  at  lesser  angles,  and 
longer  at  greater.  That  is,  if  these  waves  can  be 
manufactured  out  of  the  light  supplied  ;  if  not, 
some  directions  are  left  dark. 

Fine  wire-gauze,  or  the  silk  of  your  umbrelhi. 
held  up  to  a  sharp  distant  light,  shows  these 
diffraction  spectra,  both  sides  and  both  ways,  and  even  your  half- 
closed  eyelashes  produce  them  ;    but  the  Gratings  actually  used 


Fio.  222. 


442  LIGHT  [§  554 

are  ruled  on  polished  speculum  metal,  by  automatic  machines  of 
the  greatest  exactness,  about  15,000  lines  per  inch,  and  5  in.  or  more 
wide.  They  can  produce  a  spectrum  50  ft.  long,  with  more  than 
sixty  lines  between  the  D  lines  of  sodium  (which  differ  in  wave-length 
by  1/10%,  and  are  only  just  separable  by  the  best  little  laboratory 
prisms). 

Colours  and  lines  follow  in  precisely  the  same  order  as  with  a 
prism,  but  the  red  is  elongated ;  while  a  prism  make  this  short, 
and  draws  the  violet  out  long. 

For  whereas  the  Grating  works  in  virtue  of  its  spacing,  and  spreads 
its  spectrum  proportionally  to  Wave-length,  the  Prism  works  by 
means  of  the  controlled  frequency  of  vibration  of  the  electrons  in 
the  glass  (which  is  an  insulating  dielectric,  §  737),  and  the  scale  of 
its  single  bright  spectrum  is  roughly  like  one  of  Frequency,  the 
reciprocal  of  the  former,  as  in  Fig.  223. 

§  555.  Putting  your  doctored  bunsen  flame,  then,  before  the 
slit  of  the  properly  adjusted  Spectroscope,  you  see  now  the  same 
spectra  as  you  did  before,  only  now  you  keep  the  '  source  of  light ' — 
the  Slit — as  narrow  as  you  comfortably  can,  until  the  coloured 
images  of  it  become  mere  bright  *  lines,'  with  much-diminished  risk 
of  overlapping  and  causing  impurity  of  spectrum,  and  disclosing 
the  existence  of  certain  definite  Frequencies  of  Vibration  in  their 
source. 

Bright-line  Spectra  characterize  incandescent  gases  and  vapours. 
The  Flame  brings  them  out  from  the  alkali  metals  and  thallium 
(10-10  gm.  of  sodium  is  enough)  ;  and  from  the  alkaline -earth 
metals,  gay  in  fireworks,  and  containing  also  broader  lines  or 
bands.  The  volatile  salts  put  into  the  flame  soon  oxidize,  and  their 
non-metallic  constituents  make  little  difference  after  the  first  couple 
of  seconds  :  one  generally  dips  the  wire  repeatedly  into  hydrochloric 
acid,  for  chlorides  are  volatile.  Mixtures  give  the  spectra  of  every 
metal  present.  The  pretty  spectrum  of  the  blue-green  cone  itself 
now  shows  as  four  '  fiuted '  bands,  sharp  on  one  side,  Fig.  223, 
the  Carbon  Spectrum  of  the  CO  flame,  as  over  a  coke  fire.  The 
brighter  blue  that  results  from  throwing  salt  on  the  fire  is  a  different 
story  altogether ;  a  good  spectroscope  discloses  in  it  the  arc  lines 
of  copper,  widespread  in  all  coal. 

All  metals  blaze  out  into  magnificent  spectra  of  many  bright 
lines  when  put  into  the  crater  of  the  Electric  Arc;  some  of  calcium 
are  sketched  in  Fig.  223. 

The  greater  violence  of  the  Electric  Spark  between  points  of 
the  metal,  or  wet  with  solutions  of  its  salts,  enhances  certain  arc 
lines,  and  calls  forth  new  ones  of  its  own,  approximating  to  those 
of  the  hotter  Stars. 

Gases  are  made  luminous  in  the  way  now  familiar  with  neon,  mer- 
cury (blue  or  green),  and  sodium  (yellow),  §551,  by  passing  high- 
voltage  electric  discharges  through  them  at  a  reduced  pressure. 
These  are  the  brightest,  but  no  gas  fails  to  glow  when  enclosed 


'' 


§566] 


COLOUR 


443 


in  a  narrow  tube  at  about  1/100  atmosphere  pressure,  and  tickled 
by  a  sparking  coil.  The  red,  blue,  and  violet  lines  of  Hydrogen 
appear  in  Fig.  223,  the  C  spectrum  is  equally  that  of  COg  in  a  tube. 
In  addition  to  lines,  Nitrogen  in  a  tube,  or  in  a  Leyden  jar  spark, 
gives  many  flutings  in  the  violet,  accounting  for  that  tint  in 
lightning. 

Gaseous  Nebulae  emit  hydrogen  lines,  and  occasional  C  or  N, 
and  almost  invariably  a  line  at  0-37  micron  is  photographed,  of 
Oxygen. 

FREQUENCY      OF     VIBRATION     IN     BILUONS     (lo"^)    PER      SECOND 
^lOo  Sioo  6100  7po 


o7'o  o-6'o  o-^o  ,         cytfo 

WAVE       LENGTH      IN       MICRONS^  ; 

red  \  orange., y  el. \       ore  en      »  Hue  1  motet 

BRIGHT/^£SS  OF  SOLAR  SPECTRUM  TO  THE  EYE 

3      ZO    30  rOO  7S         ZS        6  CENTRAL       OS  Ct 

i-fi    10  35  35        7S     3S     18  P£R/PHe./iAL  z  o  s 

Fig.  223. 

Comets  show  the  carbon  spectrum,  and  sodium  when  near  the 

sun.  ^        ,         . 

All  the  foregoing  are  'Emission  Spectra,'  sent  out  by  coloured 
lights  that  shine  of  themselves  in  the  dark,  emitting  waves  that  are 
really  tiny  radio  waves,  from  atoms  instead  of  from  long  poles. 
On  a  500  million  times  larger  scale  a  *  crystal  set '  is  emulating  your 
'  lustre  ' ;  while  a  hyper-super-everything,  with  its  massed  tune<l 
circuits,  picks  out  the  spectrum  of  Europe  as  well  as  does  a  pocket 
spectroscope,  with  the  controlled  electronic  vibrators  of  its  glass, 
the  spectrum  of  an  arc.  The  spectroscope  is  the  radio-set  of  the 
atoms,  as  the  fireside  wireless  is  the  spectroscope  of  the  sprawling 
waves  manufactured  by  man. 


444  LIGHT  [§666 

§  556.  When  hydrogen  is  compressed,  its  few  bright  lines 
broaden  out  into  indistinct  bands.  Matter  being  much  closer 
packed  as  solid  or  liquid  than  as  gas,  it  is  not  surprising  to  find 
that  incandescent  solids  and  liquids  give  a  Continuous  Spectrum, 
all  characteristic  lines  being  blurred  out.  The  electrons  in  atoms 
close  together  interfere  with  one  another's  vibrations,  and  produce 
a  confused  jumble  of  indefinite  frequencies. 

On  this  view,  the  candle  flame  owes  its  luminosity  to  in- 
candescent particles  within  it,  easily  deposited  as  soot,  whereas 
the  blue  base  gives  the  C  bright  lines.  The  continuous  spectrum 
of  the  arc  lamp  is  crossed  by  bright  lines  :  by  forming  a  real  image 
on  the  slit,  the  continuous  is  seen  to  rise  from  the  hot  carbons,  and  the 
lines  from  the  faint  arc  itself.  Everything  put  into  that  furious 
little  furnace  gasifies,  and  colours  the  arc  flame,  and  so  we  got  its 
bright*  line  spectrum. 

This  Continuous  Spectrum  is  the  familiar  broad  rainbow  band  of 
colours  :  what  colours  ? 

At  first  glance.  Red,  Green  and  Blue. 

Between  red  and  green  is  a  bright  region  which  changes  its  apparent 
colour  according  as  you  have  been  staring  at  the  red  or  the  green. 
Call  it  Orange  towards  the  red,  and  Yellow  towards  the  green ; 
but  to  see  Yellow  conspicuous  in  the  spectrum  you  must  open  the 
slit  wide,  so  that  orange  and  green  overlap,  or  else  use  a  very  dazzhng 
light,  which  blurs  them  together  in  your  eye  ;  i.e.  yellow  is  an  im- 
pure mixture,  and  has  no  business  there  :   see  also  §  576. 

As  you  go  along  the  blue,  taking  care  to  shut  out  all  extraneous 
light,  for  it  is  getting  dark,  a  faint  coppery  tinge  appears.  Here  is 
what  Newton  called  Indigo,  knowing  that  crystals  of  that  deep  blue 
dye  had  a  coppery  sheen.  This  merges  into  Violet,  which  is  brighter 
the  whiter  is  the  light ;   and  then  the  dark. 

This  almost  Octave,  shown  in  Fig.  223,  from  wave-length  0*77 
micron  down  to  0-39,  or  from  frequency  390  to  770  billions  per 
second,  is  all  that  the  Eye  can  see  of  the  vast  range  of  Radiation 
discussed  further  in  Chapter  LVI ;  and  from  the  figures  at  the  very 
bottom  of  Fig.  223  you  see  how  much  its  sensitivity  has  already 
faded  off  both  ways.  This  is  not  the  place  to  treat  of  infra-red  and 
ultra-violet,  for  the  flint-glass  prism  of  the  ordinary  spectroscope 
is  opaque  to  both  of  them  ;  and  anyway  they  have  nothing  to  do 
with  Colour,  being  invisible,  see  Chapter  LVI. 

§  557.  The  room  is  in  darkness,  save  for  the  glow  of  a  flameless 
fire.  Musing  beside  it  on  what  I  shall  write,  I  reach  down  to  it 
three  books,  all  alike  black  as  they  lay  on  the  table.  The  first 
glows  red  in  the  firelight,  with  an  ornamental  pattern  plain  upon  it 
in  black,  '  The  Sun,'  by  C.  A.  Young,  a  prize  book  of  long  ago 
that  actually  got  read.  The  second  shows  dull  green,  black  and 
quite  unornamental  upon  it,  '  Handbook  of  Physics,'  by  one  W.  H. 
White.  The  third  is — black,  yes,  all  black  all  over,  until  I  strike 
a  match,  when  it  becomes  a  purplish-blue,  changing  to  royal  blue 


§  558]  COLOUR 


445 


under  the  gas-mantle,  and  declaring  itself  in  black  type  '  Sic  itiir 
ad  astra,'  a  lighter  effort  by  that  same  star  of  small  magnitude. 

I  turn  a  prism  on  the  glowing  coke — red,  and  a  fair  fringe  of  green, 
nothing  else.  Red  lit  my  Sun  gaily  ;  of  green  there  was  less,  but 
enough  ;  of  blue,  none,  to  call  forth  the  response  of  my  pamphlet's 
cover.  Coloured  things  that  do  not  shine  of  themselves  have  no 
colour  in  the  dark,  only  '  potential  colour  '  :  the  finest  church- 
window  offers  no  diversion  during  a  dull  sermon  in  winter  Evensong. 
The  gas-mantle,  far  hotter  than  the  fire,  intensely  '  white-hot,' 
shows  a  long  rich  train  of  blue  and  violet. 

In  monochromatic  light,  of  course,  everything  reduces  to  one 
colour  in  varying  brightness.  Passing  a  geranium  along  a 
spectrum,  it  is  dark  in  the  blue,  in  the  green  a  black  flower  with 
green  leaves,  then  dull  yellow  all  over  ;  and  in  the  red  a  red  flower 
with  black  leaves.  The  face  and  lips  are  sallow  and  dark  in  the 
light  of  a  salted  bunsen  flame,  with  a  trace  of  blue  from  the  flame 
itself. 

Your  mother  has  told  you  of  the  difficulty  in  matching  colours  by 
artificial  light,  especially  plum  colour  (try  chrome-alum  solution) ; 
because  it  is  so  often  deficient  in  blue  afid  violet.  We  smile  at 
the  poor  bedridden  old  vanity  who  would  have  '  rose-coloured 
curtains  for  the  doctors,'  but  green-painted  walls  in  the  bathroom, 
and  a  greenish  cast  in  the  glass  of  its  mirror,  shake  any  man's  faith 
in  his  liver. 

The  pretty  bluish  mercury- vapour  lamp,  with  its  yellow,  green, 
and  violet,  shows  up  objects  of  those  colours  vividly,  but  the 
process-engravers  introduced  it  very  cautiously  into  their  work- 
shops, fearing  an  outcry  from  the  women  workers  ;  while  a  butcher, 
who  had  been  persuaded  to  instal  it,  cast  it  out  in  haste  when  his 
customers  fled  from  the  yellow-green  fat  and  lurid  purple  flesh  that 
they  misdoubted  on  his  stall. 

All  ordinary  coloured  things,  therefore,  must  be  illuminated  before 
they  can  show  colour,  and  by  the  white  light  of  day  if  their  colours  are 
to  show  '  true.'  They  cannot  add  to  it,  they  can  only  take  away  from 
it :  their  spectra  are  therefore  called  Absorption  Spectra,  and  consist 
of  dark  blocks  and  bands  and  lines  and  shadings,  often  very  nebulous 
indeed,  all  curtaining  the  featureless  Continuous  Spectrum  of  the 
source  of  light  you  must  employ. 

§558.  Colour  by  transmitted  light.  Put  the  coloured!  glass  or 
film,  or  the  coloured  liquid  in  cell  or  test-tube,  before  the  spet*tro- 
scope  slit,  and  point  it  at  a  White  Light.  All  coloured  substances 
then  produce  Absorption  Spectra ;  that  of  the  photographer's  ruby 
glass,  for  instance,  is  a  broad  black  shadow  blotting  out  all  except 
the  red.  A  red  signal  shines  through  it  with  transparent  bright- 
ness ;  to  a  green  signal  it  is  opaque,  the  received  energy  merely 
goes  to  warming  it.  A  test-tube  of  weak  pink  permanganate 
solution  held  between  lamp  and  spectroscope  slit  produces  five 
dark  bands  in  the  green,  looking  like  your  fingers  held  up  in  front, 


446  LIGHT  [§  558 

a  stronger  solution  blots  out  the  green  altogether.  Hence  its  colour 
is  what  is  left  of  White  Light  after  the  green  has  been  removed. 
Restoring  this  would  complete  the  white  again,  and  that  is  what  is 
meant  by  the  statement  that  crimson  and  green  are  complementary 
colours. 

The  yellow  or  orange  of  a  flower  petal  is  interesting,  and  rather 
startling  ;  it  is  just  red  and  green  in  more  or  less  equal  parts ; 
there  is  no  spectrum  yellow. 

Cobalt  glass  lets  through  a  narrow  band  of  red  without  absorption, 
a  narrow  green,  and  the  whole  range  of  blue  and  violet  with  a  slight 
absorption.  There  is  much  more  of  these  last  to  start  with,  and  the 
light  appears  blue,  but  many  thicknesses  of  glass  increase  this  ■ 
veiling  of  blue  and  violet,  and  the  bundle  turns  red,  the  only  un- 
hindered colour. 

The  green  tube  of  light  of  the  advertising  signs  contains  mercury 
vapour,  as  does  the  blue,  but  its  glass  is  faintly  yellow,  and  hides 
the  violet  line,  and  that  makes  the  difference. 

Many  colours,  especially  browns,  are  disappointing,  because 
the  colours  are  only  more  or  less  shaded  :  these  have  to  be  left  to 
the  Spectro-photometer,  an  expensive  measuring  instrument,  for 
the  specialist. 

Iodine  vapour  and  NOg  have  very  complex  dark-line  spectra ; 
didymium  salts  absorb  several  scattered  portions,  with  the  result 
that  they  appear  almost  colourless.  Fig.  223. 

Chlorophyll  (for  which  crush  green  leaves  in  alcohol,  and  filter) 
contains  at  least  two  constituents  (some  make  out  six)  :  a  blue- 
green  and  a  yellow.  The  yellow  absorbs  the  blue  and  violet.  Fig. 
223,  which  are  definitely  injurious  to  plant  life  (compare  the  '  tan  ' 
which  develops  and  saves  our  skins  from  the  ultra-violet),  the  green 
absorbs  that  massive  band  of  red  from  0-64  to  0-68  micron,  and 
beyond  that,  right  down  through  the  infra-red,  it  absorbs  no  more. 
It  is  an  important  band,  for  on  that  band  of  light  the  plant  lives, 
and  therefore  so  do  we. 

Blood  (a  few  drops  in  water)  of  course  transmits  through  to  us 
plenty  of  red,  but  only  a  trace  of  blue ;  its  great  characteristic  is 
the  pair  of  bands  in  the  green.  These  can  be  moved  into  one  by 
reducing  it  to  the  venous  condition,  but  more  important  is  the  change 
from  oxy-  to  carboxy-hsemoglobin  by  coal-gas  or  exhaust  fumes 
poisoning.  This  moves  the  '  greener  '  band  nearer  its  fellow  by 
a  very  small  amount,  best  measured  in  a  '  reversal '  spectroscope, 
which  lays  two  spectra  together  side  by  side  opposite  ways  :  the 
change  is  minute,  but  sufficient  to  ascertain  the  percentage  of  CO 
in  the  blood  within  2  or  3%. 

§559.  The  Solar  Spectrum.  Pointing  the  spectroscope  at  the 
bright  sky,  or  the  sun  itself,  and  carefully  narrowing  the  slit,  you 
see,  crossing  the  bright  continuous  spectrum,  and  interrupting  its 
continuity,  the  dark  lines  which  Fraunhofer  in  18141isted  by  hundreds, 
naming  the  most  prominent  alphabetically,  and  of  which  great 


§  559]  COLOUR  447 

modern  spectroscopes  reveal  many  thousands.  [Not  lines  running 
from  end  to  end  of  the  spectrum  ;  they  are  due  to  dirt  on  the  slit  : 
clean  it  with  a  pointed  match-stick.]  Evidently  these  are  absorp- 
tion lines,  being  dark,  but  it  was  noticed  presently  that  several 
of  them  tallied  in  position  and  appearance  with  well-known  bright 
lines  of  the  laboratory,  such  as  D  with  the  sodium  yellow,  b^^  with 
the  magnesium  green  triplet,  and  C,  F,  G  and  h  with  the  hydrogen 
red,  blue,  and  violet.  Fig.  223  ;   but  why  were  they  dark  ? 

If  you  look,  through  a  small  spectroscope,  at  an  Arc  which  has 
been  dosed  with  soda,  the  sodium  line  will  flicker  bright  and  dark, 
and  if  the  instrument  be  powerful  enough  to  show  the  line  as  a 
well- separated  twin  pair,  each  will  show  a  fine  dark  line  down  its 
middle,  the  exact  frequency  of  vibration  of  the  cooler,  less-agitated, 
sodium  vapour  surrounding  the  arc,  from  which  it  is  distilling  out. 

Naturally  the  atoms  of  this  absorb,  from  the  abundant  and  varied 
radiation  pouring  through,  just  their  own  quantum  of  energy,  and 
then,  having  nothing  else  to  do  with  it,  radiate  it  out  again.  It  is 
just  what  the  piano-strings  do,  when  the  forte  pedal  is  held  down, 
and  the  dog  barks.  But  whereas  they  got  it  from  one  direction, 
they  now  throw  it  out  in  all  directions,  so  that  only  a  fraction  of 
it  now  travels  on  along  the  original  direction  towards  you,  and  that 
particular  frequency  therefore  appears  to  you  much  reduced  in 
brilliance — a  comparatively  dark  line. 

Had  the  Sun,  then,  a  cooler  atmosphere  of  hydrogen  and  metallic 
vapours  to  select  and  absorb  parts  of  the  continuous  spectrum 
emitted  by  the  dense  incandescent  sphere  ?  If  so,  at  the  moment 
when  the  eclipsing  Moon  just  blotted  out  that  dazzling  body, 
would  not  the  glowing  atmosphere  round  the  edge  show  these 
same  lines  as  bright  lines,  since  it  was  only  by  contrast  that  they 
looked  dark  ?  At  the  eclipse  of  18G8  the  red  ring  of  *  chromo- 
sphere '  that  flashed  round  the  dark  moon  showed  the  spectra  of 
hydrogen,  calcium,  and  the  element  first  discovered  and  name<l 
from  its  line  there,  Helium. 

Sometimes  the  chromosphere  is  so  thick  and  hot  that  it  actually 
radiates  visible  bright  lines  down  the  middle  of  the  heavy  black 
hydrogen  C  red,  or  calcium  H  and  K  violet — a  stage  further  than 
we  saw  from  the  arc — and  by  the  isolated  light  of  these,  hydrogen 
and  calcium  clouds  are  now  photographed  daily  all  over  the  surface 
of  the  sun,  and  round  its  edge. 

Many  thousand  solar  lines  have  been  identified  with  H,  Na,  Ca,  Ba, 
Mg,  Fe,  Mn,  Ni,  Al,  etc.,  etc. 

The  strongest  bands  in  the  red,  A,  a,  and  B,  get  darker  as  the 
sun  sinks  and  shines  through  a  longer  length  of  the  eurth's  atmos- 
phere :  they  are  due  to  our  own  water  vapour,  as  is  also  a  broad 
band  in  the  yellow,  shown  above  D  in  Fig.  223,  called  the 
Rainband,  darkening  before  rain,  and  used  by  some  people  in 
forecasting  it. 

To  see  this  extreme  red  end  of  the  spectrum  at  all  well,  you  must 
cut  off  the  dazzling  mid-parts  by  red  glass,  or  even  better,  a  piece 


448  LIGHT  [§  559 

of  the  common  cobalt-blue  glass  of  §  558,  which  transmits  the  red 
end  brilliantly. 

The  Solar  Spectrum  thus  represents  the  Continuous  Emission  of  a 
dense  incandescent  mass,  less  the  Absorption  of  the  gaseous  envelopes 
of  Sun  and  Earth. 

§  560.  Stellar  spectra.  The  Stars  are  classified  by  their  spectra. 
The  few  of  Class  O  show  bright  lines  ;  all  the  rest  have  dark  absorp- 
tion lines.  Class  B,  bluish- white  stars  such  as  Rigel,  show  heUum 
absorption,  and  hydrogen.  Class  A,  white  stars  like  Sirius  and 
Vega,  show  only  intense  hydrogen  absorption  ;  in  Class  F,  Canopus, 
faint  subsidiary  lines  appear,  Ca,  etc.  Class  G  includes  the  Sun, 
and  Capella  ;  Class  K  is  cooler  and  perceptibly  yellowish,  Arcturus 
shows  more  lines  and  stronger  than  the  Sun,  and  a  compound, 
TiOg,  appears  ;  and  grows  to  almost  dominate  the  crowded  spectra 
of  the  reddish  stars,  like  Betelgeuse,  of  Class  M,  while  some  branch 
off  into  a  Class  N  characterized  by  cyanogen  and  carbon-arc  lines. 

This  is  perfectly  definitely  a  classification  by  Temperature,  as 
the  visual  colour  suggests  ;  in  the  hotter  stars  most  of  our  familiar 
elements  are  dissociated,  in  the  cooler  ones  they  have  the  hardihood 
to  form  simple  chemical  compounds. 

Great  pressure  modifies  a  spectrum  slightly,  and  the  actual 
masses  of  some  stars  have  been  determined  from  these  minute 
changes.  Among  the  Planets,  the  absorption  of  CO2  has  been 
detected  above  the  shining  clouds  that  hide  the  surface  of  Venus, 
ammonia  in  Jupiter's  atmosphere,  and  methane  in  Saturn's,  all 
suggesting  volcanic  activity. 

Doppler's  Principle,  §  398,  finds  its  chief  application  here  :  if 
source  and  spectroscope  are  approaching  each  other,  the  frequency 
of  vibration  appears  increased — the  keyboard  of  a  piano  is  a 
spectrum  of  sound,  and  the  note  is  falsely  struck  sharp,  like  an 
approaching  railway  whistle — i.e.  well-recognized  spectrum  lines 
are  shifted  a  trifle  towards  the  quicker  violet,  and  the  speed  of 
approach  is  easily  calculated  in  terms  of  the  great  speed  of  light. 

In  this  way  the  speeds  of  approach  or  recession  of  multitudes 
of  stars  have  been  measured,  and  are  of  the  order  of  20  miles  per 
second ;  Saturn's  rings  have  been  found  to  revolve  faster  inside 
than  outside  (meteor  swarms)  ;  enormous  numbers  of  stars  prove 
to  be  '  spectroscopic  binaries,'  some  lines  showing  double  as  one 
component  goes  away  and  the  other  approaches  us  in  their  orbital 
motion  ;  and  lastly,  the  most  minute  and  distant  nebulae  that  can 
be  photographed,  appear  to  be  all  receding  from  us  at  amazing 
speeds,  proportional  to  their  distance,  up  to  12,000  miles  per  second 
at  6  X  10^0  miles  distance,  a  shift  of  lines  I /16th  the  spectrum 
towards  the  red,  '  a  semitone  flattening,'  it  looks  as  if  the  Universe 
is  expanding. 

§561.  Colour  by  irregularly  reflected  light.  Thus  colour  seen 
through  is  accounted  for,  but  what  of  the  colour  of  leaves,  flowers, 


§562]  COLOUR  449 

and  earths,  of  feathers,  fabrics,  etc.,  looked  at  and  seen  by  the  light 
they  scatter,  §  493  ? 

(1)  A  glossy  leaf,  or  varnished  picture,  when  regularly  reflecting 

light  to  the  eye,  shows  hardly  a  trace  of  its  own  colour ; 
smooth  water  reflects  noonday  blue  or  sunset  gold  impartially. 

(2)  Under  the  microscope,  by  transmitted  light,  individual 
coloured  grains,  cells,  and  fibres  are  remarkably  trans- 
parent, and 

(3)  By  reflected  light  each  shows  a  certain  amount  of  internal 

reflection,  like  cut  gems  or  rods  of  coloured  glass,   §491, 
Fig.  191  (10). 

(4)  You  write  or  paint  on  white  paper,  not  on  black  ;  i.e.  you  ask 

for  light  to  be  sent  back  through  the  colours. 
Coloured  chalks,  for  blackboard  use,  are  plaster  of  paris,  with 
a  trifle  of  '  transparent '  colouring  matter. 

That  is,  part  of  the  light  things  scatter  has  dived  through  absorb- 
ing material,  and  therefore  they  show  much  the  same  colour  as  by 
directly  transmitted  light.  The  other  part  has  come  back  uncoloured 
from  the  front  surface. 

The  proportion  of  the  two  parts  varies  greatly  ;  silk  dilutes  its 
colour  with  surface  light,  velvet  does  not,  satin  looks  either  rich 
coloured  or  merely  shiny  according  as  its  surface  light  misses  or 
catches  the  eye.  Wetting  a  sponge,  or  varnishing  wootl,  means 
filling  it  with  a  medium  of  about  its  own  refractive  index,  which 
does  away  with  the  more  superficial  reflections,  and  permits  the 
light  to  dive  deeper  and  return  more  richly  coloured. 

Conversely,  finely  powdered  copper  sulphate,  bichromate,  froth 
on  beer,  etc.,  appear  only  slightly  tinted  ;  so  many  little  surfaces 
are  flinging  back  the  light  before  it  can  traverse  any  appreciable 
thickness  of  coloured  substance. 

§  562.  Metallic  colours.  Whereas  the  surface  sheen  on  a 
'  skyed  '  picture  hides  all  the  colours,  the  golden  lustre  of  its  frame 
never  changes  :  its  sheen  is  its  colour.  The  thickness  of  the  gold- 
leaf  on  the  frame  is  1/10  micron,  not  a  thousandth  of  the  thickness 
of  this  paper.  Stuck  on  glass,  the  leaf  is  seen  to  be  translucent,  it 
lets  through  about  as  much  light  as  this  paper  does,  and  that 
light  is  blue-green,  complementary  in  colour  to  the  reflected.  To  all 
other  colours  the  metal  is  intensely  opaque  ;  double  thickness 
won't  let  even  that  through. 

The  Reef  of  §  493  and  Fig.  195  presses  into  service  again  :  one 
frequency  of  vibration  finds  it  possible  to  struggle  a  little  way  over 
the  rough  rocks ;  for  the  other  the  toll  of  energy  demanded  is  too 
heavy,  and  they  turn  back  at  the  very  surface,  with  the  loss  of  their 
favoured  companion. 

Copper  cannot  be  beaten  into  thin  leaf,  but '  diluted  '  into  solution 
transmits  the  complementary  blue  :  the  bluish  chrome  plating  of 
Q 


460  LIGHT  [§  562 

the  car  is  the  opposite  in  tint  of  the  chrome  yellow  warning  painted 
on  the  roadway. 

Many  very  intensely  coloured  dye-substances  act  similarly.  Indigo 
crystals  have  a  brilliant  coppery  sheen  :  '  Crystal  violet '  is  lustrous 
green,  purest  methylene  blue  is  golden,  blue  ink  dries  coppery  on 
the  rim  of  the  bottle,  clean  crystals  of  permanganate  can  show  that 
green  which  you  saw  refused  transmission  in  §  558. 

The  intense  black  band  in  the  red  of  the  Chlorophyll  spectrum 
leads  one  to  expect  something  of  the  sort  there,  and  a  test-tube  of 
its  filtered  alcoholic  solution  glows  like  thick  blood  in  a  red  light. 

§563.  Fluorescence  and  Phosphorescence.  Sun  shining  on  the 
'  canary  glass  '  (containing  uranium),  sometimes  used  in  ornaments, 
makes  it  glow  with  a  green  light  quite  different  from  its  ordinary 
pale  yellow  tint.  Even  if  filtered  through  blue  glass  (short  waves 
only)  it  still  excites  the  green  (longer  wave)  Fluorescence. 

Further,  by  using  a  rotating  perforated  disc  arrangement 
(phosphoroscope),  the  glow  can  be  seen  persisting  for  a  small 
fraction  of  a  second  after  the  sunlight  is  cut  off.  This  is  brief 
Phosphorescence.  All  fluorescent  solids  phosphoresce,  for  at  least 
1 /100,000th  second. 

From  Fig.  223,  top,  you  see  that  this  gives  time  for  5000  million 
vibrations  or  so,  which  suggests  that  fluorescence  and  phosphores- 
cence are  processes  in  which  energy  is  absorbed  and  stored  by  the 
substance,  for  subsequent  re-emission  in  a  form  it  selects  for  itself. 

For  any  particular  substance  the  exciting  light  vibrations  must 
be  within  certain  limits  of  frequency.  This  can  be  shown  by  using 
any  fluorescent  solution,  e.g.  very  weak  quinine  bisulphate  (blue), 
decoction  of  horse-chestnut  bark  (blue),  fluorescein  and  eosin 
[red  ink  in  water]  (green).  Holding  these  in  the  sunshine,  the 
fluorescence  does  not  spread  far  into  the  liquid,  evidently  the 
first  layers  have  used  up  the  effective  rays.  Lamplight  is  less 
efficient,  yellow  does  not  excite  the  glow,  blue  does.  This  points 
to  the  blue,  violet,  and  ultra-violet  as  usually  the  exciting  radiation. 
(Early  investigations  of  the  ultra-violet  spectrum  were  made  by 
forming  it  on  a  quinined  screen.) 

Fluorescence  spectra  are  rather  vague  emission  spectra. 

Fluorescein  is  a  brown  powder  which  the  M.O.H.  throws  down 
the  drain,  and  later  examines  the  drinking-water  supply  for  the 
green  glimmer  which  would  prove  that  traces  of  contamination  were 
seeping  through.  The  biggest  instance  on  record  occurred  in  1930, 
when  a  geological  student  and  his  sister  sprinkled  a  hundredweight 
of  it  into  the  Trou  di  Toro  in  the  Pyrenees,  and  the  Garonne 
gushed  green  all  the  next  day,  proving 

'  that  sacred  river  ran 
Through  caverns  measureless  to  man  ' 

right  under  the  main  ridge  ;  and  saving  it  from  being  drained  dry 
by  a  Spanish  hydro-electric  power  scheme  which  would  have  dis- 
charged into  the  Ebro. 


§  564]  COLOUR  451 

More  or  less  associated  with  fluorescence  are  the  '  fugitive  *  (i.e. 
easily  light-bleached)  dyes,  which  are  used  to  make  the  photographic 
film  sensitive  to  colours  other  than  the  blue  and  violet  which 
naturally  affect  it ;  and  also  the  '  visual  purple  '  of  the  Retina, 
§  603.  The  film  is  dipped  in  a  bath  of  suitable  mixed  dyes,  and 
dried,  to  become  '  orthochromatic  '  or  '  panchromatic,' — sensitive 
to  green  and  yellow  or  to  red  also  ;  or  even  a  little  beyond  into  the 
invisible  '  infra-red.'  These  cause  changes  in  the  dye,  which  in 
turn  affect  the  silver  bromide. 

Commonly  we  see  little  of  the  Red  Seaweeds,  for  their  habitat 
is  below  the  lowest  tide-marks  on  coasts  where  the  water  is  clear 
and  tides  are  deep.  Below  them  no  plant  grows,  for  lack  of  light : 
they  contain  chlorophyll,  but  through  such  depth  of  blue-green 
water  only  a  trifle  of  red  can  pass,  into  its  great  2/3  micron  absorption 
band.  Consequently  a  red  pigment,  phyco-erythrin,  is  developed, 
and  this  absorbs  just  the  green,  from  D  to  F,  Fig.  223,  which  is 
not  only  most  abundant  in  daylight,  Fig.  411  top  scale,  but  is  best 
transmitted  by  water.  This  substance  fluoresces,  converting 
this  blue-green  into  orange  and  the  red  which  the  chlorophyll 
must  have  for  life. 

Though  why  Sphcerella  {hcematococcus)  pluvialis,  '  blood  rain,' 
flourishing  in  pure  culture  in  our  bird-bath  in  broad  daylight,  should 
elect  to  develop  a  blaze  of  orange-red  which  altogether  obscures 
its  chlorophyll  absorption,  still  leaves  one  thinking. 

The  ultra-violet  makes  many  things  fluoresce ,  see  §  955 .  EggsheUs 
phosphoresce  only  when  cooled  by  liquid  air. 

The  luminescence  of  oxidizing  phosphorus,  of  course,  gives  its 
name  to  Phosphorescence.  The  little  lantern  of  the  glow-worm 
is  fanned  by  air  from  its  spiracles,  see  §  975  ;  and  that  is  how 
fire-flies  carry  on  their  conversations  in  Morse.  The  luminosity  of 
some  fungi  and  micro-fungi,  of  fish,  and  that  of  noctiluca  and  other 
small  plankton  which  cause  the  phosphorescence  of  the  sea,  can 
be  only  mentioned. 

X-rays,  cathode-rays,  and  radium  call  forth  strong  phosphores- 
cence in  barium  platinocyanide,  willemite,  zinc  blende,  etc.,  §§  884, 
917,  932. 

The  flashes  between  lumps  of  sugar  rubbed  together  in  the  dark, 
in  the  crack  of  black  insulating  tape  as  you  pull  it  open,  of  mercury 
shaken  in  vacuo,  etc.,  are  electrical  discharges. 

§564.  Interference  colours.  In  a  soap  bubble,  in  the  thin 
film  of  oil  on  water  or  a  wet  roadway,  of  oxide  on  hot  polished  metal, 
of  tarnish  on  Roman  glass,  of  air  or  water  squeezed  between  clean 
plates  of  glass,  of  air  in  cracks  in  glass,  mica,  ice,  and  opal,  there 
appears  a  play  of  *  Newton's  colours '  which  are  due  to  Interference 
(§  399).  The  thin  transparent  film  has  two  surfaces,  each  of  which 
reflects  back  a  small  fraction  of  the  incident  light.  That  reflected 
from  the  back  surface  has  had  farther  to  go  than  that  which  came 
to  the  eye  at  once  from  the  front  surface.  Suppose  it  happens  to 
be  just  half  the  wave-length  of  some  particular  spectrum  colour 


452 


LIGHT 


[§664 


behind  ;  interference  smoothes  out  its  waves  and  so  destroys  that 
colour,  and  the  light  that  reaches  the  eye  is  of  the  complementary 
tint.  Examined  by  the  spectroscope,  there  is  a  black  gap  in  its 
spectrum. 

As  the  film  thickens  the  wave-length  destroyed  must  increase. 
A  very  thin  coloured  film  removing  the  violet  appears  straw- 
yellow  ;  a  thicker  appears  orange  as  the  blue  goes  ;  then  purple 
as  destruction  reaches  the  green,  while  violet  has  reappeared 
in  the  spectrum  ;  then  blue  as  the  long  waves  of  red  interfere. 
[With  workshop  experience  you  recognize  the  tempering  tints  of 
steel,] 

Thickening  still,  more  than  one  colour  can  be  removed  at  once  by 
the  odd  half-wave-length  lag,  e.g.  2J  waves  of  red  =  in  length  3|^ 
green  =  4 J  blue.  The  complements  amount  to  pale  tints  of  pink 
and  green,  fading  away  altogether  in  thick  films  to  a  white,  which, 
however,  yields  a  spectrum  showing  many  equidistant  dark  gaps. 
The  monochromatic  light  of  a  soda  flame  continues  to  show  yellow 
and  black  bands  even  in  thick  films,  there  being  no  other  colours 
to  overlap  them. 

The  presence  of  coloured  streaks  in  a  film  evidently  means  that 
it  is  wedge-shaped,  the  brightest  tints  near  the  thin  end. 

The  tints  change  when  looked  at  obliquely  in  the  same  way  as 
by  thickening  the  film,  for  rays  penetrating  across  the  film  and 
back  have  farther  to  go  when  oblique. 

These  tints  are  exceedingly  useful  in  testing  the  truth  of  lens 
surfaces  during  manufacture,  see  §  544.  The  little  lenses  of  your 
'  sixth  '  shone  out  as  beads  of  blue  when  the  maker  capped  them 
with  the  concave  test-mould  :  if  any  part  looked  green  or  violet, 
that  was  a  bad  fit  by  millionths,  and  the  polishing  had  to  be  con- 
tinued. 


§  565.  Diffraction  colours.  In  §  554  it  was  pointed  out  how  a 
Grating  with  several  thousand  strise  or  dots  to  the  inch,  will  break 
up  white  light  and  throw  off  colours.  This 
diffraction-grating  structure  exists  in  mother- 
of-pearl,  labradorite,  etc.,  and  accounts  for  the 
lustre  of  the  feathers  of  the  drake's  head  and 
the  peacock's  tail.  Examine  these  in  oil  under 
the  highest  power  of  your  microscope,  and  see 
the  fine  wrinklings  of  their  upper  surface,  like 
a  streaky  bark. 

A  swarm  of  minute  particles,  all  the  same 
size,  scattered  in  the  path  of  light,  will  break 
it  up  in  a  similar  way,  the  angle  at  which  a 
particular  colour  is  thrown  off  now  depending 
on  the  diameters  of  the  particles.  In  this 
way  thin  cloud  produces  coloured  Coronse 
round  the  moon,  particles  not  quite  in  the  line  of  sight  diffract  off 
waves  which  reach  the  eye,  the  farther  out  of  line  the  longer  the 


' 

■ 

■    ■ 

A 

A 

Vv^ 

v-i'^\ 

'\  w 

"^^V^ 

k 

:  ; 

Vvii 

•     / 

\\\ 

^ 

r 

1 

Fig.  224. 


§566] 


COLOUR 


453 


waves,  Fig.  224,  i.e.  the  red  corona  is  outside  the  blue.  Similar 
rings  round  all  bright  lights  indicate  a  misty  deposit  in  the  humours  of 
the  eye,  and  need  for  a  holiday.  With  particles  of  various  sizes 
the  colours  blend  into  a  colourless  haze. 

A  Blood-film  on  a  slide  produces  a  corona  round  a  distant  brilliant 
point  of  light,  of  diameter  inversely  proportional  to  that  of  the 
constituent  corpuscles,  and  it  is  much  easier,  in  diagnosing  pernicious 
anaemia,  to  get  their  average  size  by  this  device,  than  by  painfully 
micrometering  a  number  of  them  under  the  microscope. 

§  566.  Rainbows  are  caused  by  refraction  and  internal  reflection 
in  myriads  of  spherical  water  drops  on  which   the  sun  (or  the 


Fig.  227. 


Fig.  226. 


Fio.  225. 


moon)  is  shining.  The  bright  primary  bow  is  returned  after  one 
reflection  {not  total,  §  491)  inside  the  drop.  In  Fig.  225  the  paths  of 
several  equi-spaced  parallel  '  rays  *  of  sunlight  meeting  the  upper 
half  of  the  drop  have  been  exactly  traced.  It  will  be  seen  that 
they  emerge  in  very  scattered  directions,  except  three  which  are 
practically  parallel,  i.e.  the  drop  throws  back  a  much  more  con- 
centrated reflection  in  this  direction  than  in  any  other.  This  is  a 
direction  of  minimum  deviation,  the  obtuse  angle  turned  back 
(between  the  dotted  lines)  being  here  less  than  for  any  of  the  other 
rays.     Hence  raindrops  lower  down  in  the  sky  will  each  reflect 


454  LIGHT  [§666 

a  little  light  to  the  eye  along  paths  such  as  PQ,  but  drops  near 
a  certain  greatest  height  reflect  a  lot  and  appear  very 
bright. 

As  the  light  has  suffered  two  refractions,  the  minimum  devia- 
tions for  red  and  violet  are,  of  course,  different  (180°— 42-1,  180 
—40-2),  i.e.  the  brightest  red  and  violet  come  from  drops  nearly 
2°  apart,  and  a  spectrum  is  drawn  out  in  the  sky. 

Referring  to  Eig.  226,  the  line  from  the  observer's  head  to  its 
shadow  is  180°  away  from  the  sun,  and  therefore  all  reflecting 
drops  lying  at  42-1°  off  this  line  appear  red,  and  all  at  40-2°  blue ; 
i.e.  the  rainbow  forms  part  of  a  circle  with  its  centre  in  the 
direction  of  the  shadow  of  one's  head,  and  with  outer  angular 
radius  42-1°  red,  and  inner  40-2°  blue.  Inside  the  bow  is  a  light 
haze,  outside  a  dark  space.  At  the  top,  inside,  are  '  super- 
numerary bows  '  caused  by  diffraction,  since  the  bow  suddenly 
limits  the  broad  reflected  waves  (cf.  §401).  Rainbows  formed  by 
drops  1  mm.  diameter  or  more  show  the  full  spectrum  from  red  to 
violet,  and  indications  of  a  couple  of  supernumeraries  perhaps. 
With  1/4  mm.  drops  the  red  arch  becomes  only  orange,  but  there 
are  several  supernumeraries  of  varied  colours.  Fog-bows  are  mainly 
white.  Each  observer  sees  his  own  bow,  built  up  from  all  the  drops 
lying  in  a  cone  from  his  eye  to  the  distant  margin  of  the  rain.  If  you 
shake  a  half-wrung  rag  in  the  sunshine,  so  as  to  make  a  dust  of 
spray,  you  may  see  two  bows,  one  for  each  eye.  The  lower  the  sun 
the  more  bow  can  be  seen  in  the  sky ;  the  rest  of  the  circle  has  a 
background  of  earth,  and  to  get  enough  drops  to  show  it  one 
must  stand  in  the  midst  of  drenching  spray  from  a  fall  or  a  hose  : 
Niagara  is  solemn  without  its  flashing  bows. 

Other  light  shining  on  the  lower  part  of  the  drop  is  twice 
reflected  inside,  and  emerges  to  give  the  larger  '  secondary  bow,' 
weaker  on  account  of  the  double  loss  on  reflection,  and  with  its 
colours  inverted,  red  of  angular  radius  50-8°  and  blue  54-5°,  having 
been  separated  as  shown  in  Eig.  227.  It  bounds  the  dark  space, 
and  outside  it  there  is  hazy  reflection  again. 

The  little  figures  interspersed  with  the  chief  drops  are  marked 
to  show  the  directions  over  which  diffuse  reflection  of  the  first  and 
second  varieties  takes  place.  There  is  a  gap  of  9°  in  which  no 
reflection  occurs  from  the  raindrops  :  between  the  two  bows  the 
observer  looks  into  this  gap  and  sees  only  the  dark  cloud. 

§  567.  The  Halo,  a  white  ring  of  22°  angular  radius  surrounding 
the  sun  or  moon,  is  due  to  refraction  at  minimum  deviation 
through  floating  ice-crystals.  Colours  are  sometimes  visible  in 
it,  the  red  inside  and  the  blue  outside  (contrast  coronse),  and 
one  occasionally  sees^  a  solitary  speck  of  cloud,  22°  from  the  sun, 
brightly  iridescent,  a'  mock  sun.' 

§  568.  The  colour  of  the  clear  sky.  Take  from  your  microscope 
case  the  test-slide  which  the  maker  is  often  kind  enough  to  supply, 


I 


§  568]  COLOUR  455 

the  common  diatom  Pleurosigma  angulatum,  mounted  dry,  and  hold 
it  in  the  sunlight,  or  to  a  bare-wire  lamp  :  it  is  a  sparkle  of  splendent 
dust,  greeny-blue  to  sky-blue. 

Put  it  under  your  low  power,  focus  your  substage  condenser 
on  the  bare-wire  lamp,  and  open  its  iris  wide  ;  round  the  outfield 
you  see  the  leaflet -like  '  frustules  '  shining  blue. 

Turn  on  your  sixth,  and  adjust  the  condenser  to  ordinary  light 
with  care,  and  you  discover  the  cause  of  the  colour  :  the  clear 
siliceous  frustules  are  strongly  striate  with  a  regular  structure,  a 
natural  diffraction  grating  much  finer  than  we  have  been  thinking 
of — its  interspace  is  about  the  wave-length  of  green  light. 

With  a  trifle  more  skill,  you  can  find  the  same  structure  accounting 
again  for  the  blaze  of  blue  in  the  resplendent  wings  of  the  great 
Guiana  butterfly,  Morpho  menelaus,  employed  in  so  many  pretty 
decorative  fancies  in  the  shop  windows  ;  the  scales  of  the  wings  are 
ribbed  and  very  finely  cross  puckered. 

That  is,  very  fine  diffraction  gratings  scatter  widely  a  light  pre- 
dominantly blue. 

Without  an  atmosphere  we  should  have  only  a  hard  sun  in  a 
black  sky,  as  the  moon  has,  but  the  little  moving  molecules  of  oxygen 
and  nitrogen  themselves  provide  a  structure  very  much  finer  than 
the  waves  of  light,  even  in  miles  thickness  no  brighter  than  the 
tenuous  diatom  or  butterfly -scale,  but  scattering,  in  all,  a  good 
fraction  of  the  incident  sunlight,  lighting  up  the  sky  all  over  the 
sunny  side  of  the  earth,  and  lapping  beyond  as  twilight ;  and 
this  scattered  light  is  preponderantly  blue. 

With  larger  structures,  the  excess  of  blue  disappears  ;  clouds, 
of  water  drops,  are  white.  You  see  this  again  in  cigarette  smoke ; 
as  it  curls  dry  from  the  burning  end  it  is  blue  as  wood  smoke, 
but  drawn  through  the  mouth,  and  moistened  there,  it  blows  out 
as  a  thick  white  cloud.  Under  the  ultra-microscope,  §  642,  the 
first  is  scarcely  visible,  the  fat  sparkling  particles  of  the  latter  are 
a  dazzling  swarm. 

Or  if  you  mix  equal  volumes  of  |%  HCl  and  J%  hypo,  solution, 
NagSgOg,  and  wait  a  little,  you  will  presently  see  a  cloud  of  colloidal 
sulphur  forming,  blue  at  first,  turning  white  later  as  it  thickens  : 
the  growth  of  the  particles  that  cause  this  can  be  watched  at  the 
same  time  in  a  drop  under  the  ultramicroscope. 

It  is  not  difficult  to  calculate  that  The  Amount  of  Scattering  of 
any  particular  radiation,  by  very  small  obstacles,  is  inversely  propor- 
tional to  the  fourth  power  of  its  ivave-length.  Red  waves  averaging 
0-7  and  blue  047  micron,  blue  is  (70/47)*  =  (3/2)*  --  5  times  as 
freely  scattered  as  red,  and  as  the  eye  is  about  equally  sensitive 
to  these  wave-lengths,  blue  preponderates.  The  shorter  waves  of 
blue  are  turned  aside  more  easily  by  the  little  obstacles  just  for  the 
same  reason  as  are  toddlers'  tiny  feet,  while  yours  brush  by. 

The  main  bulk  of  the  light,  and  the  greater  sensitivity  of  the  eye, 
lie  in  between  these  wave-lengths,  so  that  daylight  is  only  tinted, 
never  saturated,  blue. 


456  LIGHT  [§  568 

Skylight  is  polarized,  §  652,  which  again  is  a  proof  of  reflection  or 
scattering. 

But  if  so  much  of  the  Blue  in  white  sunlight  is  flung  aside,  that 
which  gets  straight  along  must  contain  an  overplus  of  Red.  If  you 
look  at  a  lamp  through  that  big  flask  of  blue  sulphur  cloud,  you  see 
it  orange-red — as  the  sun  in  a  fog. 

When  the  setting  sun  shines  through  100,  or  even  200,  miles  of 
clear  blue -scattering  air,  it  shines  red,  and  the  clouds  and  the 
mountain- tops  light  up  with  a  rosy  hue,  composed  of  the  sun's 
residual  redness  and  the  scattered  blue  of  the  air  between  them  and 
you. 

How  much  light  can  be  scattered  by  '  clear  air  '  is  hard  to  realize. 
The  Cumberland  mountains  are  usually  invisible  50  miles  across 
the  water  ;  lost  in  the  light  horizon  blue — not  all  molecular  air 
at  that  height,  but  paled  by  the  presence  of  nuclei  and  sub-micro 
droplets  from  the  sea.  The  Mountains  of  Moab  seen  from  Olivet 
lie  in  a  hot  blue  haze.  The  sandy-pink  tints  of  the  Grand  Canyon 
of  the  Colorado  deepen  to  a  rich  coloration  as  the  late  afternoon 
shadows  lengthen  over  the  distant  walls,  the  one  sending  you  the 
indigo  scattered  out  of  ten  miles  of  sunlit  air,  the  other  glowing  to 
crimson  under  the  sun's  reddening  rays. 

I  have  gone  far  afield  for  my  instances  :  London  air  is  so  choked 
with  great  lumpy  rubbish,  that  our  atmospheric  blues  and  reds  are 
hard  to  recognize  ;  yet  have  I  seen  summer  dawns  straight  across  all 
London  as  pure  as  sunsets  in  the  Western  Sea ;  and  once,  from  a 
southern  height,  the  whole  outspread  as  one  dark  vale  of  blue. 

The  very  last  end  of  sunset  red  is  sometimes  seen  in  an  eclipse 
of  the  moon,  which  looks  coppery-red  instead  of  black.  Sunlight 
has  crept  round  through  our  encircling  atmosphere,  which  must 
have  been  generally  free  from  cloud,  losing  its  blue  by  the  way, 
and  the  residue  glowers  upon  the  dark  moon  ;  from  it,  earth  would 
be  a  black  ball  ringed  with  red. 

§  569.  Of  all  coloured  glasses,  red  glass  has  the  purest  colour — 
a  narrow  band  of  unmixed  red.  But  copper  silicate  is  pale  green, 
and  not  until  it  is  heated  in  a  reducing  flame  does  this  red  develop, 
so  intense  that  it  is  only  '  flashed '  in  a  thin  layer  over  colourless 
glass.  Kept  hot  longer,  the  red  glass  becomes  '  avanturine  '  glass, 
in  which  the  particles  of  glistening  metallic  copper  can  be  seen  with 
any  pocket-lens.  Selenium,  too,  gives  a  red  glass,  and  the  ruby 
of  old  church -windows  is  often  coloured  by  gold. 

What  have  these  three  in  common  ?  This  :  that  the  ultramicro- 
scope,  §  642,  has  shown,  in  their  clear  glass,  multitudinous  minute 
particles  ;  their  uncompromising  colour  is  '  sunset  red.' 

Amber  glass  may  likewise  owe  its  colour,  almost  that  of  an 
unresolved  diatom,  §  635,  to  particles  of  silicon  reduced  by  the 
carbon  dust  stirred  into  the  melt. 

And  many  crystals  are  '  dichroic,'  i.e.  differ  in  colour  according 
to  the  direction  they  are  looked  through,  or  the  direction  of  vibration 


§  570]  COLOUR  467 

of  the  light,  cf .  §  654  ;   evidently  it  is  a  question  of  how  the  minute 
enclosures  are  '  packed  '  in  different  crystal  planes. 

You  are  almost  treading  on  the  tail  of  the  question  why  anything 
is  coloured  at  all ;  we  can't  pursue  it,  but  the  next  time  you  gaze 
into  a  pair  of  blue  eyes,  reflect  that  but  for  the  black  backing  that 
permits  the  iris  to  be  seen  so  heavenly  blue  by  the  light  its  fine 
structure  scatters,  they  would  undoubtedly  be  pink. 

§  570.  And  what  of  the  blue  of  the  Sea  ?  I  remember  quashing 
this  question  in  a  projected  examination  paper  :  said  I,  '  What 
can  they  tell  of  sapphire,  who  only  Southend  know  ?  '  and  my 
learned  brother  yielded.  I  was  fed  up  with  the  stock  reply,  '  It  is 
the  reflection  of  the  blue  sky.'  Child  of  our  island  race  :  is  your 
sea  blue  but  by  false  surface  sheen  ? 

One  must  admit  that  the  blue  that  spreads  away  to  the  distance, 
even  on  London  River,  can  be  little  or  nothing  but  this  reflected 
blue  of  the  vault  of  heaven,  for  our  inshore  waters  are  turbid,  and, 
at  best,  pale  green.  Perhaps  that  blue  is  what  most  have  to  be 
content  with,  the  blue  on  the  sea,  and  flecked  on  the  green  waters 
of  the  Channel  the  colour  combination  is  very  beautiful ;  but  it  is 
not  the  colour  called  forth  in  the  ocean  by  the  high  plunging  sun, 
the  true  blue  of  '  the  deep  blue  sea.' 

If  you  would  gain  acquaintance  with  this,  in  English  waters, 
go  down  to  Mevagissey,  where  the  thin  waste  of  the  St.  Austell 
china-clay  washings  runs  out  to  sea,  and  look  down  from  the 
headland,  and  your  eyes  shall  be  opened,  e'en  though  the  sky  be 
gray. 

Yet,  I  grant  you,  not  gray  as  the  belly  of  the  she-ass,  as  the  Spaniard 
puts  it,  for  the  sea  cannot  make  blue  light  of  itself,  and  if  the  blue 
be  not  in  the  day's  light,  then  may  it  not  show  in  the  sea.  All 
blue  comes  to  us  from  sun  or  sky,  and  intervening  cloud  undoubtedly 
obstructs  the  shortest  waves  most,  so  that  presently  it  becomes 
hopeless  to  look  for  '  violet  and  indigo.'  How  many  of  you  wear 
dark  blue  for  choice  under  our  dull  Winter  skies  ?  Who  troubles 
about  the  added  blue  of  blue-black  ink  except  as  an  antidote  to 
rustiness  ?  But  your  Summer  blazer  is  blue  (and  bluer  under 
diffuse  skylight  than  in  direct  sun),  for  then  it  has  a  colour,  little 
sensitive  though  our  eyes  are  far  along  the  spectrum.  Fig.  223, 
beneath. 

Here  on  this  high  fo'c'sle  in  mid- Atlantic,  on  an  unbelievably 
glassy  sea  of  glowing  ultramarine,  over  which  the  flying-fish  are 
putting  up  long-distance  records,  and  beside  which  the  dye  of  my 
one  garment  is  but  dull  woad,  let  me  write  down  what  I  have  made 
out  with  mine  own  wandering  eyes. 

At  the  Admiralty  they  say.  It  is  a  question  of  saltness  :  I  have 
swallowed  the  water  of  the  Swedish  sounds ;  I  shall  presently  slip 
into  the  pale  bluey-green  coolness  of  our  tank  aft,  half-a-dozen  times 
as  salt,  looking  so  incredibly  unlike  its  source  outboard  ;  I  have 
swum  as  best  one  may  in  the  dense  brine  of  a  sunken  sea,  six  times 


458  LIGHT  [§  670 

Salter  yet,  on  which  My  Lords  maintain  no  fleet ;  and  I  submit  that 
salt  has  nothing  directly  to  do  with  the  case. 

Transparent  sea- water,  of  itself,  is  a  pretty  bluey -green.  You 
see  something  of  it  in  your  bath,  you  see  it  on  your  own  limbs  as  you 
swim,  you  see  it  best  in  the  curling  heads  that  the  billows  shake 
at  the  colourless  neutrality  of  an  overcast  sky. 

This  blue  in  the  Cornish  sea,  partly  it  is  the  white  reflecting  mass 
of  the  larger  grains  of  waste  shining  back  through  the  blue-green 
water  itself — one  sees  that  in  Lake  Louise,  saltless  melted  ice  of 
a  muddy  mountain  glacier  in  the  Rockies — mainly  it  is  the  dif- 
fractive  scattering  action  over  again,  that  we  have  just  described 
in  the  sky,  with  the  very  finest  particles  of  clay,  that  subside  only 
day  by  day,  now  acting  as  scatterers.  Dip  out  a  drop,  and  contrive 
yourself  an  ultra -microscope,  §  642,  and  hunt  them  down,  minutest 
jiggling  specks. 

An  iron-stained  rivulet  trickles  into  King's  Bay,  Spitzbergen, 
contaminating  the  water  to  cafe-au-lait ;  round  this  blur  is  a  semi- 
circle of  dirty  white,  then  of  milky  green,  and  then  a  broad  belt 
of  brilliant  blue  fades  into  the  general  tint  of  the  outer  sounds. 

That  rio  Colorado,  the  Ebro,  brings  down  a  similar  tribute  from  its 
red  gorge  to  the  Mediterranean  ;  and  Rhone,  yellow  Tiber,  Litany 
from  Lebanon,  Nile  notorious  for  its  mud,  many  a  sudden  stony 
mountain  torrent,  all  originate  in  the  soft  mountain  limestone, 
all  pour  into  that  deep  land-locked  evaporating  basin  an  almost 
never  settling  '  sediment,'  and  its  name  for  blueness  is  a  magnet 
to  all  Europe — until  the  shelving  yellow  sands  show  through,  and 
then  it  is  the  familiar  green.  The  Danube  flows  into  its  settling 
pond,  the  Black  Sea ;  thence  the  swift  current  of  the  Bosphorus 
runs  clear,  and  a  less  caerulean  wave  beats  upon  the  coasts  of  the 
Turk. 

Amazon  and  Orinoco  discharge  athwart  the  westerly  equatorial 
current  flowing  into  the  Caribbean,  and  give  that  whole  sea  its 
marvellous  colour  ;  then,  sweeping  round  the  Gulf  of  Mexico,  the 
current  receives  the  mighty  Mississippi,  father  of  muddy  waters ; 
his  silt  precipitates  to  form  the  long  easterly -growing  delta,  the 
fine  residue  imparts  that  brilliant  blueness  to  the  Gulf  Stream  which 
marks  it  off  so  sharply,  half-way  across  the  North  Atlantic. 

Here  aboard  ship,  to  see  the  colour  best,  look  over  the  side  with 
the  sun  behind  you,  into  the  depths  where  radiant  streaky  shadows 
round  your  head  declare  that  the  pellucidity  is  not  perfect ;  and 
no  Neptunian  grandmother's  washing-day  could  show  a  blue-tub 
of  a  richer  hue  than  this,  flecked  with  the  yellow  gulf -weed,  flashing, 
most  exceptionally,  the  drawn-out  reflections  of  white  sunlit 
cumulus  cloud. 

Seldom  is  the  sea  so  smooth,  most  often  the  steep  near  faces  of 
the  wavelets  reflect  to  your  eye  the  darker  vault  above,  while  the 
sloping  backs  reflect  the  lower  lighter  paler  regions — make  a  diagram 
for  yourself — until  the  setting  sun  flings  over  all,  deep  water  and 
squalid  mudflat  alike,  the  overwhelming  surface  sheen  of  blue-bereft 


§  572]  COLOUR  459 

gold — now  we  are  back  to  the  colour  on  the  sea — yet,  by  the  ship's 
side,  the  water  darkens  into  blackness  still  by  way  of  blue. 

If  you  perceive  the  water  gone  a  deep  rich  green,  or  a  riper  brown, 
then  you  shall  see  small  craft,  for  here  is  organic  growth  and  food 
for  fishes.  Then  get  out  dipper  and  length  of  line,  and  centrifuge, 
and  moderate  microscope,  and  join  with  me  in  hunting  down 
golden-brown  diatoms  and  dinoflagellates,  source  of  the  vitamin- Z> 
of  the  fish  liver,  and  wild  variety  of  vegetable  life  where  the  land- 
streams  come  in  ;  and  in  the  brilliant  green  of  the  fjords  I  leave  you 
to  these  researches. 

§571.  The  Green  Ray,  or  Green  Flash.    Now  are  we  come  to 

the  Sun's  hour  of  rest,  and  §  488  and  Fig.  183  tell  us  that,  as  he 
tries  t6  set,  his  light  is  refracted  by  the  air,  by  his  full  diameter,  just 
over  half  a  degree.  From  the  table  in  §  589  there  is  a  corresponding 
Dispersion  of  Colour  1/101  of  this,  so  that  when  only  the  very  last 
bit  of  his  rim  remains  above  the  horizon,  like  a  spectroscope  sHt, 
it  should  spread  into  a  vertical  spectrum,  1/200°  long,  and  the  green 
and  blue  should  set  last,  being  by  that  much  more  bent  than  the  red. 

Only  once  have  I  seen  the  faintest  indication  of  this  in  the  disturbed 
air  of  this  country,  where  the  setting  sun,  watched  through  binoculars, 
is  apt  to  go  through  plum-pudding  and  wagon-top  shapes,  and 
ultimately  break  up  into  flakes  ;  but  watching  brilliantly  clear  and 
steady  sunsets  at  sea  will  reward  you  presently  and  again. 

Most  usually  you  must  be  content  with  green,  for  the  setting  sun 
has  nearly  always  lost  all  his  short-wave  blue  by  scattering,  §  568, 
and  has  no  smaller  change  left  than  the  green  the  spectroscope 
discloses  in  any  glowing  fire.  But  with  the  luck  of  exceptionally 
clear  sunsets,  you  may  come,  with  me,  to  expect  blue. 

For  instance,  11/8/33  the  line  of  intense  light  showed  yellowish- 
green  at  the  thin  ends,  turned  brilliant  green  for  two  seconds,  and 
was  gone  ;  running  up  the  fo'cs'le  steps,  the  whole  was  repeated 
ten  seconds  later.  And  7/9/33,  under  a  low  bar  of  cloud,  with  a 
scarcely  tinted  sun,  the  flash  went  pure  brightest  blue,  and  lasted 
full  five  seconds,  by  some  freak  of  atmosphere,  altogether  a  record. 

Piazzi-Smyth  declared  that  he  ran  up  the  Great  Pyramid  and 
kept  the  green  ray  in  view  for  half  a  minute  as  the  sun  sank ;  he 
got  well  laughed  at,  yet  a  brief  mental  calculation  bears  out  his 
tale — the  shadow  moves  very  slowly  up  at  first. 

But  in  desert  air  he  never  saw  it  true  blue — blue  of  Adams  ware  and 
Andalusian  azulejos,  blue  of  the  borage  and  the  Corsican  pimpernel, 
magic  miracle  of  colour  in  the  midst  of  the  golden  glory  of  the  west, 
ethereal  distillate  of  the  leagues  of  sleeping  sea. 

§572.  The  light  of  the  midnight  sky.  Knowing  the  country, 
well  away  from  towns,  you  know  that  the  background  of  the  stars 
does  not  appear  utterly  black.  Of  course,  a  telescope  will  show  you 
little  stars  in  it,  but  really  few  and  very  faint,  except  only  in  the 
Milky  Way. 


460  LIGHT  [§  572 

Lord  Rayleigh,  exposing  a  wide -aperture  spectrograph  for 
200  hr.  in  all,  found  continuous  spectrum  present  in  the  darkness, 
the  H  and  K  calcium  lines,  and  the  green  auroral  line  at  0-56  micron. 

The  continuous  spectrum  showed  signs  of  polarization,  §  652  ;  it 
is  daylight  leaking  round  the  world  inside  the  atmosphere. 

The  violet  lines  are  probably  scattered  starlight,  and  evidently 
the  Aurora,  §  889,  is  not  confined  to  the  ends  of  the  earth,  as 
we  imagined,  but  sheds  a  faint  steady  electric  light  on  all  the 
countryside. 

§  573.  As  the  eye  sees  colour.  It  has  been  tacitly  assumed 
throughout  that  the  eye  blends  all  the  spectrum  colours  into  a  tint. 
If  all  are  present  in  normal  proportion,  it  is  unconscious  of  tint — 
white  light ;  when  some  are  weak  or  absent,  the  tint  perceived  is 
complementary  to  that  which  the  abstracted  parts  would  add  up  to. 
Unlike  the  prism,  the  eye  has  little  power  of  analysing  colours,  but 
can  blend  them  most  exquisitely. 

Here  is  a  simple  experiment  for  you,  roughly  made  with  crude 
colours,  but  none  the  worse  for  that : 

At  the  glazier's  they  will  find  you  up  odd  bits,  big  enough  for  eye- 
glasses, of  red  and  green  glass  (full  coloured,  not  pale  tints),  and 
possibly  of  '  signal-green,'  which  is  a  much  bluer  colour  by  daylight. 
If  they  haven't  this,  get  one  of  those  celluloid  anti-dazzle  fingers 
fitted  on  windscreens,  and  made,  with  100%  stupidity,  of  the  best 
possible  colour  for  blinding  out  a  red  lamp. 

With  red  glass  covering  one  eye,  and  signal-green  covering  the 
other,  look  around  you.  At  first  everything  will  keep  going  red 
and  green,  as  one  or  the  other  eye  takes  charge,  but  this  soon 
diminishes,  and  you  find  you  have  a  very  fair  appreciation  of  the 
natural  colours  of  objects,  leaves,  flowers,  sky,  fire,  fabrics,  etc. 
Reds  and  greens  stand  out  in  a  disjointed  fashion,  owing  to  the 
chromatic  aberration  of  the  eye,  §  605,  but  the  colours  are  all  there 
and  recognizable,  and  you  perceive  that  the  light  you  are  using  is 
a  dull  nondescript  or  neutral  tint,  perhaps  as  good  as  London 
winter  daylight.  Shut  your  eyes  alternately,  and  see  what  it  is 
made  up  of  :  two  practically  complementary  colours  are  adding 
together  to  make  a  serviceable  white  light ;  the  spectroscope  can 
show  you  how. 

Now  put  both  together  to  one  eye. 

Well,  if  you  blot  out  one  half  of  the  spectrum,  and  then  blot  out 
the  other,  what  do  you  expect  ? 

Try  now  red  and  common  green  glass,  and  you  again  have  a  fair 
appreciation  of  colours,  except  blue,  but  all  white  surfaces  are  some 
shade  of  yellow ;  the  duller  lit  are  greeny,  the  brighter,  pinky 
(for  the  eye  changes  its  sensitivity  to  greens  a  little  wdth  change  of 
brightness).  Yellow  is  not  a  '  primary  '  colour,  you  saw  hardly 
a  trace  of  it  through  the  spectroscope,  and  here  you  have  manu- 
factured it  in  any  quantity  by  adding  red  and  green  ;  but  see  §  576. 

Shutting  either  eye,  neither  glass  permits  you  to  see  blue,  as 


§  574]  COLOUR 


461 


would  the  signal-green ;  so,  lacking  that  to  complete  the  mixture, 
you  make  up  only  its  complementary,  yellow. 

Again  hold  both  to  one  eye  :  this  green  cannot  stop  all  the  orange 
red. 

Try  also,  if  you  like,  with  blue  and  two  thicknesses  of  amber  glass. 

You  see  the  essential  difference  between  subtracting  colours 
from  white,  as  all  absorbing  substances  do,  and  then  adding  the 
residues  ;  and  subtracting  colours  and  then  trying  to  subtract  the 
residues  :  make  sure  which  you  mean  when  you  talk  about '  mixing  ' 
colours,  or  you  mean  nothing. 

Gelatine  films  dyed  with  various  aniline  dyes  offer  much  more 
choice  of  colours  than  do  glasses  ;  only  one  must  make  sure  that 
the  dyes  are  non-fading. 

A  particular  crimson  and  blue-green  pair  divide  the  brighter 
part  of  the  spectrum  between  them  better  than  do  the  railway  glasses, 
and  they  not  only — one  or  the  other  of  them — transmit  all  its 
colours,  but  they  also  blend  in  various  proportions  into  mixtures 
which  the  eye  accepts  as  a  wide  variety  of  colours.  The  Kinemacolor 
camera  took  alternate  pictures  through  these  two  colour  screens, 
and  then  they  were  projected  on  the  screen  also  alternately  in  the 
two  colours  :  rapid  movements,  either  in  the  picture  or  of  your 
eye,  gave  the  process  away,  but  the  gorgeous  colours  of  the  Delhi 
Durbar  came  out  quite  satisfactorily,  except  dark  blues. 

The  same  two  colours  are  now  being  used  for  simultaneously 
projected  pictures,  which  will  do  away  with  the  sudden  zebra 
effects  :  the  two  half-width  pictures  are  side  by  side  on  the  film, 
and  the  optical  system  projects  them,  each  in  its  own  colour,  in 
coincidence  on  the  screen. 

But,  as  Thomas  Young  discovered  in  1807,  much  better  effects 
are  obtained  by  the  summation  of  Three  Colours  in  varying  pro- 
portions. On  this,  being  a  Bart's  man,  he  based  a  Tri-chromatic 
Theory  of  Colour  Vision,  assuming  that  three-component  colour- 
sensations  were  stimulated  in  some  triple  mechanism  in  the  eye. 
No  such  triplicity  in  the  eye  has  ever  been  made  out,  and,  in  any 
case,  this  book  has  nothing  to  do  with  physiological  sensations  or 
psychological  impressions  ;  but  the  most  beautiful  Colour  Repro- 
duction Processes  of  the  present  day  stand  on  this  Three-Colour 
basis,  and  two  different  ones  may  be  briefly  described. 

§574.  A  three-colour-addition  projection  process.  The  three 
colours  are  centred  on  wave-lengths  0-65  [i,  0-53  jx  and  0-46  y. ;  the 
first  is  a  full  red,  that  of  the  hydrogen  red  line  ;  the  second  is  a 
brilliant  green  without  trace  of  yellow  or  blue  ;  and  the  third  is 
a  deep  blue  passing  into  violet,  probably  Newton's  '  indigo.' 

Fig.  228  shows  diagrammatically  the  essentials  of  the  Kodacolor 
movie  camera  and  projector.  Over  the  //I -9  lens  is  fitted  a  screen 
(left)  divided  into  three  horizontal  strips  of  these  colours,  of  unequal 
widths  found  to  be  correct.  Light  from  the  view  outside,  coming 
through  this  screen,  is  focussed  by  the  lens  on  the  film  (right)  which 


462 


LIGHT 


[§574 


has  its  panchromatic  emulsion  on  the  back.  The  front  of  the  film 
is  moulded  into  minute  convex  ribs,  559  to  the  inch  ;  these  act  as 
'  corneal  surfaces,'  §  602,  and  focus  linear  images  of  the  luminous 
tricoloured  window  on  the  sensitive  '  retina  '  (which  is  really  in  the 
position  of  the  eye-ring  in  a  telescope),  so  that  the  light  which  has 
just  passed  through  a  point-image  of  the  coloured  point  of  the  object 
is  now  spread  over  a  little  tricolour  strip,  across  the  film.  If  it  is 
white  light  W,  all  three  colours  will  be  equally  bright ;  if  it  is  red  R, 
none  will  have  got  through  the  green  and  blue  sections  of  the  screen, 
and  only  under  the  red  strip  can  the  emulsion  be  affected  ;  if  purple, 


Fig.  228. 

P,  there  will  be  action  under  both  red  and  blue  strips,  two  micro- 
strips  of  darkening  on  the  film  ;  if  yellow-green,  YG,  i.e.  bright 
green  and  a  little  red,  there  will  be  a  strip  of  action  under  the  green, 
a  little  under  the  red,  and  none  under  the  blue. 

The  film  is  developed  and  '  reversed  '  into  a  positive,  and  now  is 
a  succession  of  microscopic  triplets  of  more  or  less  blackened  strips, 
each  under  its  own  '  corneal  lens '  :  run  through  an  identical 
projector,  each  transmits  exactly  the  right  amount  of  light  to  its 
own  section  of  the  colour  filter,  for  projection  in  that  particular 
direction,  and  the  lens  integrates  each  lot  into  a  coloured  point  on 
the  screen. 

Thus  the  picture  is  made  by  adding  together  the  right  amounts 
of  the  three  original  colours,  in  the  right  places. 

§575.  Three-colour  printing.  The  other  process  is  that  which 
produces  the  three-colour  print  familiar  to  everybody,  and,  except 
for  gay  things  like  labels,  has  superseded  the  painstaking  old 
lithography,  where  as  many  as  seventeen  various  inks  might  be 
worked  into  one  picture,  each  distributed  over  the  areas  where  the 
artist -operator  thought  he  detected  that  tint. 

Three  separate  negatives  are  taken,  through  red,  green,  and  blue 
filters  respectively,  half-tone  screens,  a  little  way  in  front  of  the 
plates,  being  commonly  used  also  to  break  up  the  pictures  into  the 
usual  little  dots,  which  prevent  the  ink  from  forming  great  blurs 
in  printing.     From  each,  a  positive  zinc  or  copper  block  is  prepared, 


§  575]  COLOUR  463 

by  printing  on  a  sensitive  bichromated  gelatine  coating,  washing 
off  the  still-soluble  unacted-on  gelatine,  and  etching  away  that 
unprotected  metal  by  acid,  as  in  black-and-white  work ;  and 
now  there  comes  in  a  sharp  difference  both  from  that  and  from  the 
projection  process  described  above  :  the  high  lights  of  a  black-and- 
white  print  are  places  where  least  ink  is  put  on  the  paper,  and  the 
rest  is  more  or  less  blacked  in  ;  but  now  we  want  a  rose  to  have 
more  red  on  it  than  the  rest  of  the  picture. 

But  on  the  picture  taken  through  the  red  screen  the  red  rose  is 
the  highest  light  present,  therefore  that  block  has  a  blank  white 
flower  :  if  it  were  printed  in  red  ink,  everything  would  be  more  or 
less  red  except  the  rose. 

To  the  blue  and  green  lights,  however,  the  rose  was  a  poor  dull 
thing,  having  little  of  either  ;  accordingly,  their  blocks  will  print 
it  deeply  :  plainly  they  must  not  be  inked  with  blue  or  green. 

Three  new  colours  have  to  be  used  as  printing-inks,  minus-red, 
minus-green,  and  minus-blue,  complementary  to  the  screen-colours, 
each  of  them  combining  what  is  left  of  white  light  after  red,  green, 
or  blue  has  been  filtered  out :  you  see  them  with  a  pocket-lens, 
in  light  parts  of  the  three-colour  print,  as  a  pale  greeny-blue,  a 
bluish-pink  or  crimson,  and  (by  daylight)  a  pale  yellow.  Many 
things  in  the  process,  including  the  precise  shades  of  these  un- 
promising-looking colours  to  be  employed,  depend  on  the  judgment 
of  the  printer,  for  three-colour  printing  is  still  more  of  an  art  than 
a  science.  Inked  with  these,  the  blocks  are  printed  on  to  a  rubber 
sheet,  on  which  the  paper  is  then  pressed,  this  '  offset '  being  usually 
preferred  to  direct  printing. 

Thus  the  rose  is  printed  in  pale  yellow,  and,  on  top  of  that,  pink, 
and  the  white  light  on  its  way  down  to  the  reflecting  paper,  and  on 
its  way  back,  is  robbed  of  its  blue  and  of  its  green,  and  comes  out 
red ;  not  a  mixture  of  two  composite  colours,  pink  and  yellow, 
but  the  real  red  end  of  the  spectrum  of  white  light,  which  has  alone 
escaped  being  absorbed  by  this  mixture  of  inks. 

The  green  foliage  is  avoided  by  the  pink  '  minus  green,'  the  yellow 
anthers  lack  only  blue,  and  therefore  print  only  in  yellow '  minus-blue.' 

The  dark  shadows  are  minus  everything,  all  three  inks  spread 
over  them,  and  the  absorption  is  amazingly  complete,  nothing 
passes,  a  dense  black. 

Thus  the  process  is  the  reverse  of  the  former  one ;  it  is  one  of 
successive  subtraction  from  white  light,  and  the  white  reflecting 
surface  is  essential  to  it ;  it  corresponds  to  the  two  glasses  super- 
posed of  §  573. 

Super-imposition  is  apt  to  fail  in  light  parts  of  the  print,  on  account 
of  the  smallness  of  the  dots  there,  and  the  skilful  avoidance  of  false 
tints  from  this  is  again  part  of  the  art  of  the  printer. 

For  while  greeny-yellow  gamboge  and  greeny-blue  pnissian- 
blue  mix  on  white  paper  to  a  dingy  green,  which  alone  escapes 
absorption  in  either,  yellow  chrome  and  pure  blue  ultra- 
marine   mixed    make    mud,    having    no   tint    in    common ;    and 


464  LIGHT  [§575 

blue  light  and  amber  light  entering  the  church  through  panes  side 
by  side,  spread  and  add  and  blend  into  a  sufficiently  white  light  on 
your  book. 

§  576.  The  Paint  Box.  Evidently,  since  ordinary  painting  is 
done  by  overlapping,  or  else  mixing,  which  is  a  more  intimate 
overlapping,  transparent  colours  on  white  paper — the  case  of  oil- 
colours  is  not  so  plain,  but  see  §  561 — what  has  just  been  said  of  the 
three-colour  process  applies  to  them.  They  are  '  minus-colours,' 
and  the  Artist's  Primary  Colours  should  therefore  be  greenish -blue, 
yellow,  and  crimson.  H.  E.  Ives  has  lately  been  using  Chinese- 
blue  (purest  ferric  ferrocyanide),  extra  pale  cadmium-yellow,  and 
a  '  phospho-molybdo-tungstic  acid  lake  of  Rhodamine  6  G  '  ;  and 
with  zinc -white,  and  black,  these  very  permanent  colours,  although 
they  take  a  good  deal  of  blending,  adequately  replaced  the  10  to  25 
of  the  artist's  palette. 


EXAM   QUESTIONS,   CHAPTER   XXXVII 

1.  Describe  a  method  of  producing  a  pure  spectrum.  Explain  why  you 
may  call  it  piu-e.     [Nothing  said  about  any  screen.]     (  X  2) 

2.  Draw  a  diagram  of  two  lenses  and  a  prism  producing  a  pure  spectrum 
on  a  screen.     (  X  3) 

3.  Describe  the  prism  spectrometer,  with  careful  diagram  to  show  the 
formation  of  a  spectrum. 

Give  an  account  of  its  uses.     (  X  3) 

4.  Give  a  diagram  of  a  spectrometer,  and  show  how  it  is  used  to  determine 
the  refractive  index  of  the  glass  of  a  prism,  or  of  a  liquid.  If  A  be  60°  and 
minimum  D  30°,  calculate  fi.     (  X  2) 

5.  What  is  a  spectrum  line,  and  how  is  it  caused  ?  How  would  you  arrange 
to  see  the  lines  of  the  electric  arc  ? 

6.  Detail  the  arrangements  for  viewing  a  pure  spectrum.  What  are  the 
outstanding  differences  between  the  spectra  of  a  sodium  flame,  an  incandescent 
lamp,  and  the  sun  ? 

7.  How  may  an  absorption  spectrum  be  obtained  and  mapped?  What 
information  may  be  derived  from  a  study  of  absorption  spectra  ?     (  X  2) 

8.  How  could  you  detect  spectroscopically  (a)  calcium,  (b)  carbon  dioxide, 
(c)  chlorophyll  ? 

9.  Describe  a  direct-vision  spectroscope  and  explain  its  action.  How 
would  you  use  it  to  examine  a  red  liquid  for  blood  ?     (  X  3) 

10.  The  presence  of  CO  in  the  blood  is  indicated  in  its  spectrum  by  certain 
dark  bands.  What  experiment  would  you  carry  out  to  test  for  it  ?  Draw 
a  figm-e  of  your  apparatus.     (  X  3) 

11.  How  would  you  observe  the  spectrum  of  (a)  a  hot  furnace,  (6)  a  coloured 
ink  ?  Why  do  some  colom-s  appear  different  by  candle-light,  and  which  are 
most  likely  to  be  affected  ? 

12.  The  spectrum  of  the  sun  is  crossed  by  dark  lines  of  various  intensities. 
What  are  their  causes,  and  what  knowledge  of  the  sun's  structure  have  they 
afforded  us  ? 


COLOUR  465 

13.  Explain  the  lines  of  the  solar  spectrum.  How  do  they  resemble,  and 
differ  from,  those  of  electrically  excited  gases,  and  how  are  they  modified 
during  eclipses  ?     (  X  2) 

14.  Explain  the  red  colour  of  (a)  a  coke  fire,  (6)  a  rear  lamp,  (c)  a  poppy 
petal,  (d)  a  strontium  flame,  (e)  copper,  (/)  noble  opal. 

15.  What  do  you  know  of  the  red  colour  of  (a)  a  neon  shop-sign  tube,  (6) 
very  hot  copper,  (c)  cold  copper,  (d)  beef,  (e)  leaf -green  solution,  (/)  sunset  ? 

16.  Give  some  explanation  of  the  green  of  (o)  a  firework,  (6)  a  railway 
signal,  (c)  grass,  (d)  a  shallow  sea,  (e)  red  ink  and  water,  (/)  a  drake's  neck. 

17.  Explain  the  green  of  (a)  a  mercury -vapour  lamp,  (6)  gold  leaf,  (c)  the 
spectrum  seen  in  an  ultra-violet  spectroscope,  (d)  some  red  crystals. 

18.  How  do  you  account  for  the  blue  colour  seen  in  (a)  a  bunsen  flame, 
(6)  copper  solutions,  (c)  a  cornflower,  (d)  oil  spilt  on  water,  (e)  wood-smoke, 
and  (/)  dilute  quinine  bisulphate  solution  ? 

19.  Explain  how  the  blue  coloiu"  is  produced  in  (a)  the  sky,  (6)  the  sea,  (c) 
blue  glass,  (d)  blue  paint  or  cloth,  (e)  the  blue  of  tempered  steel. 

20.  Explain  the  colour  of  (a)  red  paint,  (6)  blue  glass,  (c)  a  potassium  flame, 
(d)  clear  sky,  (e)  white  froth  on  a  coloured  liquid.     (  x  2) 

21.  Discuss  the  principal  causes  of  the  colours  of  natural  objects.     (  X  2) 

22.  Discuss  two  of  the  following  observations  : 

(a)  Yellow  can  be  obtained  by  mixing  red  and  green  light,  but  not  by 
mixing  red  and  green  paint. 

(6)  Two  colours  which  match  in  artificial  light  do  not  necessarily  match 
in  daylight. 

(c)  The  sun  looks  red  when  seen  through  fog.     (  X  2) 

23.  Why  is  sunlight  said  to  be  composite  ?     What  occurs  when  it  trav< 
(a)  a  yellow  solution,  (6)  a  blue  solution,  (c)  both  in  succession  ? 

24.  Discuss  the  formation  of  Rainbows. 


PRACTICAL  QUESTIONS 

Arrange  lenses,  etc.,  to  form  a  spectroscope. 

Focus  and  adjust  a  spectrometer  and  plot  an  absorption  spectrum. 

[This  may  be  that  of  blue  glass,  for  instance.  Get  the  broad  coloured 
spectrum  of  a  white  lamp  in  about  the  minimum  deviation  position  for  the 
yellow,  focussed  to  sharp  upper  and  lower  edges,  and  the  crosswires  in  focus. 
Put  in  the  glass,  draw  what  you  see  in  pencil,  shading  it  recognizably,  name 
the  colours  and  mark  their  reading  on  the  scale  and  vernier,  to  1'.] 

Measure  the  three  angles  of  a  prism,  by  spectrometer. 

Measure  the  minimum  deviation  of  sodium  light,  and  calculate  the  refractive 
index  of  the  prism.  [This  should  be  to  three  decimal  places;  vermers  must 
be  road  acciu*ately.] 


CHAPTER  XXXVIII 


ABERRATIONS  OF  MIRRORS  AND  LENSES 


Aberation  dependent  on  Shape  of  Surfaces  of  Mirror  or  Lens 

§  581.  Spherical  Aberration.  Set  a  cup  of  tea  in  a  direct  light. 
On  the  surface  appears  the  famihar  bright  cusped  curve  of  light, 
called  a  Caustic,  reflected  from  the  semicircular  margin  of  the 
cup.  Pass  a  vertical  penholder  across  the  lamplight ;  its  pointed 
shadow  (Fig.  229,  A)  sweeps  round,  the  tip  '  rolling '  on  the 
caustic   and  in   every  position  blotting    out    a    little    bit    of    it. 

DO  THIS. 


Fig.  229. 


This  little  bit  was  evidently  the  focus  of  all  the  rays  that  fell 
on  the  now  darkened  patch  of  mirror.  The  rays  are  not  all 
reflected  to  one  hearth,  the  complete  semi-circular  mirror  has  a 
complex  succession  of  foci  instead  of  the  single  point,  though  the 
brilliance  of  the  cusp  still  tells  us  that  a  large  proportion  of  the 
light  is  condensed  thereabouts.  This  imperfection  in  focussing  of 
circular  and  spherical  surfaces  is  referred  to  as  Spherical  Aberration. 

466 


§584]  ABERRATIONS  4C7 

§  582.  Mirrors  can  be  made  that  are  free  from  this :  the  EUipse, 
Fig.  B,  has  been  referred  to  in  §  416  ;  more  important  is  C,  the 
parabola.  Fasten  one  end  of  a  thread  at  O'  on  the  edge  of  a  T 
square,  and  the  other  at  F,  keep  a  pencil  point  under  the  thread 
at  P,  and  let  it  run  along  the  edge  as  the  T  square  is  slid  up  and  down 
the  board  :  then  the  parabola  O'P  +  PF  =  constant  reflects  to 
its  geometrical  focus  F  all  light  O'P  arriving  parallel  to  its  axis. 

You  are  familiar  with  this  paraboloid  reflector  in  car  headlamps, 
in  '  spun  '  metal ;  the  middle  part,  on  a  larger  scale,  in  silvered 
glass,  is  used  in  searchlights  ;  and  a  central  patch,  not  more  than 
1 /5th/  in  diameter,  is  the  great  mirror  of  reflecting  telescopes. 

The  reflection  of  complete  waves  is  shown  in  the  lower  halves  of 
B  and  C. 

§  583.  Spherical  aberration  occurs  with  Lenses  as  well.  A 
thick  bull's-eye  held  in  a  bright  light  in  smoky  air  produces  a 
'  pulled-out '  cone,  Fig.  D,  quite  like  the  middle  part  of  the  re- 
flection caustic.  The  outer  rays  are  refracted  too  much  (the  figure 
is  accurate),  the  focal  distance  of  the  outer  '  zones  '  of  the  lens  is 
unduly  short.  Instead  of  a  sharp  cone,  and  image,  there  is  a  sort 
of  bottle-neck,  with  a  moderate  image  anywhere  within  half  an 
inch  or  so,  and  always  round  it  an  unpleasant  haze. 

A  reading-glass  (get  a  '  strong  magnifier  '  from  the  sixpenny 
stores)  forming  on  the  wall  an  image  of  a  distant  lamp  shows  this 
quite  well.  Or  looking  through  the  lens,  spherical  aberration 
accounts  for  the  distortion  and  smearing  of  the  print  all  round  the 
outside. 

§  584.  Caustic  by  refraction.  Turn  back  to  Fig.  192,  the  fish-eye 
view,  for  light  coming  from  an  object  0.  The  emergent  directions, 
produced  backwards,  all  touch  a  virtual  caustic  curve,  with  its 
cusp  where  the  object  point  would  be  clearly  seen  looking  almost 
straight  down.  The  mushroom-cap  curve,  the  geometrical  involute 
of  this  caustic,  cuts  all  the  rays  at  right  angles  ;  it  is  the  wave-front 
emerging  into  the  air ;  you  see  how  much  distorted  it  is  from  the 
dotted  sphere. 

Now,  if  0  be  3),  micro-object,  under  ON  its  cover-glass,  and  SS 
the  flat  front  of  your  l/6th  object-glass,  you  see  that  if  the  latter 
is  to  make  anything  of  the  distorted  wave,  its  successive  zones, 
counting  outwards,  have  to  be  focussed  on  points  higher  and 
higher  up.  This  is  made  possible  by  the  maker  separating  the  cor- 
recting lenses  which  succeed  the  first  magnifying -lens,  but  you 
see  that  you  are  tied  to  a  best  thickness  of  cover-glass,  l/6th  mm., 
for  a  perfect  image,  and  that  for  Uncovered  Objects,  wlien  all 
zones  must  focus  on  the  same  point  O,  your  glass  is  useless,  imless 
you  sacrifice  all  the  important  outer  zones  upon  which  its  excellence 
depends. 

Actually,  with  glass  of  y.  1-5,  the  aberration  is  even  more  marked 
than  in  Fig.  192  ;  per  contra,  in  Fig.  230,  the  air  has  been  displaced 


468 


LIGHT 


t§584 


by  oil  of  the  same  refractivity  as  glass,  so  that  right  through  from 
O  to  the  back  of  the  hemispherical  front  lens  of  your  '  homogeneous 
oil-immersion  twelfth  '  no  refraction  can  occur.    It  can  be  shown  that 

if  00  is  2/3  of  r  all  light  from  O  leaving 
the  hemisphere  appears  to  come 
accurately  from  the  aplanatic  ('  non- 
straying  ')  point  O',  where  00'  =  3/2 
of  r,  an  immense  simplification,  and 
one  cause  of  the  superiority  of 
Immersion  Lenses.  Fig.  230  shows  the 
circular  waves  spreading  from  the 
object  until  they  reach  the  back  of 
the  hemispherical  front  lens,  and  there 
becoming  perfect  circular  arcs  half  as 
far  again  apart,  travelling  half  as  fast 
again  in  air. 

§  585.  Utilization  of  spherical  aberration.  In  Fig.  229  D  light 
arriving  parallel  spreads  over  an  angle  of  50°  after  passing  through  a 
fat  lens  :  conversely,  if  you  held  up  the  lens  at  arm's  length,  light 
from  a  field  50°  wide  beyond  it  would  all  come  in  a  very  small  angle 
to  your  eye,  and  show  you  this  whole  picture,  though  with  the  outer 
parts  very  cramped  up.  You  see  this,  too,  in  a  round  flask  of  water, 
or  in  the  back  of  one  or  both  lenses  of  your  Abbe  condenser,  held 
up  to  the  light.  So  that  if  a  fat  little  bull's  eye  is  mounted  at  the 
far  end  of  a  narrow  tube,  and  a  miniature  telescope  at  the  near  end 
to  view  it  with,  a  view  '  of  sorts,'  comprising  a  considerable  angle, 
can  be  obtained  :  this  has  been  used  in  Cystoscopes.  '  Orystal 
gazing  '  might  be  described  as  the  useless  limit  in  this  direction. 

And  you  see  that  a  bright  bit  of  tin  on  the  flat  face  of  D  will  be 
struck  perpendicularly  somewhere  by  any  ray  coming  in  the  cone 
of  directions  on  the  right,  and  will  send  it  back  on  its  own  track. 
So  these  little  fat  bulls'-eyes,  backed  by  a  fiat  reflector  either  white 
or  red,  stud  Roadside  Night-signs,  and  make  the  cheapest  Rear 
Reflectors  for  push-bikes,  and  are  effective  over  a  wide  angle. 

§  586.  Means  of  reducing  spherical  aberration.  '  Stopping 
down  '  the  lens  of  §  583  to  3/4  in.  diameter  with  a  perforated  card, 
and  so  cutting  off  the  outer  rays,  removes  the  haze  and  gives  a  more 
definite  focus.  But  the  objection  to  this  way  of  reducing  spherical 
aberration  is  at  once  apparent ;  it  cuts  off  light.  It  is  all  that  can 
be  done,  however,  with  spherical  mirrors. 

Fortunately,  with  lenses,  the  fact  that  the  larger  the  angle,  by 
far  the  greater  the  aberration,  gives  another  means.  Reduce  the 
amount  of  bending  that  occurs  at  any  one  refraction  and  share 
it  equally  among  several  refracting  surfaces ;  n  may  be  needed 
instead  of  one  to  produce  the  required  total,  but  each  involves 
perhaps  only  Ijn^  as  much  aberration,  so  that  altogether  there 
is  only  about  l/wth,  e.g.  in  Fig.  G  the  same  bending  as  in  Fig.  E 


§  587]  ABERRATIONS  469 

is  shared  between  two  surfaces  about  equally ;  the  haziness  of 
the  image  is  halved.  And  see  how  the  same  idea  is  carried  farther 
in  the  Abbe  condenser,  Fig.  274,  II. 

In  complex  lens  combinations 

Focal  Power  is  mainly  a  question  of  the  extra  thickness  of  glass 
on  the  axis ; 

Chromatic  Correction  (see  further)  of  how  this  is  allotted  among 
different  sorts  of  glass  ; 

Spherical  Correction  of  '  dishing  '  the  lenses,  without  alteration 
of  strength,  from  biconvex  to  meniscus,  so  as  to  alter  the  angles 
at  which  rays  strike  them. 

§  587.  Astigmatic  beams.  The  caustic  of  Fig.  229  A  is  in  one 
plane,  a  thin  sheet  of  light  reflected  at  a  semicircle.  Rotate  the 
whole  diagram,  through  a  very  small  angle,  about  its  axis  MF,  and 
you  get  Fig.  231,  in  which  the  shaded  part  is  an  enlargement  of  the 
black  of  Fig.  229  A,  and  lies  in  the  plane  of  the  paper,  while  the  rest 
of  the  figure  is  produced  by  lifting  up  from  the  plane  of  the  paper, 
about  the  fixed  hinge  MF.  If  the 
rotation  be  continued,  the  little  key- 
stone-shaped patch  becomes  a  com- 
plete zone  of  the  mirror,  and  P  de- 
scribes a  complete  ring  round  the  axis  ; 
but,  by  symmetry,  all  the  light  passes 
through  the  axis,  somewhere.  Now 
think  of  a  little  beam  of  light  which 
would  fill  a  quarter-inch  patch  of  the 
dark  zone.  Reflected,  it  first  all  passes 
through   a  little  length   PP'   on  the 

ring,   making   a    minute    bright    lino     [m Q^^QJ" 

(standing  out  perpendicularly  to  the  yig.  231. 

paper)  which  might  be  caught  on  a 
screen.  Continuing,  it  then  all  passes  through  the  axis  between 
Q  and  Q'  ;the  screen  held  hereabouts  would  show  a  second  bright 
line  at  right  angles  to  the  former  (and  in  the  plane  of  the  paper).  ^ 

These  are  the  primary  and  secondary  focal  lines  of  an  astigmatic 
beam,  they  form  '  adze  edge  '  and  oblique  '  axe  edge  '  of  the  volume 
of  light  between  them,  which  nowhere  passes  through  a  focal  point  : 
hence  the  name  astigmatic— pointless. 

Images  built  up  of  little  lines  like  these,  instead  of  tiny  circles, 
looking  as  if  '  smudged  while  wet,'  irritatingly  impossible  to  see 
distinctly,  are  characteristic  of  Oblique  Reflection  or  Refraction. 
Turn  your  stopped-down  reading-glass  askew,  and  it  draws  out 
the  image,  either  horizontally  or  vertically,  according  to  its 
distance.  You  get  it  dreadfully  badly  from  the  concave  mirror  of 
your  microscope. 

One  may  sav  that  the  focus  of  a  large  lens  is  built  up  of  the 
little  focal  lines,  pointing  in  all  directions,  of  the  oblique  beams 
from  all  parts  of  it ;    a  sort  of  asterisk,  a  spot  with  hazy  margm, 


470 


LIGHT 


[§587 


an  image  spoiled  by  '  spherical  aberration.'  Uncover  the  reading- 
glass  while  square,  and  you  get  this ;  turn  it  askew,  and  the  total 
aberration  is  now  vastly  worse  on  one  side,  and  receives  the  apt 
name  of  Coma. 


Chromatic  Aberration,  dependent  on  Nature  of  Kefracting 

Substance 


§  588.  The  spreading  apart  or  Dispersion  of  the  spectral  colours 
which  accompanies  the  deviation  of  white  light  when  refracted, 
and  is,  of  course,  the  whole  aim  of  the  spectroscope  prism,  becomes 
a  nuisance  among  lenses.  For  these,  bending  the  blue  more  than 
the  red,  bring  it  to  a  shorter  focus,  and  a  good  image  becomes 
impossible.  In  Fig.  232  (vastly  exaggerated)  at  B  there  would 
be  a  sharp  blue  image  of  the  star  with  a  red  fringe  round  it,  and  at 
R  a  red  image  with  a  blue  fringe.  This  is  called  Chromatic 
Aberration,  and  in  ordinary  lenses  it  is  many  times  more  serious 
than  the  spherical.  Fortunately  C.  M.  Hall  discovered  in  1757 
how  to  correct  it,  and  make  images  and  lenses  a-chromatic,  not 
colouring.  Fig.  233. 


B 


^  >  <. 


Fig.  232. 


Fig.  233. 


In  Fig.  232  wg  measures,  of  course,  the  average  Deviation  produced 
by  the  prismatic  edge  of  the  lens  and  now  rb,  the  difference  of  the 
deviations  of  red  and  blue  light,  on  account  of  differences  in  refrac- 
tivity  for  these  colours,  is  the  measure  of  the  accompanying  Dis- 
persion of  the  two  colours.  Hall  discovered  what  Newton  had  missed 
a  century  before,  that  this  dispersion  is  a  different  fraction  of  the 
deviation  in  different  substances  ;  of  the  two  kinds  of  glass  available 
in  his  day  one  had  only  half  the  dispersion  of  the  other. 

These  are  a  glass  of  density  2-5  and  chemical  composition,  roughly, 
Na2O,CaO,10SiO2,  called  Crown  glass,  from  a  shape  it  assumed 
during  opening  out  from  blown  bulb  into  fiat  sheet  for  window- 
glass  ;  and  the  more  colour- dispersive  clear  '  crystal,'  of  densitj^ 
3-6  and  composition  approximating  to  K20,PbO,8Si02,  used  for 
cut-glass  tableware,  and  called  Flint,  because  calcined  and  crushed 
flints  were  formerly  used  as  a  source  of  silica  free  from  the  green- 
tinting  contamination  of  iron.  These  names  still  divide  between 
them  the  long  tribes  of  Optical  Glasses  manufactured  at  Jena  since 
1882,  and,  in  more  recent  years,  by  Chance  and  Parsons  in  England. 


590] 


ABERRATIONS 


471 


§  589.  Now,  taking  a  small  prism  angle  A,  the  Deviation  wg 
orD  =  (tx-  1)A,  §409. 

If  jxp  and  [lq  be  the  refractive  indices  for  the  blue  and  red  light  of 
hydrogen,  Fig.  223,  and  [xd  the  convenient  intermediate  sodium 
yellow,  the  Dispersion,  the  difference  of  the  deviations  for  blue  and 
red,  Dbiue  p  —  Dred  o>  IS  therefore 


The  ratio 


{[ip  —  1)A  —  ((jLo  —  1)A  =  ((xp  —  {io)A 
Deviation        jxd  —  1 


Dispersion       {xp  —  fxo 


=  V  (Gr.  nu) 


gives  therefore  the  amount  of  deviation  wg  that  accompanies  the 
production  of  Unit  Width  of  Colour  rb  in  any  particular  medium. 

Deviation  =  v  times  Dispersion. 

The  following  Table  gives  for  some  commonly  used  substances, 
the  mean  index  for  sodium  yellow,  the  difference  of  index  for 
the  brilliant  blue  and  red  hydrogen  lines  (Fig.  223  H,  left  and  middle  ; 
solar  F  and  C)  which  are  the  most  useful  colours  for  calculating  lenses 
for  visual  purposes,  and  the  corresponding  v  : 


/AD 


/*F  —  /*c 


/*D  —  1 
MF  -MC 


Fluorite,  CaFg     . 
Fused  silica,  SiOg 
Boro -silicate  crown  glass, 
Hard  crown  glass  'old,' 
Dense  flint  glass  '  old,' 
Very  dense  barium  flint. 

Air    .... 

Water,  16"  C.  . 
Canada  balsam  . 
Xylol,  20°  C.  . 
Carbon  disulphide,  15°  C. 


Chance's  646 

605 

360 

4675 


1-434 

1-458 

1-5087 

1-5175 

1-6225 

1-6683 

1-000292 

1-334 
1-526 
1-495 
1-630 


0-00454 
0-00675 
0-00793 
0-00856 
0-01729 
0-01876 

0-0000029 

0-0060 
0-0127 
0-0151 
0-0345 


96-5 

68 

64 

60-5 

36 

35-5 

101 

56-5 
41-5 
32-5 
18-5 


Plainly,  the  last  column  means  the  amount  of  bending  out  of  the 
straight  that  accompanies  a  unit  amount  of  spreading  of  colour. 

§  590.  So  that  if  prisms  be  made  (of  very  different  angles)  giving 
these  amounts  of  deviation,  any  one  of  them  clapped  on  to  any 
other,  upside  down,  will  shut  up  its  spreading  colours  into  a  colour- 
less beam  again,  forming  an  Achromatic  Prism,  retaining  a  deviation 
equal  to  the  difference  between  the  two.  That  for  common  crown 
and  flint  is  shown  in  Fig.  234  ;  for  any  other  required  deviation  the 
angles  must  be  changed  proportionally,  i.e.  for  an  Achromatic 
Prism,  individual  deviations  are  made  proportional  to  v'«,  and  they 
are  turned  opposite  ways. 


472 


LIGHT 


[§590 


The  Achromatic  Lens  is  of  enormous  importance  :  every  lens 
you  use,  except  spectacles  and  common  magnifying-glasses  and  the 
cheapest  toys,  is  achromatized. 

Galileo's  original  Optic  Tube  is  almost  as  big  a  non-achromatic 
telescope  as  is  any  real  use  ;   and  Fig.  113  would  actually  show  as 


D  (iVffVp 


Fig.  234. 


Fig.  235. 


much  as  any  microscope,  however  elaborate,  prior  to  1829,  when 
Lister  constructed  the  first  English  achromatic  micro-object-glasses. 

Deviation  in  the  Prism  formula  becomes  Dioptric  strength  in  the 
Lens 

.*.  for  an  Achromatic  Lens,  dioptric  strengths  are  made  proportional 
to  vs,  and  one  is  concave. 

Any  two  lenses,  ground  to  the  strengths  in  the  right-hand  column, 
one  -f  and  the  other  — ,  and  stuck  together,  would  make  an  Achro- 
matic Lens,  of  dioptric  strength  equal  to  their  difference. 


i 

I 

1 


Fig.  236. 


In  general,  this  would  be  s  times  stronger  than  is  wanted,  so  all 
the  curves  are  made  s  times  weaker,  i.e.  all  radii  and  focal  lengths 
s  times  longer,  and  they  make  the  lens  required. 

Naturally  it  is  most  economical  to  select  two  glasses  of  widely 
different  v,  and  you  arrive  at  the  crown-flint  combination,  in  which 
the  less-dispersive  crovm  is  the  stronger  lens,  whichever  way  you  want 
strength,  and  the  corrective  flint  wipes  out  all  the  colour,  but  only 
about  half  the  focal  power.  Usually  the  crown  and  flint  are  ground 
to  the  same  curve  and  cemented  together  with  Canada  balsam. 


§  592]  ABERRATIONS  473 

Examples  of  achromatic  lenses,  actual  sizes  and  curves,  Fig.  236. 

Curvatures.  /id  —  1-  Dioptres;  their  Ratio,     v. 

Top,  telescope  object  glass,  of  '  old  '  glasses. 

Crown  4-5    +  6-25  =      10-75  x  0-5175        =       5-6  \  60-5  60-5 

Flint  0-75  -  6-25  =- 5-5    x  0-6225       =-3-4/        ~  36^  36 


Total  2-2 


Left,  photographic  *  aplanat,'  of  new  glasses. 
Crown        12-75  —  3-75=        9-0    x  0-5087       =       4-6  \  64  64 

Flint         -7-5     +  3-75  =- 3-75  X  0-6683       =-2-5/        "35  35-5 


Total  2-1 

The  little  lenses  are  the  fronts  of  a  1-in.  micro,  object  glass,  of  old  glasses 
on  the  left,  and  new  on  the  right,  as  made  by  Messrs.  Swift,  who  kindly  gave 
me  the  particulars  of  the  four  glasses  used  above. 


§591.  The  skilful  lens-maker  has  at  disposal  many  gh 
four  curves,  two  thicknesses,  and  an  air-space,  and  with  those  he 
carries  on  the  fight  against  chromatic  and  spherical  aberrations  in 
all  their  complexities. 

Unfortunately,  if  spectra  produced  by  different  glasses  are  plotted 
all  to  the  same  length  from  F  to  C,  none  of  them  fits  exactly  anywhere 
else — the  green  is  nearer  red  C  in  crown  than  in  flint,  and  flint's 
violet  tails  out  disproportionately  beyond  blue  F.  That  means, 
that  no  two  glasses  yet  made  will  combine  into  a  truly  achromatic 
lens ;  the  blazing  red  and  blue  fireworks  are  quenched,  but  there 
remains  a  suspicion  of  greeny-yellow  and  purple  edges,  and  a  brilliant 
star  lights  up  a  deep  violet  cloud. 

With  great  telescope  lenses  this  has  to  be  endured  ;  fine  photo- 
graphic lenses,  especially  for  three-colour-process  work,  employ 
a  third  compensating  glass  ;  while  fluorite  (and,  it  is  said,  sylvine) 
of  perfect  optical  quality,  is  obtainable  in  large  enough  chips  to 
use  in  the  more  perfect,  very  expensive,  '  apochromatic  '  (apo, 
away  from)  microscope  object-glasses. 

§  592.  Direct- vision  spectroscope  prisms.  It  is  an  inconvenience 
with  the  spectroscope  prism  that  you  have  to  look  round  the  corner  ; 
why  not  therefore  combine  with  it  an  achromatic  prism  which, 
without  otherwise  interfering  with  the  spectrum,  will  bring  it  bodily 
back  into  the  straight  ? 

In  Fig.  235  there  is  an  extra  8°  of  the  spectrum-maker,  flint. 
With  jx  =  1-62  this  produces  an  average  deviation  (1-62  —  1)8® 
=  5°,  which  is  just  brought  straight  by  the  rest  of  the  prism  (=  Fig. 
234)  ;  while  corresponding  to  5°  deviation  in  '  dense  flint  '  a  spectrum 
5°  -f-  36  =  1/7°  long  is  dispersed.     [6°  m  mid-figure  should  be  5°.] 

Thus,  whatever  deviation  you  can  get  an  achromatic  prism  to 
allow  you,  fatten  out  its  flint  until  you  bend  it  back  again,  and  then 
your  spectrum  is  (this  deviation  -^  v)  in  length,  and  the  fattening 
is  about  deviation/({x  —  1).     An  exact  formula  is  useless,  anyway, 


474  LIGHT  [§  592 

for  nobody  rests  content  with  such  short  spectra,  but  uses  d.v. 
prisms  Hke  Fig.  219.  There  the  flint  in  the  middle  escapes  from  the 
critical-angle  limitation  of  Fig.  187  because  the  refraction  is  now  into 
glass,  with  a  relative  index  1-6/1 -5  only,  so  one  can  widen  it  enor- 
mously, and  get  quite  a  useful  length  of  spectrum  ;  and  double 
that  in  another  following  prism,  if  desired. 


EXAM   QUESTIONS,   CHAPTER   XXXVIII 
Elaborate  answers  are  not  expected. 

1.  A  white  stone  lies  on  the  bottom  of  a  pond.  Its  edges  are  generally- 
observed  to  be  fringed  with  colour,  blue  and  orange.  Explain  this,  and  state 
which  is  the  blue  edge,     try  it. 

2.  What  is  observed  near  the  boundary  of  total  reflection  of  white  light  ? 
Would  a  submerged  eye  have  a  larger  angle  of  vision  into  the  air  for  red  or 
blue? 

3.  Explain  in  detail  how  deviation  and  dispersion  are  caused  by  a  prism. 
Describe  how  to  measure  the  dispersion  of  a  glass.     (  X  2) 

4.  Explain  the  construction  of  an  achromatic  lens.  Why  are  different 
kinds  of  glass  necessary,  and  why  do  ordinary  pocket  lenses  or  microscope 
eyepieces  give  colourless  images  without  them  ?     (See  Chap.  XL.) 

5.  What  is  meant  by  the  dispersion  of  light  by  refractive  media  ?  Describe 
carefully  how  your  telescope  or  microscope  is  freed  from  its  objectionable 
effects.     (  X  3) 

6.  Describe  a  device  for  obtaining  (a)  dispersion  without  deviation,  (6) 
deviation  without  dispersion. 

7.  Describe  experiments  to  show  the  different  focal  values  of  a  lens  for 
blue  and  red  light.     How  can  they  be  equalized  ?     (  X  3) 

8.  Explain,  by  means  of  diagrams,  the  colours  seen  on  a  white  screen  which 
is  moved  a  small  distance  from  the  image  of  a  bright  white  source,  either 
towards  or  away  from  the  convex  lens  which  produces  the  image. 

9.  By  what  means  can  it  be  arranged  that  the  light  passing  through  a 
spectroscope  retains  very  nearly  its  original  direction  ? 

10.  Describe  the  principal  defects  of  lenses,  and  explain  with  the  aid  of 
diagrams  how  these  defects  may  be  overcome. 


CHAPTER   XXXIX 


THE  EYE 


t 


§601.  An  earthworm  seems  sensitive  to  light  anywhere  near 
its  anterior  end.  In  several  animalculae  this  sensitiveness  is 
concentrated  in  a  red  '  eye-spot.'  In  the  *  compound  eyes  '  of 
insects  better  provision  is  made  for  localizing  light  and  shade  ; 
the  central  nervous  tissue  sends  a  fibre  into  each  of  surrounding 
hundreds  of  long  narrow  tubes,  like  so  many  gun-barrels,  radiating 
in  most  directions  of  the  sphere.  Along  each  comes  the  light 
gathered  solely  from  the  direction  in  which  it  is  aimed,  to  help 
build  a  patchwork  or  mosaic 
picture  of  the  world  without. 

A  mosaic  has  been  obtain- 
ed in  the  pearly  nautilus  by 
packing  nerve -endings,  like 
a  velvet  pile,  on  the  back  of 
a  hollow  chamber,  in  the 
front  of  which  is  a  small 
hole — a  pinhole  camera.  To 
gain  more  illumination  the 
pinhole  is  enlarged  and 
covered  with  a  lens,  and 
there  result  the  eye  of  verte- 
brates. The  nervous  '  pile  ' 
of  the  retina  is  so  fine  that 
the  '  mosaic  grain  '  becomes 
unnoticeable. 

In  a  fish's  eye.  Fig.  237,  a  dense  spherical  lens  has  to  do  all  the 
refraction.  In  land  animals  the  clear,  hard,  spherically  bulged, 
front  of  the  '  Cornea  '  does  most.  The  Lens  separates  the  anterior 
'  aqueous '  and  posterior  '  vitreous '  '  humours,'  both  of  them 
jeUies  which  are,  optically  speaking,  water.  It  is  less  curved  ;  and 
is  variable  in  curvature  and  position,  to  ensure  the  clear  focussing 
on  the  retina  of  light  from  different  distances,  and  so  to  '  accom- 
modate '  vision. 

§  602.  Hence  a  first  approximation  to  the  action  of  the  human 
eye  is  obtainable  by  regarding  it  as  a  case  of  Refraction  at  a  single 
spherical  surface.  Fig.  238.  Let  us  find  how  its  focal  strength 
depends  upon  its  curvature. 

Suppose  a  wave  of  light,  plane  because  coming  from  a  great 
distance,  would  have  advanced  to  the  position  ACB  had  it  not  been 
for  the  bulged  refractive  surface  AGB.    The  bulge  GC  measures 

475 


BIRO 
Fig.  237. 


476 


LIGHT 


[§602 


Fig.  238. 


the  curvature  1/R  of  this  cornea.  The  centre  of  the  wave,  struggling 
along  through  the  refractive  medium,  reaches  only  to  E,  where 
GE  =  GC  -^-  (x.  Therefore  EC,  which  is  the  curvature  of  the  plane 
wave  after  refraction,  and  is  therefore  the  principal  focal  power 
l//=GC-GE  =  GC(l-l/(x). 

fnTirr)  7  =  —1  — ture  ^  X  (1  -  Jv 

[The  figure  shows  how  the  waves  have  shortened,  from  GO  to 
GE,  in  the  slow  refractive  medium.     You  can  also,  of  course,  get 

the  formula  by  considering 
ray  refraction  at  A.] 

To  this  approximation  the 
Eye  may  be  considered  as  a 
bulk  of  water,  {jl  =  4/3,  with 
a  refracting  cornea  of  radius 
5  mm.  =  1 /200th  m.  There- 
fore its  focal  power  is  200  X 
(1  -  3/4)  =  50  D. 

Such  an  eye  would,  of 
course,  lose  all  refracting 
power  in  water. 
What  sight  the  actual  eye  retains  under  water  is  due  almost 
entirely  to  the  denser  '  crystalline  lens,'  (x  1-45.  How  imperfect 
this  vision  is  you  know  quite  well :  you  hardly  recognize  those  greeny- 
white  lumps  on  the  bottom  of  the  tank  as  your  own  feet,  until  they 
move. 

On  the  other  hand,  in  Cataract,  when  the  lens  becomes  opaque 
and  has  to  be  removed,  the  patient  is  given  a  10  D  spectacle-lens  to 
make  up  for  its  loss.  (Thus  under  water  you  have  about  one-fifth 
of  normal  vision.) 

§  603.  The  Retina  lines  the  back  of  the  eye,  like  white  velvet, 
its  pile  pointing  outwards,  away  from  the  incoming  light.  Its 
fabric  is  woven  of  nerve-fibres  and  -cells,  and  over  its  inside  surface 
ramify  blood-vessels,  of  which  you  get  a  rather  terrifying  glimpse 
if  you  reflect  sunlight  very  obliquely  on  to  your  eye  while  looking 
at  a  dark  background. 

Ordinarily  you  never  see  them,  because  they  lie  some  distance 
from  the  sensitive  layer,  and  the  cone  of  light  coming  from  the 
pupil  is  wider  than  they  are,  and  shines  past  both  sides  of  them. 
The  same  can  be  said  of  the  drainage  rubbish  that  accumulates, 
with  years,  in  the  '  vitreous '  humour,  but  if  you  narrow  the  cone  by 
looking  through  a  pinhole,  or  by  using  a  badly-illuminated  over- 
powered microscope,  specks  and  polywogs  and  lace  curtains  float 
across  your  view  :  Recipe  grey  powder  gr.  2,  etc.  ;  but  later  on  you 
have  to  put  up  with  them.  Your  eyes  are  freely  irrigated  and 
drained  :  squeeze  them  during  the  Litany  and  you  can't  see  the 
prayer-book  again  until  more  lymph  flows  in  and  plumps  them  out. 


§  604]  THE  EYE  477 

All  the  structures  of  the  eye,  however  pellucid,  are  cellular  (as  is 
the  flesh  of  a  grape),  so  that  incoming  light  encounters  many  chances 
of  very  slight  refractions  and  reflections,  §  493.  Mostly  these  are 
swamped,  and  unnoticeable,  but  they  surround  a  dazzling  star  in 
the  darkness  with  those  long  flashing  '  rays  '  which — in  spite  of 
their  variation  from  eye  to  eye  and  moment  to  moment — most 
people  believe  to  have  a  physical  existence  around  it.  They  have 
not,  they  are  purely  '  entoptic  '  ;  the  usual  long  evening  reflections 
of  lamps  in  a  quiet  river  or  dock  originate  on  the  imperfectly  smooth 
water  much  as  these  rays  in  the  imperfectly  '  smooth  '  surfaces 
and  media  of  the  eye.  Twinkling  of  stars  arises  from  imperfect 
and  varying  atmospheric  smoothness,  §  488. 

Older  people  shade  their  eyes  from  skylight,  when  gazing  keenly 
into  distance,  you  don't  need  to  :  with  age  comes  a  precipitation 
of  minute  rubbish  in  the  media,  and,  lit  up  by  skylight,  this  fills 
the  eye  with  luminous  haze,  obscuring  all  else. 

The  pile  of  the  velvet  is  formed  by  the  sensitive  nerve-endings, 
the  Rods  and  Cones  (rather,  spindles)  which  are  3  microns  diam.  in 
both  man  and  frog.  They  are  bathed  in  '  visual  purple  '  secreted 
from  the  black  choroid  layer  behind,  a  '  fugitive  dye  '  which  is 
bleached  by  light ;  this  electrochemical  change  is  picked  up  by  the 
nerve  and  sent  to  the  brain.  Too  bright  a  light  bleaches  too  much 
purple,  and  leaves  you  locally  blind  until  more  diffuses  in. 

On  account  of  the  aberrations  of  the  curved  surfaces,  only  one 
spot  of  image  is  well-formed,  and  this  falls  on  the  minute  fovea 
centralis,  where  alone  the  retinal  filaments  all  terminate  in  '  cones,' 
and  are  quite  unobscured.  Small  as  is  the  fovea,  the  eye  is  per- 
petually making  little  excursions  ;  if  not,  bright  points  looked  at 
would  fade,  from  the  too  localized  bleaching  of  the  purple. 

Thus,  although  the  whole  field  of  vision  is  large,  only  a  very 
small  portion  is  perfectly  sharp.  This  is  a  great  advantage,  for  it 
compels  attention  to  one  thing  at  a  time. 

§  604.  Binocular  vision  enhances  the  advantage.  Looking  at 
a  jumble  of  things  with  one  eye,  you  will  find  its  attention  wanders 
from  one  attraction  to  another  much  more  than  does  that  of  both 
together.  Two  eyes,  looking  at  the  world  from  different  points 
of  view,  form  slightly  different  pictures ;  you  make  these  coincide 
in  the  point  looked  at,  but  they  fit  together  nowhere  else  ;  every- 
thing else  is  blurred,  and  in  fact  doubled — hold  up  two  fingers  in 
line,  look  at  either  and  the  other  appears  on  both  sides  of  it — but 
vision  off  the  axis  is  so  imperfect  that  this  doubling  usually  passes 
unnoticed. 

Incidentally,  two  eyes  relieve  each  other,  the  chief  attention 
changing  over  from  one  to  the  other  every  few  seconds. 

Judgment  of  distance,  Stereoscopic  vision.  Shutting  one  eye, 
the  effort  of  focussing  the  other  on  near  objects  may  give  some 
estimate  of  their  comparative  distance,  but  you  have  probably  found 
out  at  Christmas  parties  that  it  is  a  feeble  one.     The  one-eyed 


478  LIGHT  [§  604 

have  to  put  up  with  it,  but  manage  by  moving  the  head  ;  two  eyes 
give  us  simultaneous  solid- seeing,  stereoscopic  vision,  and  the 
means  of  judging  distance — that  at  which  the  two  lines  of  sight  cross 
— from  a  few  inches  to  a  good  many  yards.  Prismatic  binoculars 
with  wider-apart  object-glasses  enhance  the  effect,  and  are  in- 
comparably better  than  a  monocular  when  any  sort  of  tangle  has 
to  be  looked  into. 

Range-finders,  §  628,  are  the  same  things  with  '  eyes '  up  to 
90  ft.  apart,  working  to  correspondingly  greater  distances  :  without 
them,  experience  and  environment  are  the  guide. 

In  the  Stereoscopic  Camera,  two  camera-lenses  replace  the  two 
eyes,  and  their  two  slightly  differing  pictures  are  fixed.  Looked  at 
afterwards  by  two  eyes,  aided  by  simple  magnifying-lenses,  more  or 
less  de-centred,  the  two  pictures  blend  into  one  view,  giving  the 
illusion  of  the  original  solidity  (though  see  §  516). 

§  605.  Chromatic  aberration  of  the  eye  accounts  for  the  '  standing 
out '  of  colours  in  front  of  a  pattern  showing  violent  contrasts. 
You  get  this  very  strikingly  in  the  experiment  of  §  573.  The  distant 
purple  light  of  a  shunting  engine  appears,  to  an  eye  slightly  out  of 
focus,  as  a  blue  dot  with  a  red  ring  round  it.  Window-bars,  seen 
only  through  the  edge  of  the  pupil  when  a  book  is  held  close  so  as 
to  obstruct  most  of  the  eye,  are  margined  with  blue  and  orange. 

§  606.    Accommodation  of  vision. 

There  are  two  c's  and  two  m's  in  accommodate. 

At  rest,  the  normal  eye  is  adapted  for  plane  parallel  light  from  a 
distance.  For  near  objects,  a  combination  of  muscular  effort  and 
the  natural  elasticity  of  its  containing  capsule  causes  the  rather 
flat  front  of  the  lens  to  bulge,  and  thus  makes  it  stronger.  See 
Fig.  237,  which  is  drawn  to  natural  size  and  correct  curvatures. 

In  birds  the  lens  is  forced  forward  by  the  hydrostatic  pressure 
of  the  vitreous  humour  when  encircling  muscles  squeeze  inwards  the 
overlapping  bony  plates  which  surround  the  eye.  There  is  also 
a  highly  vascular  organ,  the  pecten,  into  which  blood  can  be  forced 
so  as  to  increase  the  total  contents  of  the  eye  and  again  to  drive 
the  lens  forward. 

In  fish,  the  retina  is  farther  away  from  the  spherical  lens  aft,  than 
on  the  beam  :  as  the  fish  turns  and  swims  up  to  the  object,  its 
conjugate  image  recedes,  Fig.  204,  and  remains  in  focus  on  the 
retina. 

Distinct  vision  is  possible  only  in  the  interval  between  limits 
of  distance  E  and  E',  called  the  near  and  far  points  of  the  eye. 
Both  can  be  found  with  the  old-fashioned  optometer,  which  is  just 
a  convex  spectacle-lens  with  an  object  sliding  beyond  it  on  a  gradu- 
ated bar.  The  nearest  and  farthest  distances  e  and  e'  are  noted  at 
which  the  object  can  be  seen  clearly  by  the  eye  close  behind  the 
lens  ;  from  their  reciprocals  is  subtracted  the  focal  power  of  the 
lens,  the  remainders  are  the  limiting  focal  powers  of  the  eye  itself, 


§606] 


THE  EYE 


479 


1/E  and  1/E',  whence  you  get  E  and  E'.  The  accommodating 
power  of  the  eye  is  1/E  —  1/E',  it  is  the  blackened  angle  in 
Fig.  240. 

Distances  having  been  kept  in  Metres,  the  Accommodating  Power 
is  in  Dioptres. 

Fig.  239  shows  the  course  of  change  of  accommodation  with  age. 
The  accommodation  is  given  in  Dioptres,  the  reciprocals  of  the 
figures  of  the  upper  curve  are  the  near  points,  E,  in  metres  ;  those  of 
the  lower  curve  (either  infinity  or  negative)  give  the  '  far  pints.' 
The  vertical  distance  between  the  two  curves  is  the  Accommodating 
Power  at  that  age.  The  diagram  is  figured  for  a  normal '  emmetro- 
pic '  eye,  but  the  only  difference  for  other  eyes  is  that  the  whole 


1'* 

D 

.^^■■^"  J 

\ 

- 

< 

/         1 

ris 

■~ — -^                  1 

?°L 

^ 

1 — 

= 

^ '                   *•        ^ 

8 

V 

1 — 

— 



~~^ — -^    \ 

\ 

_ ■                   i 

6 

\, 

■== 

3      1 

S 

\ 

^^^^^1 

4 

\ 

X 

*  1 

a. 

\ 

N 

»«- — ■ 

Q_ 

\ 

^ 

^^ 

N 

5 

-2. 

A 

G 

E 

^ 

1^ 

^w 

10 

2 

0 

3 

0 

^ 

0 

S 

0 

6 

0 

7 

0 

EC 

F 

IG. 

23 

9. 

Fig.  240. 

vertical  scale  slides  up  or  down,  the  shape  of  the  curves  remaining 
fixed,  so  that  while  a  patient  may  be  long-  or  short-sighted  at  aiiy 
age,  the  determination  of  both  his  limits  of  vision  tells  his  age  with 
more  certainty  than  do  a  horse's  teeth. 

Beside  it,  in  Fig.  240,  are  five  eyes  ;  1,2,  and  3  belong  to  your 
age,  4  and  5  to  mine.  The  accommodating  power  is  measured  by 
the  black  angle,  0-6°  per  dioptre,  which,  taking  the  pupil  as  3  mm. 
diam.,  is  a  six-fold  exaggeration.  The  Normal  '  emmetropic  '  eye 
at  your  age  has  such  strength  and  elasticity  that  it  can  converge 
to  a  focus  on  its  retina  light  diverging  as  in  1,  just  as  well  as  the 
parallel  beam  bounding  the  black  angles  on  their  outside. 

The  short-sighted  '  myopic  '  eye,  2,  has  just  the  same  A.P., 
the  black  angles  are  the  same  as  before,  but  its  refraction  as  a  whole 
is  too  strong,  and  can  never  slacken  to  deal  with  distant  light  :  you 
see  both  E  and  E'  for  it ;  this  particular  specimen  has  5  D  of  myopia, 
or  in  Fig.  239  the  vertical  scale  has  to  slip  down  until  5  lies  on  the 
present  zero  line,  the  range  being  from  5  to  15  D. 


480  LIGHT  [§606 

No  3  is  '  hypermetropic  '  or  long-sighted,  he  can  actually  relax 
his  eyes  until  the  stars  go  fuzzy,  for  him  the  vertical  scale  moves 
up  2  ;  his  range  is  from  —  2  to  +8.  He  doesn't  hold  his  book 
quite  so  close  ;  while  youth  lasts  he  gets  on  perfectly  well,  but  starts 
glasses  five  years  early  :  they  were  my  eyes,  with  2D  of  long  sight, 
i.e.  lens  as  a  whole  flatter  than  fits  the  eye,  —  2D,  and  as  it  takes 
4D  to  accommodate  up  from  zero  to  25  cm.  reading  distance,  you 
see  that  for  intricate  jobs  I  had  to  start  glasses  by  thirty-five,  when 
only  6D  was  left. 

No  4  is  the  '  emmetropic  '  eye  grown  '  presbyopic,'  able  to  see 
stars  and  recognize  distant  faces  perfectly,  but  unable  to  converge 
on  a  page  of  print ;  even  at  arm's  length,  when  print  had  need  be 
large,  it  is  apt  to  jump  about  and  cheat  him.  That  awful  frown 
the  Head  bent  on  you,  as  his  stern  eye  pierced  you  through  and 
through  :  the  good  man  was  but  straining  his  remaining  accommoda- 
tion in  the  effort  to  see  at  least  who  the  culprit  was — and  may  the 
hint  be  useful  yet. 

As  Fig.  239  shows,  the  near  limit  goes  towards  the  far,  never  the 
other  way,  so  of  course  my  eyes  are  now  like  5,  unable  to  deal  clearly 
even  with  distant  objects  :  not  for  me  the  first  pale  glimpse  of  the 
evening  star,  I  want  the  comfort  of  1-5  or  2D  glasses  out  of  doors 
all  day  long,  I  cannot  compete  with  old  schoolmates  who  crow 
about  their  perfect  sight  for  distance  ;  '  Bifocals  '  are  my  daily 
wear,  and  their  '  flexibility  '  makes  me  scarcely  conscious  of  my 
limitations.  Fortunately,  both  eyes  are  nearly  enough  alike,  and 
3-5D  piano-spheres  that  cost  me  only  a  trifle  have  carried  me 
through  most  of  the  close  work  of  this  book  and  its  diagrams. 

§  607.  Spectacles.  A  spectacle-lens  is  an  optical  instrument  : 
through  it  one  sees,  not  the  object,  hut  the  Virtual  Image  of  it,  and  the 
lens  must  be  such  as  to  form  this  virtual  image  at  a  distance  within 
the  wearer's  range  of  accommodation.  You  cannot  '  make  the  eye 
see  things,'  you  must  make  images  where  the  eye  can  see  them.  Glasses 
are  so  familiar  that  one  is  apt  to  forget,  or  even  disbelieve,  this 
statement ;  but  borrow  a  pair  and  try  to  walk  downstairs,  looking 
at  your  feet. 

A  student's  first  acquaintance  with  '  glasses  '  is  often  made  when 
he  finds  reading  at  night  unduly  trying  to  the  eyes.  He  consults 
'  Eyes,'  and  is  told  it  is  '  tired  sight,'  and  is  prescribed  a  pair  of 
+  JD  spheres.  The  fact  is,  near-distance  vision  is  normally  effected 
by  muscular  action,  and  after  a  day's  bench-work  at  medium  distance, 
short  reading -distance  at  night  strikes  the  eye  as  a  bit  too  much 
of  a  good  thing,  so  you  have  to  humour  it  a  little.  It  is  most  desirable 
that  these  glasses  fit  well  for  bridge-height  and  eye-width,  be 
'  cranked  '  down  so  that  they  are  looked  through  squarely,  and  do 
not  pinch  the  nose  :  their  actual  strength  is  of  little  more  importance 
than  their  shape  and  size  and  general  get-up,  which  are  matters  of 
individual  taste.     Use  a  good  light  well  placed. 

Fortunately,  you  are  not  so  fussy  that  you  must  have  those 


§  608]  THE  EYE  481 

little  meniscus  lenses  stuck  in  actual  contact  with  the  cornea  that 
are  now  available  for  those  who  must  not  be  seen  wearing  glasses. 

Ten  inches,  or  25  cm.,  1/4  m.,  is  taken  as  normal  Reading  Distance, 
this  being  conditioned  by  the  usual  size  of  print.  To  the  emmetro- 
pic eye,  with  normally  zero  accommodation  for  distant  vision,  this 
means  an  accommodation  of  -f  4D  all  the  time. 

But  if  the  Refraction  of  the  Eye  be  always  abnormally  Strong, 
as  you  see  in  Fig.  240  (2),  then  the  incoming  rays  must  be  divergent, 
or  else  they  will  be  bent  to  an  image  before  reaching  the  retina,  as 
in  Fig.  242  M,  where  they  are  parallel ;  i.e.  only  near  objects  can 
be  seen  clearly.  This  is  short  sight  or  myopia.  The  trouble  of  the 
short-sighted  is  that  he  cannot  see  distant  objects  :  provided  with 
a  lens  which  makes  parallel  light  diverge,  so  as  to  be  within  his  outer 
visual  limit,  he  will  be  content,  i.e.  a  concave  lens  is  added  to  his 
eye  to  give  a  normal  combined  refraction.  If  his  far  point  is  E' 
cm.  away,  his  eye  already  possesses  100/E'  dioptres  of  refraction, 
and  this  must  be  taken  completely  off  by  a  lens  of  —  100/E'  dioptres. 

If  the  Refraction  be  abnormally  Weak,  rays  divergent  from  near 
points  cannot  be  brought  to  an  image  by  the  time  they  reach  the 
retina.  A  convex  lens  must  be  added  to  render  such  rays  more 
nearly  parallel,  and  thus  to  enable  the  long-sighted  or  hypermetropic 
eye  to  see  clearly  at  the  near  distance.  For  the  trouble  now  is,  that 
nothing  is  clear  until  2  or  3  ft.  away  from  the  eye,  a  distance  at 
which  most  print  is  too  small  to  read.  Fig.  242  H  shows  an  extreme 
case  in  which  even  parallel  light  is  far  from  being  focussed. 

These  abnormal  refractions  are  almost  entirely  due,  not  to  the 
eye-'  ball '  being  too  '  long,'  but  to  the  spherical  curvature  of  the 
Cornea  being  excessive  or  deficient ;  and  now  it  can  have  another 
trouble  : 

§608.  The  cornea  of  an  astigmatic  eye  is  elliptical,  curved 
differently  in  the  vertical  and  horizontal  planes,  like  a  big  dish- 
cover.  Any  image  it  produces  is  distorted,  like  your  face  in  a  tea- 
spoon. A  pattern  of  radiating  lines  cannot  be  seen  clearly  all  at 
once,  when  some  are  distinct  those  at  right  angles  are  blurred,  and 


.^^i>     n^m»     -.13115.-      w^^^^^E 
Fio.  241. 

require  re-focussing.  Print  becomes  illegible  from  the  blurring  of 
the  horizontal  strokes.  Astigmatism  is  unhappily  common  among 
students;  the  ellipticity,  if  slight,  is  practically  corrected  by 
appropriate  stress  in  healthy  eye-muscles,  but  if  more  serious, 
eye-strain  and  headache  drive  one  to  the  oculist. 

He  may  test  with  an  elaborate  and  ornamental  instrument  called 
a  Keratometer,  Ch.  XXXIV,  Q.  23,  which  actually  measures  the 

B 


482  LIGHT  [§  608 

curvatures  of  the  cornea  ;  or  else  he  holds  up  before  your  eye  a 
pattern  of  concentric  rings,  and  through  a  lens-peephole  in  the 
middle,  examines  the  little  reflection  of  it  in  your  cornea,  and 
judges  the  amount  of  astigmatism  from  the  elliptic  distortion  this 
shows.     (Or  see  Retinoscopy.) 

Thence  he  prescribes  for  each  eye  separately  a  compensating  lens. 
This  is  plane  or  spherical  on  one  surface  and  cylindrical  on  the  other, 
a  '  sphero- cylinder  ' ;  along  the  straight  axis  (dots  in  the  lenses 
of  Fig.  241  C)  of  the  cylinder  there  is  only  the  sphere's  curvature, 
at  right  angles  there  is  sphere's  +  cylinder's.  Thus  one  gets  the 
effect  of  an  ellipse,  and  distorts  the  light  ready  for  the  afflicted  eye^s  use. 

The  trouble  with  these  lenses  is  that  one  side  has  to  be  straight, 
along  the  Astigmatic  Axis,  and  they  cannot  be  given  the  periscopic 
'  bow- window  '  shape,  sectioned  both  ways  in  Fig.  241  P,  which 
most  people  prefer,  because  it  enables  the  wearer  to  look  out  less 
askew  through  their  outer  zones,  and  see  things  less  distorted  jbhere — 
doorposts  vertical,  or  pictures  hanging  straight — and  so  to  aim  better 
at  golf. 

Astigmatism  and  golf  being  both  common  afflictions,  toroidal 
lenses,  which  can  be  ground  one  by  one  on  a  special  machine,  to  a 
dish-cover  shape,  having  curvatures  different  in  the  two  directions, 
with  sufficient  accuracy  for  visual  purposes,  have  come  into  vogue, 
although  fragile  and  expensive. 

[The  Torus  is  the  '  cushion '  at  the  foot  of  a  column  :  originally 
the  soft  bag  of  sand  upon  which  its  rough  end  rested,  bulged  by 
the  weight ;  it  is  now  an  ornamental  moulding,  its  radius  horizontally 
that  of  the  column ;  vertically,  anything  the  architect  fancies.] 
Fig.  241  T  shows  sections  at  right  angles  of  such  a  lens  ;  the  inner 
surface  is  spherical. 

A  patient  who  cannot  bring  both  eyes  to  bear  on  one  point, 
without  strain,  sees  double — diplopia.  If  the  eyes  are  too  badly 
askew,  this  becomes  squint,  and  the  patient  habitually  disregards 
one  eye,  the  vision  of  which  degenerates  by  disuse. 

Evidently  simple  thin  prisms,  of  angles  from  1°  up  to  12°,  can  be 
employed  to  deflect  light  into  the  axial  direction  of  the  oblique  eye, 
and  so  retain  useful  binocular  vision. 

Their  strengths  are  described  in  prism-dioptres,  a  1  D  prism 
would  apparently  displace  an  object  at  1  metre  distance  1  cm. 
sideways,  and  so  on. 

Any  necessary  lens-curves  can  then  be  ground  on  the  prism  faces, 
but  it  is  almost  always  easier  to  take  advantage  of  ordinary  lens 
faces  being  already  inclined  to  each  other,  everywhere  except  just 
in  the  centre,  and  simply  de-centre  the  lens. 

Inspection  of  Figs.  196,  202,  will  soon  show  you  that  light  striking 
a  1  D  lens  1  cm.  from  the  centre  is  bent  1  cm.  aside  in  the  metre, 
striking  at  2  cm.  out  it  is  bent  2  cm.,  etc.  An  nT)  lens  bends  it  as 
much  at  only  1/nth  the  distance  from  the  centre ;  thus  a  4  D  lens 
de-centred  5  mm.  corrects  2  P-D  of  obliquity,  and  so  forth. 


609J 


THE   EYE 


483 


The  lenses  of  common  Stereoscopes  are  usually  strongly  decentred  • 
hold  them  in  the  sun.  ' 

As  to  the  necessary  accuracy  of  figure  of  spectacle  lenses,  see  §  526  ; 
yet  Galileo,  in  his  telescopes,  used  spectacle  lenses  ;  there  were  no 
others. 

§  609.  How  does  the  oculist  *  test  refractions  '  ?  He  invests  in 
a  large  expensive  trial  set  of  many  lenses  of  marked  dioptric  strengths, 
and  a  frame  adjustable  for  width  of  eye  and  height  of  *  bridge,' 
and  with  circular  rotating  lens-holders  for  astigmatism. 

He  sets  you  to  read  well-lighted  test-types  6  m.  away,  nearly 
enough  '  infinite  '  distance. 


Fia.  242. 


Fig.  243. 


Then,  one  eye  at  a  time,  he  tries,  on  a  definite  system,  lens  after 
lens  (including  '  cylinders ')  until  you  read  the  type  best.  Then  the 
marked  strengths  of  the  lenses  in  front  of  your  eye  add  up  to  make 
the  lens  you  require  for  distant  vision. 

If  you  are  young,  accommodating  power  takes  care  of  near 
vision  through  the  new  glasses.  If  you  are  older,  then,  says  he,  light 
diverging  from  an  object  J  m.  from  the  eye  will  be  made  parallel 
by  a  2  D  lens,  and  will  therefore  be  seen  by  the  patient's  eye,  as  now 
armed  for  distant  vision,  without  accommodation.  For  a  page 
at  J  m.,  reading  distance,  a  4  D  lens  gives  parallel  light.  H  by  your 
age.  Fig.  239,  you  still  have  2  or  more  dioptres  accommodating  power, 
he  will  '  add  +  2  D  for  reading,'  and  leave  you  to  accommodate 
up  the  remaining  2  D.  With  short-sighted  patients  he  takes  off 
—  2D,  which  comes  to  the  same  thing.  If  you  are  older,  he  may 
have  to  make  it  3  D  additional,  or  in  age  the  full  4  D. 

This  is  either  for  a  special  pair  of  glasses  for  reading,  or  else  for 


484  LIGHT  [§  COO 

little  sectors  of  this  strength  to  be  stuck  on  (easiest  to  alter),  or 
welded  in,  or  ground  more  curved,  or  somehow,  over  the  lower 
halves  of  the  glasses,  to  make  '  bifocals,'  suitable  for  both  purposes. 

Or  else,  with  children  and  people  of  poor  judgment,  he  uses  the 
Retinoscope,  Fig.  242.  He  paralyses  the  accommodation  and  opens 
the  pupils  by  the  beautifying  belladonna,  or  other  mydriatic  drug. 
He  puts  a  -j-  1  D  lens  on  the  patient's  eye,  and  sits  him  down  at 
1  m.  distance,  in  a  dimly -lit  room.  Then,  waggling  a  perforated 
mirror  in  front  of  his  own  eye,  he  flashes  the  light  of  a  lamp  across 
the  patient's  face  and  eye,  so  that  it  will  travel  by  paths  1  2  3  in 
succession.  Fig.  242.  These  being  1  m.  long,  the  +  1  D  lens  lays 
them  parallel  into  the  eye  (which  is  what  it  is  for),  and  now  if  the 
patient's  unaccommodated  eye  is  normal,  the  parallel  light  will 
all  focus  accurately  on  one  point  of  the  retina.  Whether  the  light 
goes  by  1,  2  or  3  makes  no  difference,  the  same  point  of  the  retina 
is  brightly  illuminated,  and  shines  back,  along  all  the  paths  at  once, 
to  the  observer's  eye.  That  is,  as  long  as  your  light  shines  on  the 
eye  at  all,  its  pupil  blazes  at  you  with  a  motionless  glare,  which 
flashes  in  or  out  instantly  as  your  mirror  flickers. 

If,  however,  the  patient's  eye  is  faulty,  as  in  the  lower  figures, 
and  parallel  light  does  not  focus  on  its  retina,  then,  as  you  trail 
the  flash  across  the  eye  12  3,  you  see  a  small  flash  move  across 
the  pupil  one  way  or  the  other. 

Therefore,  behind  the  +  I  D  lens,  put  in  trial  lenses  until  you  get 
the  motionless  flash  condition  :  evidently  these  make  the  eye  normal, 
and  are  what  it  requires  for  distant  vision.  Add  for  near  vision  as 
before. 

Astigmatism  is  disclosed  by  the  flash  being  on  the  slant,  and  is 
measured  by  working  both  along  and  across  the  slant. 

§  610.  The  inside  of  the  eye,  which  has  first  had  its  pupil  opened 
up  by  belladonna,  is  inspected  by  the  Ophthalmoscope.  This  used 
to  consist  of  a  concave  mirror,  for  collecting  light  from  a  lamp  and 
concentrating  it  into  the  eye,  into  which  the  observer  looked  through 
a  central  hole  in  the  mirror,  but  nowadays  the  light  comes  from  a 
bulb  of  the  smallest  size  fed  by  a  dry  cell  in  the  handle  of  the  instru- 
ment, and  in  Fig.  243,  upper  figure,  is  collected  and  directed  by 
two  little  lenses,  one  of  which  can  be  slid  up  and  down  so  as  to 
spread  the  light  or  concentrate  it  on  one  spot  as  desired.  The 
wavy  dotted  line  quite  sufficiently  indicates  this  supply  of  light 
reaching  the  retina. 

The  illuminated  retinal  area  shines  forth,  sending  out  light,  to 
be  refracted  at  the  lens  and  corneal  surface,  together  equivalent 
to  a  lens  of  10  +  40  D,  and  to  issue  thence  in  a  more  or  less  parallel 
beam  towards  the  observer's  eye.  The  parallelism  would  be  exact 
if  the  patient's  eye  were  normal  (emmetropic),  and  the  beam  would 
then  focus  on  the  retina  of  a  normal  relaxed  observing  eye,  which 
would  see  the  illuminated  retina  clearly  magnified  as  by  a  50  D  lens. 


§610]  THE   EYE  485 

i.e.  about  twelve  times,  As,  however,  neither  eye  is  likely  to  be 
quite  normal,  a  wheel  or  chain  of  little  lenses,  of  all  strengths  from 
+  or  —  0-5  D  upwards,  can  be  run  round  into  position  at  L,  the 
observer  putting  his  thumb  to  the  wheel  until  he  arrives  at  one 
which  clears  up  the  picture. 

From  the  marked  strength  of  this  lens  it  is  possible  to  prescribe 
a  spectacle-lens  for  the  patient,  but  most  people  prefer  to  do  one 
thing  at  a  time,  reserving  the  ophthalmoscope  for  visual  inspection, 
and  relying  on  the  methods  of  §  609  for  refraction  determinations.  ' 

Measurements    are    made,    however,    of    varicosities — humps 

on  the  retina,  such  as  bring  its  sensitive  focal  surface  nearer  to  the 
lens,  and  keep  persuading  the  owner  that  the  printer  has  made  a 
mistake  and  set  up  three  words  in  the  next  line  in  smaller  type. 

Noting  the  strength  of  the  adjusting  lens  he  is  using  on  the 
retina,  the  observer  runs  the  lenses  round  until  one  focusses  sharply 
on  the  top  of  the  swelling,  and  then  quotes  its  height  as  (the 
difference)  '  dioptres,'  which  conveys  just  as  much  to  him  as  if 
he  went  on  and  calculated  out  its  actual  height  in  mm. 

By  bringing  additional  stronger  lenses  into  use  in  the  ring  at  L, 
specks  in  the  vitreous  humour  can  be  examined,  or  the  lens,  or 
cornea — or  a  blocked-up  keyhole,  for  an  ophthalmoscope  has  un- 
orthodox uses. 

In  the  '  indirect '  way  of  using  it,  a  3-in.  convex  lens  is  held  in 
front  of  the  patient's  eye,  and  converts  the  nearly  parallel  streams 
of  light  coming  from  it  into  pencils  converging  in  its  focal  plane 
on  the  right.  Fig.  243  lower,  and  this  aerial  real  image  the  observer 
looks  at  from  a  foot  distance,  just  as  you  watch  for  the  pins  in  the 
daylight  conjugate  foci  method  of  measuring  a  convex  lens.  The 
picture  is  now  magnified  only  about  four  times  instead  of  twelve, 
and  this  indirect  examination  is  commonly  made  first,  as  by  moving 
the  big  lens  to  and  fro,  and  watching  changes  of  size  of  the  image, 
a  good  preliminary  notion  of  the  refractive  errors  can  be  obtained. 

I  have  just  been  shown  an  immense  potential  improvement  in 
Ophthalmoscopes,  one  of  those  simple  things  that  one  kicks  oneself 
for  not  thinking  of  first,  and  one  which  it  is  to  be  hoped  the  instru- 
ment-makers won't  spoil.  Along  the  bottom  of  Figs.  242,  243 
stretches  the  radius  of  curvature  of  a  good-sized  spherical  concave 
mirror,  of  moderate  optical  quality,  into  which  you  and  a  tiny 
lamp  L  look  together,  and  so  does  the  patient  beside  you. 

You  need  no  cat's-cradle  diagram,  the  idea  would  never  have 
struck  anyone  tangled  up  in  them  ;  the  mirror  makes  real  images  of 
lamp  and  your  eye  in  front  of  the  patient's  eye,  the  real  image  of 
the  lamp  shines  into  it  and  the  real  image  of  your  eye  looks  into  it  : 
what  you  perceive  is  the  mirror  suddenly  blazing  full  of  a  great 
picture,  not  a  thing  seen  in  small  bits,  as  in  the  last  '  lens  '  method. 
It  gives  one  a  most  vivid  idea  of  the  capabilities  of  Real  Images. 


486  LIGHT 


EXAM   QUESTIONS,   CHAPTER   XXXIX 

Anatomical  details  are  not  expected,  nor  the  calculation  of  §  602.  We 
are  all  living  through  Fig.  239 ;  this  leads  up  to  §  609,  which  you  should  read, 
because  that  is  the  simple  way  it  is  done.  Questions  11,  12,  13,  16  would 
madden  an  oculist.  The  ophthalmoscope  is  asked  for,  modern  retinoscopy 
has  not  yet  appeared. 

Here  is  a  question  I  set  while  this  chapter  was  in  the  printer's  hands  : — 

A  patient  can  see  the  stars  clearly  only  by  the  aid  of  —  3  D  glasses :  he 
possesses  6  Dioptres  of  Accommodation ;  what  is  his  nearest  distance  of 
distinct  vision  through  his  glasses  ? 

One  candidate  answered  correctly ;  74  fumbled  with  formulae,  *  learned ' 
doubtless  at  great  effort  and  expense. 

What  nobody  ever  told  them  is  that  we  wear  glasses  to  make  our  vision 
just  like  that  of  *  everybody  else ' :  to  be  normal — which  is  why  we  do  most 
things. 

The  wearer  of  these  glasses  sees,  perfectly  normally,  to  infinity,  without 
accommodation,  i.e.  to  (1/0)  metres.  Therefore,  exerting  6  Dioptres  of 
Accommodation,  he  sees  at  (1/6)  nietre.     that  is  all. 


1.  Draw  carefully  a  diagram  explaining  the  construction  and  showing 
the  path  of  two  parallel  rays  of  light  passing  through  a  glass  sphere  of  radius 
5  cm.  and  refractive  index  1-5.  Explain  how  this  may  be  used  as  a  sunshine 
recorder. 

2.  Why  does  a  goldfish  appear  unmagnified  and  nearer  the  surface  of  water, 
but  magnified  as  seen  through  the  glass  of  its  bowl  ? 

3.  Prove  that  an  air  bubble  in  a  glass  ball  or  a  goldfish  in  a  globe  will  appear 
nearer  than  it  really  is,  at  its  true  distance,  or  farther  off,  according  as  it  is 
nearer  the  surface  than  the  centre,  at  the  centre,  or  beyond  it. 

4.  Describe  how  the  retina  of  the  eye  may  be  illuminated  by  concave 
mirror  and  lamp,  and  inspected.  Explain  the  terms  Accommodation,  Blind 
spot.  Least  distance  of  distinct  vision.     (  X  2) 

5.  Describe  the  Ophthalmoscope,  giving  diagrams  of  its  use.     (  X  3) 

6.  Describe  briefiy  the  parts  of  the  eye,  regarded  as  an  optical  refracting 
instrument,  and  explain  the  variations  from  normal  vision  known  as  '  long 
sight  '  and  '  short  sight.' 

7.  Explain  the  eye  as  an  optical  instrument,  accounting  for  (i)  the  forma- 
tion and  character  of  the  image  on  the  retina;  (ii)  the  focussing  for  objects 
at  different  distances;    (iii)  the  defect  known  as  '  astigmatism.'     (  x  2) 

8.  Describe  the  optical  system  of  the  eye.  If  it  be  regarded  as  a  lens  of 
2  cm.  focal  length,  what  alteration  in  back  focal  distance  would  enable  clear 
focussing  at  25  cm.  reading  distance  ? 

9.  Explain,  with  diagrams,  what  defects  of  vision  can  be  corrected  by 
convex  and  concave  lenses. 

A  short-sighted  patient  has  a  range  of  accommodation  from  10  to  20  cm. 
from  the  eye.  What  lens  should  he  use  for  distant  vision,  and  what  would 
be  the  new  limits  of  accommodation  ? 

[Reduce  everything  to  Dioptres  : — 


THE   EYE  487 

The  normal  eye  has  accommodation  zero  for  distant  vision. 

100/10  =  10  D  for  vision  at  10  cm. 
100/20  =    5  D   „         „      „  20    „ 

The  difference,  5  Dioptres,  is  the  patient's  range  of  accommodation. 

To  convert  his  farthest -adapted  eye  into  a  normal  eye  unacconunodated 
take  off  its  5  D,  i.e.  give  him  a  —  5  D,  a  20-cm.  concave. 

His  range  is  then  from  (5  D  —  5  D)  to  (10  D  —  5  D),  i.e.  from  infinity  to 
20  cm.] 

10.  The  same  question  with  8  and  16  cm.  ? 

1 1 .  What  lens  would  enable  an  eye  with  far  point  8  in.  to  see  at  48  in.  ? 

12.  Ditto  1-8  m.  and  6  m.  ? 

13.  Ditto  2  ft.  and  20  ft.? 

14.  Ditto  5  in.  and  star.  ?     [Evidently,  5  in.  concave,  —  8  D.] 

15.  Range  4  in.  to  10  in.,  lens  0-5  in.  from  eye,  calculate  lens  for  remote 
vision  [evidently,  9-5  concave]  and  nearest  point. 

16.  A  man,  who  can  see  distinctly  only  between  5  and  8  in.  from  his  eye, 
wishes  to  read  a  notice  15  ft.  away;  what  spectacles  would  you  recommend, 
and  what  would  be  the  limits  of  his  distinct  vision  ? 

17.  Why  cannot  a  swimmer  see  plainly  under  water  ?  Water-tight  goggles 
are  procurable,  to  keep  his  eyes  dry  and  enable  him  to  do  so ;  should  their 
thin  glasses  be  flat,  or  bulged  ? 

18.  What  additional  lens  would  enable  a  patient  with  —2D  myopia  to 
read  at  25  cm.  ?  If  his  near  point  is  15  cm.,  find  his  far  point  and  the  power 
of  the  additional  lens. 

[This  question  has  escaped  the  censor.  The  -|-  2  D  myope  of  course  reads 
with  ease  between  15  and  50  cm. ;   all  he  wants  is  a  —  2  D  for  distance.] 

19.  Show  in  diagrams  the  two  common  defects  of  vision.  A  patient  who 
cannot  see  closer  than  60  cm.  wishes  to  read  at  25  cm. ;  what  lenses  should 
he  wear  ? 

[He  possesses  100/60  =  1-67  D,  and  the  demand  is  for  100/25  =  4  D,  he 
must  therefore  wear  +  2-5  D,  the  nearest  available  strength.] 

20.  Ditto  50  cm.  to  be  reduced  to  20  cm.  ? 

21.  Ditto  100  cm.  to  '  reading  distance  '  ? 

22.  Ditto  3 — 1  ft.  ?  and  compare  the  optical  properties  of  the  eye  with 
those  of  a  photographic  camera. 

23.  A  person  cannot  see  objects  clearly  which  are  nearer  than  1  m.  or  further 
than  4  m.  What  defects  do  you  diagnose  ?  What  spectacles  would  you 
prescribe  (a)  for  distance,  (b)  for  reading  at  25  cm.  ? 

[Presbyopia,  accommodation  only  from  1/1  to  1/4  =  0-75  D.  For  distance 
—  0-25  D;   for  reading  +  3  D.] 

24.  A  patient  unable  to  see  at  less  than  1-5  m.,  desires  to  read  a  book; 
what  spectacle  lenses  would  you  prescribe  ?  If  he  now  complains  that  the 
print  appears  streaky,  the  upright  strokes  sharp  but  the  cross  strokes  blurred, 
what  modification  of  the  lenses  is  called  for  ? 

25.  Using  a  lens  of  focal  length  7-3  cm.  the  limits  of  distinct  vision  were 
found  at  7-6  and  4  cm.  Calculate  the  limits  without  the  lens  and  the  accom- 
modating power. 

26.  How  would  you  test  an  eye  for  astigmatism  and  how  decide  what  kind 
of  spectacle  lenses  to  recommend  ?     (  X  3) 


CHAPTER   XL 


OPTICAL  INSTRUMENTS 


§  61L  Historical.  The  use  of  a  flask  full  of  water  to  concentrate 
the  light  of  a  lamp  upon  their  work  was  well  known  to  the  Roman 
cameo-cutters,  and  persists  to  the  present  day  among  engravers. 
Three-legged  '  candle-stools,'  Fig.  244,  from  the  Kew  museum,  with 
a  central  candlestick  surrounded  by  stumps  carrying  inverted 
flasks  of  water,  were  in  use,  in  the  eastern  Midlands,  to  throw  a  strong 
light  on  the  pillows  of  the  lace-makers  who  spent  long  evenings 
round  them. 

Pliny  mentions  the  use  of  globes  of  water  for  cauterization,  by 
focussing  the  sun's  rays,  but  never  for  magnifying  purposes ;  and 
although  Seneca  states  that  '  letters,  though 
small  and  indistinct,  are  seen  enlarged  and 
more  distinct  through  a  globe  of  glass  fllled 
with  water,'  he  merely  concludes  that  all  objects 
seen  through  water  appear  larger.  Defects  of 
vision  were  discoursed  upon,  but  even  up  to 
the  thirteenth  century  were  dismissed  as 
incurable. 

Roger  Bacon,  1214 — 1294,  Franciscan  friar, 
of  Ilchester,  showed  the  efficacy  of  crystal 
lenses,  to  show  things  larger,  and  '  to  make 
an  instrument  useful  to  old  men  and  those 
whose  sight  is  weakened,'  but  this  initiator  of 
experimental  physics  having,  of  course,  been 
imprisoned  as  a  magician,  a  Florentine,  Salvino 
degli  Armati,  ca.  1280,  had  the  credit  of  in- 
venting spectacles  ;  a  Pisan  monk,  Alexander 
Spina,  immediately  afterwards  giving  away  the 
secret  of  their  construction  and  use,  both  for  long  and  short  sight. 

Bacon's  '  Opus  Majus  '  had,  of  course,  to  be  hidden,  to  save  it 
from  destruction,  and  only  came  to  light  in  1733  ;  in  it  he  further 
wrote  :  '  Glasses  may  be  so  formed  that  the  most  remote  objects 
may  appear  just  at  hand,  so  that  we  may  read  the  smallest  letters 
at  an  incredible  distance,  and  may  number  things,  tho'  never  so 
small,  and  may  make  the  stars  also  appear  as  near  as  we  please.' 
This,  however,  to  anyone  who  uses  telescopes,  reads  unconvincingly, 
and  rather  too  much  like  the  sort  of  stuff  that  the  hopeful  examination 
candidate  serves  up,  from  dim  recollection,  to  the  examiner,  who 
skips  it. 

The  invention  of  the  first  Telescope  is  ascribed  to  Hans  Lippershey, 

488 


Fia.  244. 


§611]  OPTICAL   INSTRUMENTS 


489 


spectacle-maker  of  Middelburg,  in  Walcheren,  1608.  It  is  related 
that  his  children,  playing  with  odd  lenses,  discovered  a  combination 
which  brought  distant  objects  closer.  Now,  spectacle  lenses  are 
of  longish  focal  length,  and  children's  arms  are  short ;  they  cannot 
have  held  a  2-ft.  lens  and  a  1-ft.  lens  3  ft.  apart,  but  they  might 
hold  a  1-ft.  short-sight  lens  a  foot  behind  the  2-ft.  long-sight. 

Galileo,  who  had  been  greatly  interested  in  a  new  star  in  Sagit- 
tarius in  1604,  heard  of  the  discovery  in  June  1609,  tried  it  out,  and 
speedily  mounted  lenses  in  a  leaden  tube  to  make  an  instrument 
magnifying  three  times.  He  made  and  presented  one  magnifying 
8  diam.  to  the  assembled  Venetian  Senate  ;  then,  settling  in  Florence, 
and  acquiring  skill  in  the  grinding  and  polishing  of  plano-concave 
lenses,  he  made  many  hundreds  of  occhiali  magnifying  up  to  33 
diam.  (really  the  maximum  useful  in  this  type),  very  shortly  dis- 
covering the  lunar  mountains,  sunspots,  the  stellate  nature  of  the 
Milky  Way,  and  the  four  satellites  of  Jupiter,  the  '  Medicean  Stars.* 

These  were  the  first  undoubtedly  new  things  ever  seen  in  those 
heavens  the  incorruptibility  of  which  was  an  Aristotelian  axiom, 
and  they  provided  him  with  a  little  working  model  of  the  banned 
Copernican  System,  of  the  revolution  of  the  planets  round  the  Sun, 
of  which  he  was  the  great  protagonist. 

Although  his  opponents  succeeded,  upon  a  legal  fiction,  in  com- 
pelling his  formal  recantation  of  this  theory,  the  nominal  sentence 
passed  upon  him,  for  disobedience,  never  received  papal  ratification  ; 
and  the  true  verdict  of  the  Roman  Church  is  better  seen  in  the  long 
cupola-crowned  wall  of  the  Vatican  Observatory,  which  shuts  in 
the  view  behind  the  great  dome  of  Michel  Angelo. 

Trifling  as  was  his  instrumental  power,  it  was  many  times  more 
than  the  keenest  eyesight,  and  could  put  an  end  in  a  moment  to 
discussions  such  as  had  dragged  on  for  centuries  about,  say,  a 
'  lost  Pleiad,'  and  the  claim  is  a  perfectly  fair  one,  that  the  release 
from  thraldom  to  the  a  priori  authority  of  the  Peripatetics,  the 
liberty  of  thought  and  research  and  mental  activity  which  we  now 
all  enjoy  as  a  matter  of  course,  first  fell  upon  mankind  from  heaven 
through  the  narrow  optic  tube  of  Galileo. 

It  seems  pretty  clear  that,  by  1610,  he  had  taken  a  step  further, 
and  by  varying  the  strengths  of  the  lenses  had  been  able  to  show 
'  the  organs  of  motion,  and  of  the  senses  '  in  some  minute  insects. 

In  1611  Kepler  proposed  the  use  of  a  convex  eye-lens  in  a  telescope 
in  order  to  have  a  larger  field ;  this  was  taken  up  in  Holland,  and 
we  shall  see  how  readily  it  must  have  led  to  the  microscope. 
Galileo  returned  to  this  only  in  1624,  and  then  produced  occhialini 
which  '  magnified  50,000  times,  making  a  fly  as  big  as  a  hen,  or  a 
mite  the  size  of  a  pea,  and  to  walk  east  instead  of  west,'  i.e.  inverting 
microscopes  magnifying  3  dozen  diameters. 

L'Accademia  degli  Lincei  introduced  the  names  Telescopium 
and  Microscopium. 

The  object-glasses  of  Galileo's  telescopes  were  only  such  as  the 
spectacle-makers  could  supply,  and  for  150  years  the  telescope 


490 


LIGHT 


[§611 


languished,  first  because  of  the  difficulty  of  getting  good  enough 
glass  for  larger  lenses,  and  then  on  account  of  its  colour  troubles, 
a  way  of  escape  from  which  was  found  only  in  unwieldy  length, 
telescopes  being  slung  as  yards,  from  masts. 

Dollond,  in  London,  began  the  manufacture  of  achromatic  tele- 
scopes in  1760,  but  from  1800  to  1850  Fraunhofer's  firm  at  Munich 
guarded  the  secret  of  the  production  of  satisfactory  flint  glass 
in  pieces  large  enough  for  8-in.  lenses. 

The  reflector  had  no  colour  trouble,  and  in  1789  William  Herschel 
set  up  at  Slough  the  first  really  great  telescope,  with  a  speculum- 
metal  mirror  4  ft.  diam.,  to  be  followed  in  1848  by  Lord  Rosse, 
father    of    the    inventor    of    the    steam-turbine,    with    the    6-ft. 


Fig.  245. 


speculum  now  in  the  Science  Museum.  But  this  beautiful  metal 
tarnishes,  and  must  be  cleaned,  and  this  practically  meant  'refiguring ' 
every  time ;  and  the  renewable  chemically-silvered  glass  mirror 
came  as  a  great  improvement.  Two  of  the  earliest  of  these — a 
3-ft.  set  up  at  Halifax,  and  the  great  5-ft.  made,  and  mounted 
accurately  enough  for  photography,  by  Dr.  Common,  at  Ealing — 
are  now  at  work  all  night  long  under  clearer  skies  than  ours. 

Hooke  (§142)  had  published  his  '  Micrographia  '  in  1665,  figuring 
his  microscope  and  its  illuminator  as  in  Fig.  245  ;  one  like  it,  though 
without  the  lamp,  water-globe,  and  bull's  eye,  dated  1675,  fetched 
£160  in  1925.  There  followed  the  succession  of  beautiful  instru- 
ments of  which  you  can  see  a  wonderful  and  very  typical  selection 
in  the  ^cieBce  Museum ;    all  had  good  eyepieces  with  field  lenses, 


§  612]  OPTICAL   INSTRUMENTS  491 

but  until  Lister's  day,  §  588,  all  had  minute  biconvex  lenses,  strongly 
stopped  down,  for  object  glasses. 

Accordingly,  for  the  very  sufficient  reason  of  §629,  serious 
workers  used  these  lenses  alone  as  simple  microscoixis,  Rol)ert 
Brown,  in  1825,  preferring,  above  all  else,  the  tiny  poUshed  pip 
of  glass.  Fig.  113,  scarcely,  if  at  all,  better  than  those  with  which 
Antony  van  Leeuwenhoek,  whose  shrewd  keen  face  looks  out  on 
you  from  marble  in  the  north  aisle  of  the  Old  Church  at  Delft,  had 
in  1675  observed,  figured,  described,  and  named,  the  bacteria, 
bacilli,  micrococci,  and  other  micro-fungi,  the  further  study  and 
differentiation  of  which  you  will  presently  be  invited  to  regard  as 
a  problem  for  the  highest  powers  of  your  modern  microscope. 

§612.  Apparatus    for    projecting   an    intense    beam    of    light. 

Familiar  to  all  of  us  is  the  ordinary  Bull's  eye.  Fig.  246,  where  L  is 
a  fat  lens  in  a  position  of  minimum  spherical  aberration,  and  FF' 
the  flame,  or  filament,  at  its  principal  focal  distance. 

Since  each  point  of  the  flame  is  approximately  a  principal  focal 
point,  the  light  passes  out  in  many 
'parallel  beams,'  slightly  inclined 
to  one  another.  Really,  in  a  solid 
cone  of  angle  FLF',  but  truncated 
at  the  broad  face  of  lens,  or 
mirror.  This  spreading  causes  it 
to  follow  approximately  the  in- 
verse-square law,  §  475,  at  any  yiq.  246. 
considerable  distances. 

The  ideal  *  parallel  stream  '  of  light,  that  retains  its  brightness 
undimmed  by  distance,  cannot  therefore  be  artificially  produced. 
The  best  we  can  do,  in  the  absence  of  the  mathematical  point  source, 
is  to  use  as  source  a  pinhole  in  a  plate,  intensely  lit  from  behind. 

Now,  the  '  intrinsic  brightness,'  in  candle-power  per  sq.  era.,  of 
various  bright  radiants  is:  Paraffin  flame  1-5;  acetylene  5'5 ; 
oil-gas-mantle,  high  pressure,  for  lighthouse,  50  ;  tungsten  filament 
in  argon-filled  bulb  1200 ;  carbon-arc  crater  17,000 ;  the  summer 
sun  100,000. 

In  no  possible  way  can  these  be  '  concentrated  '  to  any  greater 
brightness,  and  as  a  1/32-in.  pinhole  has  an  area  only  1/200  sq.  cm., 
it  is  plainly  useless.  So  would  be,  in  the  distance,  a  patch  of  light 
no  larger  than  the  lantern :  we  want  some  spreading,  though  not  nuich, 
i.e.  we  want  FF'  small  and  FL  large,  to  make  FLF'  narrow ;  the 
source  should  be  concentrated  and  brilliant,  and  the  lenses  or  mirrors 
distant  from  it. 

Further,  in  Fig.  246,  the  whole  amount  of  light  falling  on  the  lens 
is  a  quite  small  fraction  of  all  that  the  flame  gives  out  in  all  directions. 
This  serves  for  a  pocket-torch  (which  can  also  be  '  spread  *  by 
pushing  L  in)  or  a  railway  signal  lamp  (with  tungsten  filament,  in 
daylight),  but  for  a  lighthouse  it  would  be  better  to  collect  all  the 
light  from  a  whole  hemisphere. 


492  LIGHT  [§  612 

That  means  that,  as  it  is  agreed  also  that  the  working  distance  is 
to  be  large,  a  lighthouse  lantern  is  going  to  be  a  very  big  affair, 
as  you  doubtless  know.  The  lens  L,  Fig.  247,  is  first  of  all  ringed 
round  by  zones  ab,  virtually  the  edges  of  larger  lenses  also  focussed 
on  F,  their  angles  modified  from  the  '  spherical,'  to  reduce  aberration. 
'  Echelon  '  lenses  like  this,  moulded  in  one  piece,  are 
common  on  ships'  lights.  Outside  these  are  con- 
centric rings  of  glass,  of  wide-angled  prism  section, 
§491,  all  most  carefully  angled  and  placed  so  as  to 
catch  and  totally  reflect  the  light  into  one  direction. 

The  East  Goodwin  has  one  of  these  facing  each  way, 
and  revolving  in  30  sec,  the  South  Foreland  has  a 
merry-go-round  of  sixteen  narrow  panels.  Down 
Channel  Alderney  and  Portland  shake  four  fingers  at 
each  other,  having  four  partial  panels  in  each  hemi- 
sphere. The  curvature  of  the  earth  sets  the  distance 
limit  to  lighthouses ;  candle-power  combats  haze. 
Suppose  a  1000-c.p.  lamp  illuminating  the  inner 
Fig.  247.  surface,  6-28  million  sq.  m.,  of  a  kilometre-radius 
hemisphere.  Concentrating  this  outflow  of  light  into 
a  cone  of  angle  3°  means  concentrating  it  into  a  50-m.-diam.  patch 
on  this  hemisphere,  or  6,280,000/2000  sq.  m.  =  3140  times,  more 
than  3  million  effective  c.p.,  and  a  smaller  angle  would  evidently 
produce  a  yet  more  brilliant  and  penetrating  flash. 

These  are  '  dioptric  '  lanterns ;  now  for  '  catoptric  '  lanterns, 
which  depend  on  mirrors. 

Since  the  mirror  has  to  collect  from  a  hemisphere,  more  or  less, 
the  angles  concerned  are  large,  and  compel  the  use  of  a  parabolic 
mirror.  Fig.  229  C,  which  has  no  spherical  aberration. 

But  it  has  a  single-point  focus,  about  which  it  is  very  particular, 
and  the  24-c.p.  filament  of  a  headlamp  bulb  is  by  no  means  a  '  point  ' 
to  its  short-focus  parabolic  reflector.  The  result  of  that,  and  of  the 
imperfections  of  make  of  the  mirror,  can  be  seen  by  looking  at  a 
Headlamp  through  a  smoked  glass,  at  any  convenient  distance  ; 
you  will  see  that  it  gives  nothing  like  the  full  flash  commonly 
supposed,  but  shows  patches  and  wriggles  of  brightness  not  totalling 
a  tenth  of  its  surface ;  but,  then,  nobody  wants  to  waste  light  a 
mile  ahead. 

The  17-in.  focus  silvered-back  glass  paraboloid  mirror  of  a  36-in. 
Searchlight,  Fig.  248,  faces  the  crater,  2/3  in.  diam.,  of  a  150-ampere 
arc  (shown  twice  too  big  in  the  diagram)  of  which  the  thin  negative 
carbon  is  kept  as  well  out  of  the  way  as  may  be.  The  cone  is  only 
2°,  and  the  19,000  c.p.  of  the  arc,  even  allowing  60%  discount  for 
numerous  losses,  concentrates  to  40  million  c.p. 

Parabolic  mirrors  are  difficult  to  make  with  any  accuracy,  whereas 
spherical  surfaces  are  easily  made  true,  §  544.  Light  has  further 
to  go  to  reach  the  outlying  parabolic  surface  than  a  spherical  one  : 
in  the  '  Mangin '  mirror  the  same  effect  is  obtained  by  packing  glass 
(in  which  it  travels  slower)  in  increasing  thickness  towards  the  edge, 


§  613]  OPTICAL   INSTRUMENTS  493 

between  two  spherical  surfaces,  Fig.  249,  and  up  to  IT-in.  diameter 
these  mirrors  are  very  accurate  indeed,  far  more  so  than  paraboloids  : 
they  are  used  for  signalling  projectors,  even  a  6-in.  one  works  up 
to  250,000  c.p.  from  a  tungsten  bulb. 

Flood-lights  make  small  attempt  to  focus. 

An  Amber  Disc  does  not  enable  a  light  to  penetrate  fog  any 
farther  :  what  it  does  is  to  cut  off  the  bluer  shorter  waves  which 
are  being  scattered  in  the  fog  inversely  as  the  fourth  power  of  their 
wave-lengths,  §  568,  and  are  thereby  making  a  cloud  of  light  through 


Fig.  248.  Fio.  240. 

which  the  driver  cannot  see — unless  he  wears  amber  glasses,  which 
have  just  the  same  effect. 

Great  searchlight  mirrors,  turned  towards  the  Sun,  have  been 
used  to  focus  his  heat  into  a  little  '  Vacuum  Furnace,'  wherein  at 
an  estimated  5000°  (1500"  hotter  than  the  arc,  our  best  artificial 
effort)  all  metals  hastily  gasify,  with  the  production  of  spectra 
uncontaminated  by  lines  of  any  other  substances. 

§  613.  The  photographic  camera  lens.  A  convex  lens  projects 
a  real  inverted  image  on  a  plate.  In  Fig.  204,  etc.,  the  image  of 
a  straight  line  has  been  drawn  as  a  straight  line  parallel  to  it,  but 
if  the  diagram  be  constructed  carefully,  for  three  or  four  distinct 
points  of  the  object,  it  will  be  found  that  the  image  is  actually  curved ; 
the  image  of  a  flat  sheet  would  be  in  focus  on  the  inside  of  a  saucer. 

This  difficulty  was  overcome  in  the  Landscape  Lens,  a  meniscus, 
hollow  towards  the  view  and  with  a  limiting  circular  hole  or  *  stop  ' 
about  0-15/  in  front  of  it.  Fig.  250.  This  gives  a  flat  image,  in  focus 
all  over  a  good- sized  plate,  but  distorted  so  that  a  square  has 
bulging  sides.  Turned  the  other  way  round,  it  makes  a  square 
'  cushion  shaped.'  Hence  the  symmetrical  Rapid  Rectilinear, 
in  which  a  pair  of  meniscus  lenses  face  each  other  (front  lens 
dotted.  Fig.  250),  with  a  stop  midway  between,  and  give  a  flat 
undistorted  image. 

All  photographic  lenses  are  achromatized,  §  590,  but  as  the  film, 
quite  unlike  the  eye,  is  blind  to  red  and  very  sensitive  to  violet, 


494  LIGHT  [§  613 

the  spectrum  colours  chosen  in  calculation  are  sodium  yellow,  D, 
and  deep  blue,  G,  these  bringing  visual  focus  and  '  actinic  '  focus 
into  best  coincidence.  To  adapt  visual  telescopes  and  microscopes 
for  photography,  orthochromatic  film  and  a  yellow  screen  must  be 
used,  to  kill  the  violet  which  their  lenses  let  run  wild. 

The  Stop,  nowadays  an  '  iris  diaphragm  '  of  variable  aperture, 
is  an  essential  part  of  the  complete  lens,  it  removes  the  haze  in 
which  '  spherical  aberration  '  would  otherwise  envelop  the  picture. 
*  Stopping  down  '  also  reduces  focussing  difficulties  for  objects  at 
all  sorts  of  distances  away.     For  suppose  the  cone  of  light  from  the  j 

lens  is  not  coming   to   a  focus 
,^y-HF^p-----_______^^  until  F,  Fig.  250;    evidently  a 

(u. J^-r1t__^^^^^^L=:^^^^=^'       smaller    stop    makes    the    cone 

*^-i — *^ "^  narrower,  and  the  circular  patch 

in  which  it  strikes  the  plate  (the 
Fig.  2o0.  cross  line)  smaller,  and  more  like 

a  true  focus. 

Unfortunately,  cutting  down  the  size  of  the  window  in  this 
way  necessitates  a  lengthened  exposure.  The  diameter  of  the 
aperture  is  stated  as  a  fraction  of  the  focal  distance  (which  is  usually 
nearly  enough  distance  of  plate),  e.g.  f/S,  f/ll,  etc.  The  light  it 
transmits  from  a  given  outside  brightness  is,  of  course,  proportional 
to  its  area,  or  to  (//ll)^,  etc.  Hence  the  illumination  on  the  plate, 
c.p.  -^  (P,  §475,  is  (//ll)^  ^/^  etc.,  and  the  exposure  to  catch  a 
given  quantity  of  light  is  inversely  proportional  to  this,  i.e.  directly 
proportional  to  (11)^,  etc.,  so  that  the  run  of  stops //5-6,  8,  11,  16,  22, 
32  demands  exposures  1,  2,  4,  8,  16,  32,  while //2- 5  is  10  times  as 
fast  as//8,  and//l-8  20  times. 

These  last  two  amazing  '  apertures  '  are  usable  in  small  sizes, 
and  there  only,  on  account  of  the  image-flattening  action  described 
in  §  516  ;    how  successful  they  are  then  you  see  in  the  cinema. 

This  action,  and  a  narrow  stop,  not  over  //li, 
account  also  for  the  popular  cheap  small  cameras 
of  completely  fixed  focus. 

Modern  *  anastigmat '  lenses  are  of  many  types, 
mostly  developed  from  Dennis  Taylor's,  Fig.  251, 
where  three  different  glasses  are  used  instead  of 
the  two  older  ones,  §§  590,  591  ;  they  have  done 
away  with  streaky  astigmatism  in  the  out-field, 
and    widened    and    flattened  it,   have    improved  Fig.  251. 

achromatism,  and  have  increased  speed  ten-fold. 

For  Telephoto  Lenses  see  §  627. 

§614.  The  Epidiascope  is  the  modern  Magic  Lantern.  In  the 
Epi- (upon) -scope  part  of  it  an  opaque  object  lies  flat  in  the  fierce 
light  (and,  unfortunately,  heat),  conserved  and  concentrated,  of 
500-watt  lamps  LL,  Fig.  252.  It  therefore  radiates  to  an  extent 
in  all  directions  represented  by  the  sphere  of  Fig.  174  (reproduced 
here).     Looking  down  upon  it,  at  the  top  of  this  sphere  of  radiation, 


614] 


OPTICAL   INSTRUMENTS 


495 


IS  a  very  good  and  expensive  anastigmat  camera  lens,  of  aperture 
//4,  in  Its  focussing  sleeve  (anything  beyond  that  is  hopelessly  costly), 
which  therefore  takes  in  the  radiation  filling  the  inverted  coiie  shown, 
the  base  of  which  is  1/4  the  diameter  of  the  sphere  ;  and  sends  it] 
by  way  of  a  45°-mirror,  to  form  an  image  on  the  distant  screen. 

Now,  the  volume  of  this  cone  =  base  x  1/3  height  =  7r(r/4)«  x 
2r/3,  or  1 /32nd  the  volume  of  the  sphere ;  and  all  the  rest  of  the 
light  is  utterly  wasted.  It  is  therefore  quite  a  feat  to  get  even 
1  candle-foot  brightness  of  lighting  on  a  sizeable  screen,  instead  of 
the  4  or  6  common  in  the  cinema,  and  this  severely  limits  the  utility 
of  the  episcope  in  daytime  demonstrations.  For  the  brightness 
outdoors  may  be  even  10,000  candle-feet ;   and  it  takes  a  quarter 


Fig.  252. 


of  an  hour  in  darkness  for  the  eye  to  attain  its  customary  hundred- 
fold greater  sensitivity  of  the  evening,  §  478. 

In  the  Dia-( through) -scope  on  the  right,  when  the  reflector- 
shutter  is  turned  out  of  the  way,  the  lamplight  shines  through  a 
condenser,  usually  consisting  of  a  pair  of  4J-in.-diam.  cheap  plano- 
convex lenses  back  to  back  (minimizing  spherical  aberration  by 
spreading  the  refraction  over  four  steps),  and  forms  a  rough  image  of 
the  radiant  inside  the  projecting  lens,  which  is  very  much  smaller 
and  cheaper  than  the  other,  but  passes  abundance  of  light  from  the 
direct  glare  of  the  close-wound  filament.  This  lens  therefore  sees 
the  slide  S  full  of  light  all  over,  and  projects  its  image  on  the  screen 
at  the  distant  conjugate  focus. 

The  '  home  '  Cinema  Projector  is  very  much  the  same ;  the 
full-size  machine  differs  in  detail.  Its  lamp  is  an  SO-ampere  arc. 
arranged,  with  a  small  mirror,  searchlight  fashion.  Fig.  248,  and 
what  with  its  large  crater  and  the  aberrations  of  mirror  and  con- 
denser, a  neck  of  uniform  brilliant  light,  4  cm.  diam.,  bigger  than 
the  picture,  forms  about  1  ft.  to  the  right  of  the  condenser.  Here 
the  film  is  run,  with  the  5-in.  focus  projection  lens  at  that  equivalent 
distance  beyond  :  on  account  of  numerous  losses,  only  an  eighth  of 
the  light  produced  ultimately  reaches  the  familiar  screen. 


496  LIGHT  [§615 

§  615.  The  Telescope.  We  are  going  to  develop  the  Telescope 
not  only  for  its  own  sake  as  an  instrument  of  inestimable  value,  in 
its  many  varieties,  but  also  because  by  a  quick  transformation  we 
shall  convert  it  into  our  great  ally  the  Microscope,  many  properties 
of  which  are  more  easily  understood  by  the  way  we  shall  take, 
lengthy  though  it  be. 

The  old  notion  of  a  Telescope  as  a  spear  piercing  the  depths  of  the 
skies  is  long  since  abandoned,  and  now  it  is  regarded  as  a  basin, 
or  a  funnel,  with  which  to  gather  up  as  much  as  we  can  of  the  light 
which  pours  into  it  from  far-distant  sources. 

The  greatest  basin  of  them  all  is  on  the  summit  of  Mount  Wilson, 
a  6000-ft.  eminence  overlooking  the  residential  plain  of  California, 
wherein  lies  Hollywood,  a  place  noted  for  stars  of  its  own. 

It  is  a  5-ton  disc  of  glass,  100  in.  in  diameter  and  I  ft.  thick, 
hollowed  out,  about  1  in.  deep  in  the  middle,  to  a  concave  mirror 
of  50  ft.  focal  length.  Not  a  spherical  curve  ;  that  stage  was  care- 
fully gone  through  ;  but,  then,  by  a  couple  of  years'  cautious  grinding 
and  polishing  and  proving,  the  spherical  cup  was  deepened  in  the 
middle,  or  flattened  towards  the  edge,  nearly  two  thousandths  of 
an  inch,  to  form  as  perfect  a  paraboloid  as  skill  and  patience  could 
compass,  the  true  curve  to  collect  at  one  focus  all  the  light  from 
a  star,  §582  (its  actual  semi-diameter,  to  scale,  being  1/3  of  the 
distance  from  axis  to  first  ray  above  it  in  Fig.  229  C).  Then  it  took 
the  bumpy  mountain  road,  up  which  we  will  follow  it  on  '  the  stage,' 
and  now  it  lies  on  its  back  at  the  lower  end  of  a  great  cradled  skeleton 
tube,  facing  the  stars  of  the  clear  Calif ornian  sky,  except  when, 
twice  a  year,  it  gets  a  night  off,  to  have  its  face  washed  with  nitric 
acid,  and  resilvered  chemically  from  a  solution  of  ammonio- 
nitrate  of  silver  and  sugar.  Presently  they  will  give  it  a  more 
permanent  aluminium  coating  by  the  new  process. 

It  is  the  type  of  all  telescope  mirrors,  it  is  the  largest  in  use  yet, 
it  cost  a  lot,  it  is  good  enough  for  us  to  start  on.  You  can  see  its 
great  ancestor  near  the  door  in  the  Science  Museum  at  South 
Kensington — Lord  Rosse's  6-ft.  speculum-metal  mirror. 

Come  up  with  us  into  the  dark  vault  of  its  100-ft.  dome,  and  stand 
to  look  at  what  it  makes  of  the  Moon.  '  Pull  the  eyepiece  right 
out,'  says  the  Director,  '  you'll  get  a  fine  view.'  Before  your  eye, 
close  at  hand,  hangs  a  youngster's  football,  but  a  ball  of  brightest 
gleaming  silver,  carven  and  chased  all  over  with  the  most  intricate 
and  beautiful  patterns.  '  Hey,  let's  look  at  this,'  and  out  comes 
your  pocket-lens,  and  you  scrutinize  it  all  over,  bit  by  bit,  your 
lens  a  more  mobile  weapon  than  the  proper  eyepiece — which,  after 
all,  is  nothing  more. 

So  the  chiefest  Telescope  in  all  the  world  is  just  a  Concave  Mirror, 
forming  a  real  image  in  its  principal  focal  plane,  and  you  can  look 
at  it  with  or  without  a  magnifier,  as  you  choose.  Or  you  can  lay 
a  photographic  plate  flat  on  it  and  expose  it,  half  a  second  perhaps 
for  the  moon,  half  the  night  for  a  faint  little  nebula  ;  develop  and 
fix,  magnify  or  enlarge  it  how  you  like  ;  print  and  distribute  copies, 
and  let  who  will  examine  them  at  leisure. 


§615] 


OPTICAL   INSTRUMENTS 


497 


Only  one  little  complication  one  may  mention,  a  small  flat  mirror 
at  45  turns  the  point  of  the  focussed  cone  of  light  at  a  right  ancle 
so  as  to  reach  the  side  of  the  tube,  and  one  looks  sidewav^  instead 
of  straight  down  in  front.  ' 

Newton  did  this,  in  a  telescope  he  made  ;  if  he  hadn't,  his  head 
would  have  obstructed  the  whole  of  the  light ;  here  at  Mt  Wilson 
one  would  have  to  sprawl  out  over  that  yawning  50-ft.  pit  and  my 
friends  would  not  thank  you  if  you  fell  in  and  scratched  their  mirror 
So  the  whole,  Fig.  253,  N,  is  called  a  Newtonian  Reflector. 
Mirror-making,  in  moderate  sizes,  has  long  been  a  pastime  of 
men  of  astronomical  tastes  ;  it  is  not 
dreadfully  difficult  once  you  know  how, 
it  is  a  minor  craze  in  the  U.S. ;  but 
Mt.  Wilson  is  essaying  something 
serious,  a  200-in. 

This  has  been  cast  of  20  tons  of  a 
'  super-pyrex '  glass,  of  very  small 
expansibility,  is  cooling  slowly  month 
by  month,  so  that  temperature  strains 
may  soak  out,  and  should  be  ready  for 
inspection  about  the  same  time  as  this 
book.  This  small  expansibihty  is  in- 
valuable, for  the  drawback  of  great 
mirrors  has  always  been  that  the  slight- 
est local  warming  swells  and  bulges 
the  glass,  and  the  500,000th-inch  per- 
fection of  figure  is  lost,  and  lost  for 
hours  in  these  great  sizes,  §241.  Mt. 
Wilson  started  a  thermostat,  but  one 
had  to  forecast  on  Monday  morning 
what  would  be  the  right  temperature  for 
Thursday  night,  and  keep  the  great 
mirror  under  cover  meanwhile,  so  that  particular  form  of  prophecy 
was  soon  abandoned. 

These  Calif ornian  hilltops  are  noted  for  a  very  steady  temperature 
day  and  night ;  the  mirror,  close  hidden  by  day,  works  all  night 
long ;  but  temperatures  do  vary,  fortunately  the  bulk  of  the  work 
is  spectrographic,  where,  if  the  star  can  be  kept  on  the  slit  at  all,  it 
is  good  enough  ;  only  once  in  a  while  can  the  observer  snatch  that 
perfect  hour  or  two  when  the  giant  is  in  his  best  temper  and  all's 
right  with  the  sky. 

The  '  tube  '  of  a  great  telescope  need  not  be  light-tight,  as  it  is 
always  used  by  night,  and  these  Reflectors  have  only  a  light  skeleton 
framework  to  carry  the  little  mirror.  They  are  therefore  balanced 
quite  near  the  heavy  great  mirror,  and  while  the  long  upper  end  is 
sweeping  round  the  dome  after  the  revolving  stars,  the  lower  end 
moves  but  little,  and  is  a  much  handier  place  to  observe  at,  or 
attach  heavy  spectrographs,  etc.  Most  of  them  therefore  substitute 
for  the  little  Newtonian  *  flat '  a  '  Cassegrain  '  convex  mirror,  a8  in 
Fig.  253  C,  and  this  reflects  the  light  straight  down  the  tube  again. 


Fio.  253. 


4§S  LIGHT  [§  616 

its  slight  curvature  being  sufficient  to  prevent  it  reaching  its  focus 
until  I,  where  it  forms  an  image  as  much  larger  than  the  original 
at  F,  as  I  is  farther  away  than  F  (F  being  virtual  object  and  I  real 
image  for  a  convex  mirror ;  see  Question  19,  below).  I  is  then 
caught  on  a  plate,  or  eyepieced,  or  spectrographed,  etc.,  as  required. 

§  616.  But  jump  with  me  to  another  friendly  hilltop,  300  miles 
north — Mount  Hamilton,  where  has  stood  the  Lick  telescope  since 
1888,  of  the  type  you  have  always  known,  with  a  great  clear  glass 
eye,  3  ft.  across,  up  aloft,  while  you  stand  below  and  gaze  skyward  ; 
a  Refractor,  the  '  funnel '  as  opposed  to  the  '  basin.'  For  long  the 
largest  lens  in  the  world,  it  is  still  nearly  so,  and  claims,  with  all 
its  immense  advantages  of  almost  perfect  site  and  climate,  to  be 
the  keenest  seer  of  them  all. 

The  lens  has  troubles,  but  they  are  reduced  ;  it  is  so  much  less 
bulky  that  it  can  respond  to  temperature  changes  in  l/20th  the  time, 
and,  anyway,  equal  distortion  spoils  the  working  of  a  lens  only  one- 
eighth  as  much  as  a  mirror  of  the  same  size.  Unfortunately,  no 
maker  has  yet  succeeded  with  a  lens  much  larger  than  this  :  one 
special  trouble  is  that  it  can  be  held  only  by  the  edges,  and  sags  in  the 
middle  as  it  is  turned  about,  the  other  is  that  it  must  be  of  flaw- 
less glass  throughout. 

And,  as  you  know,  there  must  be  two  glasses,  because  all  glass 
refracts  different  colours  differently,  and  a  single  convergent  lens 
brings  the  red  and  blue  of  a  star  to  different  foci,  §  588,  Fig.  232  ; 
if  you  call  B  the  focus,  there  is  a  red  ring  round  it ;  if  R,  the  red 
has  come  home,  but  now  the  blue  has  spread  out  into  a  haze.  You 
see  this  when,  in  the  laboratory,  you  start  to  rig  up  a  simple 
telescope,  using  an  ordinary  lens  ;  the  firework  display  strikes  your 
eye  almost  before  what  you  set  out  to  see. 

One  way  of  getting  rid  of  this  you  can  see  in  your  great -great- 
grandfather's heirloom  of  Trafalgar  :  its  glass  eye  is  stopped  down 
to  a  little  hole  not  much  bigger  than  your  own  pupil  :  try  it  against 
your  father's,  which  has  a  much  wider  open  achromatic  lens — both 
by  day  and  by  night,  on  something  really  difficult,  not  just  brick 
walls  and  neighbours'  windows — and  you  will  have  an  intelligent 
anticipation  of  much  that  I  am  going  to  say. 

And  never  use  a  telescope  through  a  pane  of  glass,  excepting  only 
the  very  best  plate  ;  for  common  window  glass  is  far  from  flat  and 
true,  and  ruins  the  seeing  of  the  big  eye. 

To  correct  these  colours,  then,  you  must  have  a  concave  lens  of 
dispersive  flint  glass,  behind  the  convex  crown  the  rounded  face  of 
which  looks  outwards,  Fig.  236,  top,  and  then  you  have  an  achromatic 
object-glass,  of  two  flawless  lenses,  free  from  all  strains,  perfectly 
shaped  and  polished  on  all  four  surfaces — amateur  telescope- 
makers  seldom  tackle  lens  making,  §  544. 

Horatio  Nelson,  in  Norfolk,  and  the  achromatic  telescope,  in 
Essex,  came  into  the  world  almost  together  ;  his  historic  glass 
changed  hands  in  1933  for  £360. 


§617] 


OPTICAL   INSTRUMENTS 


409 

The  colourless  image  in  the  focal  plane  of  the  achromatic  object- 
glass  can  be  treated  exactly  as  you  did  the  mirror's  image  ;  excepting 
only  that,  as  no  two-glass  lens  can  be  completely  achromatic,  §  591, 
the  straying  violet,  seldom  noticeable  by  the  eye,  is  smothered  by 
a  yellow  glass  for  ordinary  purposes  of  photography,  using  ortho- 
chromatic  plates.  For  specialized  photography,  numerous  dodges 
have  been  developed. 

§  617.  Fig.  254,  I  (with  which  compare  Fig.  203),  shows  plane 
waves  of  light  X  arriving  from  the  top  of  an  object  very  far  away 
to  the  left,  meeting  the  object-glass  G,  being  converged  by  it  to 


Fig.  254. 


a  point  X  in  its  principal  focal  plane,  spreading  from  that  point 
again  to  encounter  eyepiece  E,  and  being  reconverted  by  it  into 
plane  waves  which  enter  the  eye.  Y  is  a  similar  train  of  wavea 
coming  from  the  bottom  of  the  distant  object. 

In  II  a  perforated  screen  has  been  put  into  place,  and  you  see  only 
the  short  bits  of  wave  passing  through  its  holes  :  and  the  Y  train 
I  have  cut  down  to  a  single  stream. 

In  III  these  streams  are  narrowed  still  more,  and  packed  tight 
with  all  their  waves,  really  broad  processions  about  500  wave-lengths 
wide,  singled  out  from  all  the  horde  to  yield  us  a  clear  plan  of  the 
manoeuvres.  Plane  waves  appear,  of  course,  as  parallel  lines, 
'  parallel  light,'  and  the  telescope  taking  in  '  streams  of  parallel 
light '  from  the  distant  object  is  naturally  arranged  to  give  out 


600  LIGHT  t§6l7 

streams  of  parallel  light  to  the  eye;  which  remains  at  rest  for  distant 
vision ;  for  it  would  be  silly  to  have  to  waste  time  and  effort  re- 
focussing  your  eye  every  time  you  put  your  telescope  to  it  or  took 
it  away. 

Therefore,  just  as  the  Real  Inverted  Image  xy  lies  at  principal 
focal  distance  from  G,  so  it  must  lie  at  principal  focal  distance  from 
E  ;  the  two  focal  planes  coincide  ;  for  then  light  from  x  or  y  will 
become  '  parallel '  passing  through  E. 

But  parallel  to  what  ? 

You  know  that  the  stream  that  passes  through  the  Optical  Centre 
of  a  thin  lens,  §  504,  goes  straight  on,  quite  undeviated.  Draw 
therefore  x  E  and  y  E,  the  short  dotted  lines,  and  fill  in  your  two 
bundles  of  rays  parallel  to  their  continuations  through  E.  You  see 
that  if  G  were  a  bit  broader,  the  dotted  lines  would  be  included  in 
the  actual  bundles ;  as  drawn,  they  are  only  potentially  so ;  that 
doesn't  matter. 

The  diagram  shows  that  it  pays  to  keep  your  eye  back  a  bit  from 
E,  at  the  eye  ring  or  '  Ramsden  circle,'  the  bright  round  spot  you 
see  behind  any  telescope  or  microscope  by  drawing  your  head  back 
a  foot ;  if  not,  it  may  fail  to  catch  all  the  light,  you  won't  see  the 
whole  field  of  view. 

Recollect,  please,  that  object-glass  G  is  a  large  weak  achromatic 
lens  costing  you  £2  an  ounce  (and  glass  is  heavy),  while  E  is  just 
a  little  magnifying -lens  no  bigger  than  actually  printed,  or  more 
usually  one  of  a  lavish  supply  of  eyepieces  of  different  strengths, 
G's  servants,  just  microscope  eyepieces,  costing  ten  shillings  apiece. 

So  Fig.  254  III  is  all  you  need  draw  in  an  exam  ;  but  if  any  be 
content  to  '  learn '  '  telescope '  as  just  that,  then  next  time  he  goes 
home  and  whistles  up  the  dog,  may  its  dry  bones  jump  up  and  bite 
him.     Read  on. 

§  618.  We  are  home  again,  so  let  us  take  a  walk  with  a  chum, 
along  the  hills,  where  the  field-glasses  may  be  worth  while  carrying. 
Down  in  the  vale  is  a  farmyard,  and  you  spy  poultry  in  it,  and 
what  looks  like  straw ;  but  the  straw  seems  to  be  moving. 
'  Glasses,  please.  Ah  !  they're  little  chicks  ;  the  Buff  has — ^nine, 
and  the  Light  Sussex — no,  hers  are  too  close  packed,  can't  pick 
them  out.'  Kiddy  brother  bags  the  glasses.  '  Ooh  !  those  old 
hens  look  as  big  as  ostriches  ' ;  whereat  you  tell  him  off,  and  resume 
possession. 

You  have  already  put  more  sound  science  into  that  than  there  is 
in  any  ordinary  students'  text-book  of  optics. 

Firstly,  you  dismiss  the  youngster's  observation  as  worthless  : 
you  knew  they  were  hens,  you  saw  that  without  a  telescope  ;  you 
don't  care  how  big  they  look,  his  statement  is  merely  misleading — 
you  don't  care  in  the  slightest  how  much  the  glasses  magnify  : 
you  do  NOT  value  their  Magnifying  Power. 

You  do  care  that  they  showed  you  new  details  which  you  could 


§619]  OPTICAL   INSTRUMENTS  501 

not  see  without  their  great  eyes.  What  looked  like  continuous  straw 
breaks  up  into  separate  lumps  which  you  surmise,  by  their  movement, 
are  chickens. 

Where  they  are  far  between,  you  can  count  them,  and  do  so. 

Where  they  are  closer  together,  analogy  still  suggests  that  they 
are  chickens,  but  you  are  definitely  unable  to  distinguish  them 
individually,  and  to  count  them. 

The  black  markings  of  the  white  hen  are  sufficiently  distinguish- 
able for  you  to  call  her  a  Light  Sussex. 

The  Buff's  legs  are  badly  lighted,  and  you  cannot  see  whether  they 
are  yellow  or  pink — a  slight  contrast,  anyway — so  you  don't  commit 
yourself  as  to  her  breed. 

You  do  care  for  the  power  of  making  out  detail  which  the 
glasses  confer  on  you,  you  value  them  for  their  Resolving  Power. 

The  sun  sets,  and  in  the  gathering  dusk  you  start  a  star-hunt, 
to  enliven  the  long  tramp  home.  Little  brother  quite  likely  comes 
into  his  own  now ;  but  you  don't  play  fair,  the  great  eyes  of  your 
glasses  pick  up  very  many  stars  too  faint  for  his  keen  sight,  though 
he  will  try  to  beat  you  by  looking  askew,  for  the  outer  parts  of  the 
retina  are  so  enormously  more  sensitive  to  faint  starlight . 

You  value  the  glasses  for  their  Light-gathering  Power. 

Darkness  deepens,  all  but  concealing  some  small  moving  objects 
on  the  hillside.  Glasses  again — rabbits ;  Night  Glasses,  Captain's 
glasses  for  use  at  sea,  or  for  poaching. 

Now  let  us  set  to  work  to  unravel  these  intertwined  mysteries, 
starting  with  these  last  two,  and  taking  them  together,  though  they 
differ  as  between  looking  at  the  fireworks  and  at  the  men  letting 
them  off. 

§  619.  In  the  dusk,  the  iris  of  your  eye  relaxes  and  opens  wide, 
and  leaves  the  pupil  5  mm.  diam.  (or  even  8  mm.  in  youth,  but  not 
proportionately  effective).  Like  a  cat's  eye,  it  is  opening  to  gather 
all  the  light  it  can,  for  so  cats  wander  in  the  moonlight ;  they  stay 
at  home  on  dark  nights. 

But,  by  that  same  token,  the  great  clear  eyes  of  your  field-glasses 
collect  much  more  ;  suppose  they  are  30  mm.  diam.,  this  is  (30  5)*, 
or  36  times  the  area  of  your  own,  and  stars  36  times  fainter  ought  to 
pop  into  view.  Or  rather,  allowing  a  discount  for  reflections  at 
all  the  lens  surfaces  in  them,  and  for  the  imperfect  transi>arency 
of  the  glass,  probably  quite  20  times.  That  is,  provided  that  your 
glasses  are  so  constructed  that  the  light  the  big  eyes  collect  really  is 
poured  into  your  own,  the  *  Star  Glass  condition,'  and  not  spilt 
around  them  outside  :  no  one  would  pretend  that  merely  putting 
on  a  pair  of  large  spectacles  is  going  to  multiply  the  amount  of  light 
entering  his  eye. 

It  does  not  matter  in  how  narrow  a  stream  this  optical  funnel 
delivers  its  light,  so  long  as  it  all  streams  into  your  pupil ;  how  this 
is  done  we  will  see  later. 


602  LIGHT  [§619 

Contrariwise,  for  Night  Glasses  it  is  desirable  that  the  whole 
surface  of  your  wide-open  pupils  be  kept  illuminated  ;  i.e.  that  the 
emergent  stream  of  light  be  at  least  5  mm.  diam.  For  this  stream 
cannot  possess  greater  brilliance,  seen  in  any  direction,  than  the 
entering  light — it  would  mean  heating  the  sun  hotter  if  it  did, 
§  974 — and  therefore  we  must  not  pull  down  a  blind  over  any  part 
of  the  pupil.  That  means  that,  in  all  probability,  part  of  the  stream 
will  overflow  the  edge  of  the  pupil  and  be  wasted,  the  best  Night 
Glasses  will  not  be  the  best  Star  Glasses. 

Night  Glasses  do  confer  an  amazing  power  of  seeing  in  a  bad  light — 
things  actually  look  brighter.  This  they  cannot  be,  it  is  the  mass 
action  of  the  Zarg^er-image-not-robbed-of -its-brightness,  just  as  it 
is  easier  to  see  a  lost  collar  under  the  bed  than  a  lost  collar- 
stud.  This  mass  action  is  great,  and  modern  binoculars  not  even 
half-filling  the  night-glass  condition  are  very  serviceable  in  the 
dusk. 

§  620.  Take  now  Resolving  Power.  When  the  eye  is  intently 
studying  detail,  as  it  can  do  only  in  a  good  light,  it  contracts  to 
2  mm.  diam.  You  hold  to  it  an  instrument  made  as  perfectly  as 
man  with  skill  of  hand  and  eye  and  brain  has  learnt  to  make  it, 
contrived,  in  the  '  star-glass  '  condition,  so  as  to  pour  all  its  light 
into  your  eye,  and  it  enables  you  to  look  out  on  the  world  with  eyes 
much  larger  than  your  own — with  a  30-mm.  lens  (30/2)2  =  225 
times  the  size — and  every  little  individual  pupil  area  has  as  much 
right  to  cry  '  I  spy  '  as  you  had,  for  no  harm  whatever  has  been  done 
to  the  light  with  which  it  faithfully  reports  to  you. 

Your  eye  looks  out  through  225  eyes  as  good  as  itself,  therefore 
you  should  see  225  times  as  much  detail  in  whatever  you  examine — 
provided  the  detail  is  there  to  see.  Fifteen  black  streaks  on  the 
neck  of  the  Light  Sussex,  where  before  you  saw  only  a  grey  blur  : 
she  isn't  marked  the  other  way,  I  ought  to  have  chosen  a  speckled 
breed. 

My  160  mm.  is  a  very  minor  thing  in  telescopes,  but  it  should 
show  (160/2)2  =  6400  times  as  many  details  on  the  Moon  as  the 
unaided  eye  can  descry,  and  in  fact  one  can  take  many  an  evening 
stroll  among  her  landscapes  without  ever  a  lack  of  fresh  points 
of  interest. 

With  telescope,  and  microscope,  and  always  trying  to  see  your 
best,  you  will  gradually  learn  what  Resolving  Power  means ;  we 
shall  return  to  it  more  precisely  later. 

§621.  What  of  the  'also  ran,'  Magnifying  Power?  He  is  the 
secretary  of  the  firm,  and  holds  the  keys  of  the  situation. 

What  do  we  mean  by  magnifying  power  of  a  telescope  ?  We  don't 
use  it  to  magnify  a  thing  larger  than  life,  we  don't  want  to  see  a 
horse  as  big  as  the  wooden  horse  of  Troy,  we  merely  want  the 
image  to  bulk  bigger  to  the  eye  than  a  horse  naturally  does  when 
half-way  down  the  course.     Instead  of  occupying  a  visual  angle 


§621]  OPTICAL   INSTRUMENTS  6O3 

a.  Fig  255,  of  our  total  field  of  view,  we  get  him  to  occupy  visual 
angle  A,  just  as  he  will  when  he  comes  closer,  and  then  the  Magnifvina 
Poiver  IS  the  Angular  Magnification  A/a.  ^    Jif    v 

Look  at  Fig.  254  ;  without  the  telescope  the  eye  would  be  viewimr 
the   object  under   the  natural  ^ 

angle  yGx,  which  the  two 
streams  of  light  make — any- 
where on  the  page,  it  doesn't 
matter,  for  the  object  is  very 
distant ;  with  the  telescope  it 
sees  the  image   occupying  the  ^'°-  ^^^• 

angle  yE-r. 

Gyx  and  ^yx  are  narrow  triangles  based  on  yx ;  the  angles  at 
G  and  E  are  simply  proportional  to  their  shortness,  i.e.  inversely 
to  their  length 

<^^9leatE  _  ^^  y  •  p  _  01  _  F  _  focal  length  of  object  glass 
angle atQ-  ^^^mjying I^ower  _  ^^  _  ^  _  ^ocaHeng^A  0/ eye;>i€C€  * 

=  1//-^  IfF  =  strength  of  e. p. /strength  of  o.g. 

If  you  turn  the  telescope  wrong  way  round,  it  remains  a  telescope, 
and  minifies  in  the  same  ratio  ;  you  know  that  quite  well. 

Fig.  254  IV  shows  a  shorter  focus  eyepiece,  magnifying  twice  as 
much,  you  see  the  dotted  lines  spread  out  twice  as  fast. 

Now  follow  the  solid-lined  stream  of  light  entering  the  whole 
breadth  of  G,  through  its  focus  at  x,  on  until  it  strikes  E  and  becomes 
the  parallel  stream  emergent  to  the  eye.     By  similar  triangles 

Width  of  entering  stream       Gx       F       ^,       ...       „ 
Width  of  emergent  stream  ^  ^  =  /  =  ^"g"'/y'"g  P'^''' 

Notice  that  the  double-strength  eyepiece  emits  a  stream  only 
half  as  wide. 

Based  on  this  is  a  one-ej^e  Method  of  Measuring  Magnifying 
Power,  as  opposed  to  the  everyday  method  of  keeping  both  eyes 
open,  viewing  a  brick  wall,  and  counting  how  many  courses  of 
brickwork  one  magnified  brick  covers.  Focus  your  telescope, 
turn  its  little  end  to  the  broad  sky  at 
5'  arm's-length,   and   measure  with   dividers 

the   illuminated   diameter   of   the   object- 
^  ^'      glass ;    turn  it  round,  and   measure  with 
1/        aid  of  pocket-lens  the  clear  round  eye-ring. 
^  just  where  you  usually  put  your  eye,  a 

^  little  back  of  the  eye-lens  :    ratio  of  dia- 

FiG.  256.  meters  =  m.p.     This  is  perfectly  general, 

and  can  be  seen  another  way,  Fig.  256. 
Let  G^  and  Gg'  be  the  patches  of  light  waves,  from  bottom  and  top 
of  distant  object,  as  they  enter  the  object-glass ;  and  Ee  and  Ee' 
the  same  as  they  leave  the  eye-lens.  They  travel  throughout  at 
the  same  speed,  /.  ee'  =  gg',  and  plainly  the  angle  eEe'  between 


504  LIGHT  [§621 

the  two  waves  entering  the  eye  is  as  many  times  greater  than  gGg' 
at  which  they  enter  the  instrument,  as  Gg,  the  width  of  the  enter- 
ing wave,  is  greater  than  Ee,  the  width  of  the  emergent  wave. 
Actual  angles  are  smaller  than  shown. 

For  Night  Glasses,  Ee  was  to  cover  the  wide-open  eye,  5  mm. 

/.  m.p.  is  kept  down  to  Diam.  of  o.g./5  mm. 
For  Star  Glasses,  Ee  was  to  enter  the  eye  complete,   .*.  m.p. 

should  exceed  this,  and  may  well  be  Diam.  of  o.g./2  mm. 
For  Kesolving  Power  the  Star  Glass  condition  holds,  for  all 

the  light  must  enter  the  eye,  all  the  lens  must  be  in  use. 

For  if  the  eye-ring,  which  is  the  image  of  the  o.g.  formed  by 

the  e.p.  lenses,  is  larger  than  your  eye,  evidently  a  smaller 

o.g.  would  have  done  ;  you  are  looking  with  only  part  of  its 

big  eye. 

Over-magnification.  On  the  other  hand,  it  is  little  use  much 
exceeding  this  normal  m.p.,  diam.  of  o.g.  -^  2  mm.,  for  once  you  have 
got  the  whole  of  the  o.g.  in  use,  telling  you  all  it  can  gather,  doubling 
the  m.p.  will  merely  pinch  down  the  emergent  light  to  1  mm.  diam., 
make  things  twice  as  big  each  way,  twice  as  shaky,  a  quarter  as 
bright  (only  1/4  the  pupil  being  illuminated),  and  with  no  more 
detail. 

With  my  160-mm.  telescope  100  m.p.  is  as  good  as  any,  at  140  the 
image  is  larger  and  easier  to  observe  idly,  at  200  blurring  on  the 
edges  and  vibration  become  vexatious ;  and  yes,  well,  you  saw 
that  before — it's  not  new. 

What  is  more,  as  the  stream  of  light  entering  your  eye  is  pinched 
narrower,  by  forcing  up  m.p.,  the  little  bits  of  floating  muck  in  it, 
which  as  yet  you  hardly  suspect,  but  which  will  accumulate  as 
years  go  on,  stand  a  better  and  better  chance  of  obstructing  it,  so 
that  an  over-magnified  image  in  telescope  or  microscope  is  not  only 
dull  and  coarse,  but  is  also  infested  with  moving  specks  and  polly- 
wogs — '  entoptic  images  ' — ^which  sadly  spoil  the  real  picture. 

The  Resolving  Power  of  the  Lick  Refractor  is  914  mm.  lens  -^ 
2  mm.  =  457  times  that  of  the  eye,  each  way  ;  and  its  observers, 
with  years  of  experience  under  the  most  perfect  conditions  in  the 
world,  agreed  that  the  X  500  eyepiece  gives  them  the  best  all- 
round  results,  a  nasty  jolt  to  amateurs  who  were  thirsting  to  hear 
them  say  2000  ;  for,  whether  with  telescope  or  microscope,  it  is  so 
hard  to  give  up  the  dazzling  illusion  of  '  high  magnifying  power.' 

§  622.  Cross- wire    and    micrometer    telescope    and    microscope. 

When  measurements  are  to  be  made  with  telescope  or  microscope, 
the  measuring  mark  must  be  laid  in  contact  with  the  real  image. 
It  is  therefore  in  the  position  xy,  Fig.  254,  and  it  is  here  that  the 
Chinaman  put  the  spider,  to  persuade  the  astrologer  that  a  terrible 
monster  was  about  to  devour  the  moon  ;  an  old  tale  as  good  as 
any,  had  there  been  telescopes  in  China. 


§623] 


OPTICAL   INSTRUMENTS 


605 


The  astronomer  puts  the  spider's  line  there  instead,  and  the 
Meridian  of  Greenwich  has  long  found  its  only  material  representa- 
tion in  a  vertical  half -inch  of  spider  line,  stretched  across  the  focal 
plane  of  the  telescope  of  the  great  meridian  circle  instrument. 

The  line  from  the  crossing  of  two  spider-lines  to  the  optical  centre 
of  the  object-glass  is  the  Optic  Axis  of  the  telescope  ;  and  is  a  very 
definite  direction  indeed. 

Parallel  spider-lines,  on  little  frames  moved  apart  by  micrometer 
screws,  constitute  a  Ramsden  micrometer,  much  used  in  stellar 
measurements,  but  demanding  more  steadiness  than  a  microscope 
often  possesses  ;  for  them  one  more  commonly  employs  hundredth- 
inch  or  tenth-mm.  scales  diamond-ruled  on  a  glass  disc,  or  chequer- 
board  '  graticules  '  of  various  patterns,  for  blood-counts,  or  all 
sorts  of  purposes. 

The  Effective  Value  of  the  scale-divisions  or  screw-turns  has 
to  be  found  by  measuring  with  them  the  length  of  an  object  of 
known  size  viewed  through  the  instrument. 

This  is  particularly  necessary  in  the  microscope,  where  every  change 
of  object  glass  or  tube-length  alters  their  value ;  a  Stage-Micrometer, 
a  glass  scale  diamond-ruled  in  0-01  and  0001  in.,  or  0-1  mm., 
should  always  be  at  hand  to  check  them. 

§623.  Eyepieces  for  telescopes  and  microscopes  are  special 
varieties  of  magnifying-glass,  usually  consisting  of  two  lenses 
spaced  apart.  They  are  used  to  magnify  the  real  image  formed  by 
the  object-glass,  or  great  min'or,  instead  of  the  single  convex 
eye-lens  of  simple  theory,  which  gives  only  a  small  field  of  view, 
badly  blurred  and  coloured  all  round  the  out-field. 

The  guiding  principles  of  their  construction  are  :  that  sharing 
the  deviation  fairly  equally  among  four  surfaces  minimizes  spherical 
aberration,  §  586  ;  and,  secondly,  that  two  lenses  of  the  same  sort 
of  glass,  separated  by  about  half  the  sum  of  their  focal  lengths, 


Fig.  257. 

form  an  achromatic  combination,  as  you  will  see  sufficiently  from  the 
diagrams. 

The  Ramsden  eyepiece,  Fig.  257,  has  two  plano-convex  lenses, 
flat  sides  outwards,  of  equal  strength,  and  2/3/  apart,  which  leaves 
a  working  distance  of  1/4  as  much,  to  the  focal  planes  outside  the 

ends.  ,      r       •  *u 

The  uppermost  ray  shows  you  the  four  equal  refractions ;    tne 


606 


LIGHT 


[§623 


lowest  shows  how  the  '  field-lens,'  on  the  left,  refracts  the  longer 
waves  of  red  light  less  than  it  does  the  blue,  but  that  when  these 
separated  colours  arrive  at  the  '  eye-lens  '  the  red  hits  it  farther  out, 
where  the  prismatic  angle  is  just  so  much  greater  that  it  is  refracted 
more,  and  enters  the  eye  parallel  to  the  blue,  and  they  blend  per- 
fectly, since  the  eye  is  normally  focussed  for  the  '  parallel  light ' 
of  distant  vision. 

These  eyepieces  are  preferred  in  measuring -instruments,  for 
the  cross-wires,  etc.,  in  the  focal  plane  on  the  left  remain  fixed  in 
the  instrument  when  the  eyepiece  is  moved  in  or  out,  or  exchanged 
for  a  stronger  one. 

Whenever  you  handle  any  instrument  with  crosswires  or  micrometer 
scales,  first  of  all  pull  out  or  push  in  its  little  eye-end  until  you  see  them, 
clearly  :  it  is  made  adjustable  for  your  convenience,  and  this  does  not 
upset  the  general  focus  of  the  instrument,  as  would  turning  the  main 
focussing  screw,  which  does  not  focus  the  wires  for  you. 


Fig.  258. 

The  Huyghens  eyepiece,  Fig.  258,  gives  a  rather  better  picture, 
and  is  the  most  widely  used.  It  has  two  plano-convex  lenses, 
bulged  sides  towards  the  incident  light ;  the  little  eye-lens  is  from 
2  to  3  times  stronger  than  the  larger  field-lens,  and  their  distance 
apart  is  half  the  sum  of  their  focal  lengths.  As  before,  from  the 
upper  ray  parallel  to  the  axis,  you  see  how  the  spherical  aberration 
is  minimized,  and  the  downcast  cross-over  shows  the  blue  light 
now  more  bent  in  the  eye-lens,  and  emerging  parallel  to  the  red. 

The  field-lens  receives  rays  from  the  object-glass  before  they 
have  come  to  a  focus,  and  brings  them  in  more  quickly  to  form  an 
image  in  the  plane  of  the  field-of-view  diaphragm  inside  the  eye- 
piece, lying  in  the  focal  plane  of  the  eye-lens,  and  responsible  for 
the  familiar  black  circle.  On  it  are  placed  pointers,  micrometer 
scales,  etc.,  but  don't  trust  the  edges,  where  the  field-lens  has  dis- 
torted the  image  a  good  deal.  The  eye-lens  then  magnifies  every- 
thing in  the  usual  way. 

The  field-lens  enlarges  the  field  of  view,  for  without  it,  rays  such 
as  shown  would  continue  their  dotted  paths  and  miss  the  eye-lens 
altogether,  only  a  small  middle  of  the  field  would  be  visible.  But 
it  reduces  the  magnification,  and  for  examining  fine  detail  you  can 
remove  the  field-lens  of  your  microscope  eyepiece,  and,  holding 
your  eye  back  a  little,  gain  50%  in  size,  in  mid-field. 

These  eyepieces  vary  a  trifie  in  glass,  and  a  good  deal  in  brass, 


§  624]  OPTICAL   INSTRUMENTS  507 

between  maker  and  maker,  but  all  are  interchangeable  optically. 
Two  of  my  most  treasured  telescope  eyepieces  were  filched  from 
century-old  microscopes,  while  a  high-power  compens-okular  from 
an  apochromatic  micro-series  consorts  perfectly  with  another  smaller 
tele-o.g.,  at  which  the  rest  of  that  aristocratic  family  won't  look. 

Astronomical  telescopes  change  their  magnification  by  changing 
eyepieces  :  amateurs  have  a  whole  box-full,  but  the  great  ones  of 
the  earth  are  so  busy  now-o'-nights  photographically  that  their 
dark  domes  are  apt  to  resound  to  a  deal  of  shouting  before  '  that 
eyepiece  '  turns  up. 

More  modern  and  expensive  varieties  of  eyepiece  are  numerous  ; 
they  make  little  or  no  difference  in  mid-field,  but,  dealing  with  more 
recondite  aberrations,  they  improve  the  definition  over  the  rest  of 
it  very  considerably,  and  enable  it  to  be  enlarged.  The  eye- lens  of 
the  prismatic  binocular  is  usually  achromatized,  '  solid  '  telescopies 
are  triple  aplanats,  like  Fig.  266  (below)  ;  and  micro-compensation- 
oculars  may  have  a  separate  fourth  lens  as  well. 

Other  things  being  equal,  choose  eyepieces  tvith  a  long  working 
distance ;  you  are  going  to  wear  glasses  sooner  or  later,  and  it  is 
a  nuisance  to  be  always  putting  them  off  and  on. 

§  624.  Erect-image  telescopes.  As  you  can  see  from  Fig.  254, 
the  simple  refracting  telescope  gives  an  inverted  view — you  have  to 
look  down  for  up  and  right  for  left.  This  matters  nothing  to  an 
astronomer  (so  it  gets  called  the  Astronomical  Telescope),  but  it  is 
a  great  nuisance  when  trying  to  follow  rapid  movements  of  animals 


Fig.  259. 


or  birds.     There  are  two  ways  of  erecting  the  view  again;    but 
let  us  look  first  at  Galileo's  original  telescope,  which  never  inverts 

it  at  all.  .        ».  I 

In  fact,  it  intercepts  the  light  converging  from  the  object-glass, 
and  by  a  concave  eye-lens  makes  it  into  plane  parallel  light  again, 
ready' for  the  eye  to  deal  with.  Evidently  the  eye-lens  E  must 
stand  at  its  focal  distance  in  front  of  the  focal  plane  ary  of  the  e.g., 
for  that  is  a  concave  lens'  job  in  life— to  prevent  light  converging 
to  its  focus,  by  making  it  parallel. 

Put  your  finger  over  E  and  eye  in  Fig.  259,  and  you  see  G  forming 
image  yx  as  in  Fig.  254,  only  that,  for  clearness,  the  rays  converging 
on  y  have  been  omitted,  cf.  Fig.  II.  Put  in  E,  and  the  rays  are  turned 
from  their  dotted  courses  towards  x,  to  become  parallel— to  what  ? 
To  the  short-dotted  line,  which  is  one  of  themselves,  and  goes  straight 


608  LIGHT  [§  624 

through  the  optical  centre  of  E  to  point  x.  So  the  parallel  emergent 
stream  has  been  drawn,  and  there  is  a  similar  one  parallel  to  E^/. 

The  eye  has  to  squeeze  as  close  as  it  can  to  E,  and  catch  what 
it  can  of  them,  for  now  they  don't  cross  in  a  convenient  '  eye-ring.' 
But  it  does  look  up  for  light  coming  from  above,  and  right  for  right ; 
the  view  is  erect  and  true. 

Again  the  focal  planes  coincide,  and  again  the  Magnifying 
Power  is  the  ratio  of  focal  lengths,  or  of  strengths,  for  the  visual 
angle  x^y  is  greater  than  the  natural  xQ^y  in  that  ratio. 

Comparing  Gx  in  the  two  figures,  you  see  that  this  telescope  is 
short,  and  it  is  simple  and  cheap  and  clear,  for  the  aberrations  of 
the  two  lenses  partly  neutralize  one  another. 

Simpler  still  is  Lord  Baden-Powell's  Unilens.  Taking  a  very 
weak  long- sight  lens,  that  first  prescribed  you  for  '  tired  sight,' 
§  607,  he  held  it  at  arm's  length,  and  found  it  a  serviceable. substitute 
for  a  telescope.  His  friends  contradicted  him  in  the  Post,  but  try 
it  for  yourself,  and  you  will  quite  likely  succeed ;  it  all  depends  on 
whether  your  eye  can  relax  '  beyond  infinity,'  possesses  any  hyper- 
metropia,  as  in  Fig.  240,  3.  Take  my  own  case,  rather  an  extreme 
one.  Fig.  240,  5;  gradually  moving  my  1-5  D  'outdoors'  glasses 
forward,  I  see  the  distant  view  enlarge,  for  my  long-sighted  eyes 
can  slacken  to  perhaps  2-5  D  below  normal  '  infinity  focus,'  i.e.  to 
the  equivalent  of  a  normal  unaccommodated  eye  with  a  —  2-5  D 
lens  in  front  of  it.  Virtually,  I  have  a  pair  of  GaHleo  telescopes 
of  power  2-5/1-5  =  1-67,  which  act  as  well  as  any  opera-glass. 

Looking  back  through  my  glasses,  you  would  see  my  eyes  enlarged, 
i.e.  their  Resolving  Power,  without  which  magnification  is  nothing 
worth,  is  increased  to  correspond. 

Some  of  you  will  find  this  tip  useful  later  on. 

The  outstanding  drawback  to  the  Galilean  telescope  is  the  small- 
ness  of  its  field  of  view.  You  are  looking  at  a  round  porthole 
through  a  strong  diminishing  lens  :  in  practice  this  limits  the 
Magnifying  Power  to  1-5 — 2  in  opera-glasses,  3  in  field-glasses, 
and  5  for  those  marine  binoculars,  or  captain's  glasses,  where  the 
goggles  have  been  made  so  awkwardly  big  that  the  eye-lenses  are 
too  wide  apart  for  weaker  folk,  and  the  pictures  never  blend  stereo- 
scopically,  nor  give  their  full  night-glass  effect. 

§  625.  Accordingly,  telescope-makers  long  since  took  advantage 
of  the  inverting  effect  of  the  common  convex  lens  forming  a  real 
image,  cut  the  astronomical  tube  amidships,  and  put  one  in,  to  con- 
vert original  image  I  into  erect  image  J,  Fig.  260. 

Only,  as  in  pre-achromatic  days  they  knew  enough  to  keep  down 
aberrations  by  going  to  work  gently,  in  practice  they  split  up  both 
this  strong  lens  and  the  eye -lens  into  thinner  lenses  of  perhaps 
2 -in.  focal  length,  and  strung  them  along  the  draw- tube  until  they 
arrived  at  the  effect  desired.  You  will  find  this  arrangement  in 
the  wooden-cased  family  heirloom,  which,  if  as  long  as  your  arm, 
will  give  you  a  fairly  colourless  clear  view,  X  5  about — but  again 
of  a  very  small  field. 


626] 


OPTICAL   INSTRUMENTS 


509 


With  Achromatism  came  great  increase  in  Aperture,  and  Fig 
260  IS  a  1/3  scale  drawing  of  a  pocket  telescope,  hardly  bigger 
than  a  cigar,  x  10  diam.,  and  working  well  up  towards  the  theoretical 
resolving  power,  of  81  eyes,  of  its  18-mm.  o.g.  The  lenses,  of  focal 
lengths  14,  2,  2,  2-2  and  1-6  cm.,  are  bunched  into  Erector,  and 
Huyghenian  eyepiece;  I  and  J  are  marked,  and  this  smallest 
joint  of  the  3  draw  is  really  a  serviceable  little  Compound  Microscope, 
X  20,  achromatic  on  account  of  the  separation  of  its  lenses,  although 
all  are  of  crown  glass. 

This  is  typical  of  the  *  Long  Glass '  of  to-day  :  in  some  the  small 
joint  can  itself  be  pulled  out  longer,  thus  increasing  the  microscope 
tube-length,  §631,  and  therefore  magnification,  for  exceptionally 
clear  weather.     This  is  called  a  Pancratic  (all-powered)  Eyepiece.  * 

These  telescopes  are  long  and  awkward,  are  adversely  afifected  by 
the  aberrations  of  the  additional  lenses — producing  the  too-familiar 
haziness  of  definition — and  are  exceedingly  fussy  about  exact 
focussing,  §  631. 


Fig.  260. 

§  626.  By  far  the  best  erect-image  telescopes  are  Prismatlcs, 
where,  by  total  reflections  in  right-angled  prisms  fixed  at  right  angles, 
the  inverted  picture  is  first  folded  over  top  for  bottom,  and  then  left 
to  right,  Fig.  261.  The  prisms  are  made  of  very  clear  colourless 
glass  of  [L  1-57,  which  is  high  enough  to  ensure  total  reflection  at  all 
parts  of  a  wide  field. 

These  Prismatic  instruments  are  compact  and  easy  to  handle,  and 
to  focus,  and  are  beautifully  free  from  aberration  blur.  As  binocu- 
lars, with  enhanced  stereoscopic  effect  due  to  wider-apart  object 
glasses,  they  far  excel  the  older  patterns. 

Their  one  drawback,  apart  from  price — and  the  cheap  ones  are 
fuzzy  in  the  outfield — is  that  a  fall  or  a  sharp  knock  may  shift 
a  prism,  and  then  they  see  double.  When  that  happens,  do  not 
take  them  to  the  local  optician,  for  he  has  only  one  chance  in  four 
of  spotting  the  culprit,  and  if  he  doesn't,  the  result  can  only  be  a 
cobbled-up  imitation,  always  straining  your  eyes. 

Send  them  straight  back  to  the  maker,  who  has  the  necessary 
adjusting  apparatus,  and  for  Customs  reasons  let  him  be  English, 


510  LIGHT  [§  626 

or  have  an  English  agency.  You  can  get  a  thoroughly  good  English 
pair  of  '  8  X  30  '  (which  is  an  abbreviation  for  8-magnifying, 
30-mm.  aperture),  centre-wheel  focussing,  for  £6,  unless  you  want 
extra-large  field  of  view,  which  costs  more. 

And  do  not  be  tempted  beyond  this  size,  for  anything  bigger 
is  too  big  for  the  pocket,  and  gets  left  at  home,  or  swung  on  its 
sling  against  the  rocks,  and  anything  stronger  is  altogether  too 
badly  affected  by  vibration,  of  deck,  or  wind,  or  your  own  heart. 
Of  course,  if  you  always  have  a  caddie  and  a  stone  dyke  with  you 

A  pair  of  8  X  24  sealed-up  eyepiece-focussing  prismatics  has 
travelled  in  my  pocket  through  vacations  these  twenty-nine  years 
without  mishap,  or  needing  internal  cleaning. 


To  clean  glass,  dust  lightly,  then  breathe  on  and  wipe  round  once 
with  a  pad  of  clean  handkerchief.  Never  lay  glass  on  glass.  It  is 
siliceous  grit,  backed  by  another  unyielding  surface,  that  scratches 
glass. 

Here  let  me  warn  any  would-be  star-gazer,  that  the  lenses  of 
coastguard,  or  other  largish  telescopes  sold  for  terrestrial  use,  are 
very  seldom  of  '  astronomical  quality.'  Residual  aberrations  that 
throw  only  a  faint  veil  of  haze  over  a  daylight  picture  are  enough 
to  make  a  star  a  foggy  firework,  and  the  use  of  such  telescopes  is 
sadly  disappointing.  Dark-ground  illumination  is  a  drastic  test 
for  a  micro-o.g. 

Very  slight  experience  will  suffice  to  show  you  how  unsteadiness 
of  the  atmosphere,  due  to  wriggling  up-currents  of  warmer  air,  sets 
limits  to  the  utility  of  telescopes,  §  488. 

§  627.  Focussing  a  telescope.  For  near  objects.  All  telescopes, 
as  the  object  comes  nearer,  have  to  be  pulled  out  longer,  for  the 
real  image  retreats  from  principal  focal  distance  to  focal  distance 
conjugate  to  that  of  the  object,  and  that  is  always  longer.  Draw 
your  own  diagram. 

The  maker  usually  skimps  you  in  brass  ;  that  little  telescope  of 
Fig.  260,  lengthened  30  mm.  by  a  bit  of  brass  tubing  that  came  to 
hand,  enables  one  to  sit  in  the  garden  and  watch  the  activities  of 
ants  and  such-like  small  folk  in  the  grass  beside  the  chair,  without 
overshadowing  and  scaring  them  as  would  a  pocket  lens,  close  to. 


§  628]  OPTICAL   INSTRUMENTS  611 

Expensive  contraptions  trading  as  '  super- telescopes,'  etc..  are 
nothing  more  than  this. 

For  near  sight.  A  short-sighted  user  pushes  the  eyepiece 
in,  to  get  nearer  the  image  he  wants  to  see. 

For  far  sight.  A  long-sighted  observer  pulls  the  eyepiece 
out. 

These  are  common  sense,  and  need  no  diagram,  but,  by  way  of 
variet3%  Fig.  260  is  drawn  for  an  object  only  a  few  feet  away  ;  I  haw 
come  back  from  the  principal  focus,  and  the  user  is  at  present 
accommodating  up  to  near  vision,  instead  of  troubling  to  pull  out 
the  draw-tube  any  more.  (There  is  no  need  for  you  to  study  this 
diagram  with  a  view  to  reproduction.) 

To  produce  a  real  image  on  a  screen.  The  long-sighted  eye 
wants  light  convergent  already  :  converge  it  a  bit  more  and' it 
will  come  to  a  focus  of  itself,  on  a  screen. 

Therefore,  to  look  for  Sunspots,  turn  the  telescope  to  the  sun, 
and  throw  a  bright  patch  on  a  white  card  half  a  yard  or  more 
behind  the  eyepiece  ;  draw  that  out  some  distance,  and  the  image 
clears  :  it  is  in  the  focus  conjugate  to  the  first  real  image,  for  the 
eyepiece  lenses,  and  the  condition  is  similar  to  that  of  the  microscope 
in  Fig.  273. 

The  Telephoto  Lens  is  a  Galilean  telescope,  with  a  very  large  and 
achromatic  '  eye  '-lens,  used  in  just  that  same  drawn-out  condition. 


Fig.  262. 


By  altering  its  length  it  can  be  made  to  focus  a  real  image  at  any 
camera-extension  available,  the  longer  the  larger  the  magnification, 
but  much  the  worse  the  light.  Fig.  262  shows  the  drawmg  back  of 
the  negative  lens,  and  you  see  it  is  exactly  the  same  as  l-ig.  211, 
IX  ;  also  as  Fig.  214,  III,  which  directly  shows  you  its  long  equiva- 
lent  focus,  LF,  and  short  camera-length,  LjF. 

In  a  fixed-focus  high-speed  form,  x2,  you  have  scon  it  in  use  on 
sports  grounds,  by  press  photographers. 

§  628.  Very  special  uses  of  the  Telescope  are  found  in  Periscopes, 
including  Bronchoscope,  Cvstoscope,  Laryngoscope,  etc.,  all  of  them 
contrivances  for  seeing  a  good  broad  view  through  a  long  and  very 
narrow  tube  ;  and  in  Range-finders.  .    .  .^     . 

You  know  that  when  you  focus  a  telescope  so  that  it  gives  >ou 
a  magnified  picture  of  a  small  field  of  view,  and  then  turn  it  the  wrong 


612  LIGHT  [§  628 

way  round  and  hold  it  at  arm's  length,  you  see,  framed  in  its  object- 
glass,  a  minified  picture  of  a  large  field  of  view. 

How  it  does  it,  you  can  make  out  from  Fig.  254,  where  the  angular 
sweep  of  the  parallel  streams  on  the  right  is  F//  times  those  on  the 
left. 

In  Fig.  263,  x  i^  such  a  reversed  telescope,  minifying  6  diam., 
enabling  1-7°  to  look  out  over  10°.     Above  it  is  an  inverted 


C:^] 


\A      Galilean  field-glass  ^,  minifying  4,  increasing  this  field 
{A    to  the  very  useful  one  of  40°, 
1"^  right-angled  prism,  over  the  sea. 


to  the  very  useful  one  of  40°,  looking  out  through  the 


n 


At  the  lower  end  of  the  long  tube  is  telescope  T,  mag- 
nifying the  tiny  picture  aloft  24  diam.,  or  more,  so  that 
an  eye  looking  into  its  diagonal  eyepiece  sees  the  40° 
field  natural  size  again,  or  a  little  more. 

If  the  navigator  wants  to  scan  the  sky,  a  turn  of  his 
hand  tilts  the  top  prism,  as  shown,  cf.  Fig.  191  (1)  ;  while 
if  he  wants  to  examine  any  object  more  closely,  a  twist 
of  the  other  hand  flips  the  two  Galilean  lenses  out  of 
the  way,  and  leaves  a  magnification  of  4. 

There  are  numerous  accessory  appliances. 

Gastroscope,  cystoscope,  sigmoidoscope,  laryngoscope, 
bronchoscope,  may  retain  the  very  effective  device  of  the 
old-fashioned  erecting-telescope  maker,  §  625,  and  string 
a  number  of  convex  lenses  along  their  tubes,  not  so  thick 
as  a  pencil,  much  shortened  in  Fig.  264.  These  lenses 
simply  hand  on  little  images  along  the  tube,  inverting 
them  without  letting  them  grow  any  bigger,  a  sort  of 
optical  '  lazy-tongs.'  At  the  end  come  reflecting  prism 
and  suitable  inspecting  lens — very  short  focus  in  the  long 
Introscope  for  examining  rifle-bores — and  miniature  lamp 
and  reflector. 

A  pattern  of  Range-finder  will  be  understood  from 
the  plan,  Fig.  265.  Two  '  optical  squares '  (silvered 
Fig.  263.  blocks  of  glass  which  always  turn  light  exactly  90°)  at 
the  ends  of  the  cross  tube,  from  3  ft.  to  90  ft.  apart, 
reflect  their  views  into  two  telescope  o.g.'s,  which,  through  upper 
and  lower  reflecting  prisms,  feed  the  upper  and  lower  halves  of  the 
field  of  view  of  the  central  eyepiece.  The  lines  of  sight  are  adjusted 
to  be  parallel,  so  that  top  and  bottom  halves  of  the  bisected  moon 


^ 


Fig.  264. 


would  fit  perfectly  ;  but  parallax  of  nearer  objects,  i.e.  the  conver- 
gence of  the  lines  of  sight  to  them  from  the  ends  of  the  base, 
disjoints  them. 

One  of  the  two  thin  prisms  P,  which  are  normally  set  at  their 


§629]  OPTICAL   INSTRUMENTS  613 

position  of  minimum  deviation,  2°  (by  which  the  central  parts  of  the 
mstrument  are  actually  set  askew,  but  not  shown  in  diagram),  is  now 
rotated  so  as  to  increase  its  deviation  (30°  changes  it  only  15') 


m 11^^ — \^ — M)        lii 


Fio.  265. 

until  top  and  bottom  of  the  distant  mast  are  in  line  again  ;  and  the 
range  is  read  off  on  the  graduated  turning  gear  of  P. 

Aircraft  range-finders  are  made  binocular,  and  the  stereoecopio 
power  of  trained  eyes  is  made  use  of,  down  to  20"  of  arc. 

§629.  The  Simple  Microscope  or  Magnifying-Glass  held  close  to 
the  eye.  Things  look  larger  as  they  come  nearer  to  the  eye  and 
obstruct  a  greater  visual  angle,  but  the  trouble  is  that  within  its 
limiting  '  nearest  distance  of  distinct  vision  '  the  eye  cannot  see 
their  detail  clearly. 

This  may  be  counteracted  to  some  extent,  on  the  *  stop  *  principle 
of  §  613,  by  looking  through  a  pinhole,  but  this  reduces  the  light, 
and  presently  actually  increases  fuzziness. 

A  simple  convex  lens  gives  the  eye  the  necessary  increase  of 
accommodating  power  to  focus  clearly  on  the  near  object,  already 
magnified  by  its  nearness.  Since  light  emanating  from  the  principal 
focal  plane  of  a  convex  lens  becomes  parallel  after  refraction,  an 
object  placed  4  in.  in  front  of  a  4-in.  focus  lens,  or  placed  2  in.  in 
front  of  one  of  2-in.  focus,  will  send  parallel  light  to  the  eye,  and  if 
we  may  ignore  the  gap  between  lens  and  eye — and  we  are  going  to— 
the  thing  will  fill  twice  as  large  a  visual  angle,  will  bulk  twice  as 
wide  across,  when  only  2  in.  away  as  when  4,  a  comparative  magni- 
fication of  2  '  diameters.' 

From  where  are  we  to  start  reckoning  magnification  ?  Naturally, 
from  the  nearest  distance  of  distinct  vision,  because  everyone  brings 
a  small  object  up  as  close  to  his  eye  as  he  comfortably  can,  before 
hunting  around  for  a  magnifier  to  assist  him. 

But  different  people  have  very  different  puncta  proximo,  and  this 
might  lead  to  disputes  with  the  optician  ?  Therefore  the  Nearest 
Distance  of  Distinct  Vision  is  conventionally  made,  10  in.  or  25  cm., 
and  then  the  magnification  obtainable  with  a  lens  (gauged,  as  in 
telescopes,  by  the  visual  angle  occupied),  as  compared  with  the  object 
at  this  distance,  is  called  the  Magnifying  Power  of  the  lens. 

So  that  the  4-in.  lens,  permitting  the  object  to  come  10/4  timet 
nearer,  has  M.P.  2-5  ;  and  the  2-in.  lens  has  M.P.  10/2  =  5. 

But  that  makes  a  10-in.  lens  have  M.P.  1,  which  means  no  inoreMe 
in  size  at  all  ? 

The  cause  is  this,  that  we  are  letting  our  lenses  send  liffht  to  an 
unaccommodated  eye,  whereas  it  had  been  accommodated  up 
s 


614  LIGHT  [§629 

4  Dioptres  to  see  the  object  at  10  in.  The  10-in.  lens  just  replaced 
this ;  but  using  it  and  the  accommodation  together  =  4  +  4  —  8  D, 
a  5-in.  lens,  the  object  is  brought  up  to  5  in.,  and  thus  magnified 
by  2. 

This  1  has  to  be  added  all  through,  it  is  the  shop  allowance  for  the 
eye  supposed  to  remain  accommodated  for  its  nearest  distance, 
although  in  practice  it  probably  seldom  does. 

.-.  M.P.  =  1  +  10  ^' If ^'  or  1+  25cm.//cm. 
or,  since  100//  cm.  is  the  Dioptric  Strength  D  of  the  lens 

M.P.  -  1  +  J  D 

On  its  practical  measurement,  see  §  631. 

Graphically,  in  Fig.  266,  the  image  XY,  which  is  a  virtual  one, 
purely  your  own  property,  must  lie  between  the  lines  from  the 
ends  of  the  object  straight  through  Optical  Centre  C  of  the  lens. 
Drawing  from  the  upper  end  a  '  ray  '  parallel  to  the  lower  one  of 
these,  it  is  bent  down  to  meet  it  at  F,  and  the  eye  sees  the  top  of  the 
image  in  the  directions  COX  and  FAX,  which  meet  at  X,  since  CO 
is  less  than  the  focal  length  ;  and  CX  is  the  conventional  10  in., 
for  which  the  eye  is  accommodated. 


Fig.  266. 


If  the  eye  relaxed,  O  would  have  to  be  moved  out  to  F',  the 
principal  focal  distance,  CO  would  be  parallel  to  FA,  and  the  visual 
angle  would  diminish  a  little  ;  you  lose  that  bonus  of  1  ;  but  it 
makes  no  appreciable  difference  with  lenses  of  any  strength. 

If  you  like  to  apply  the  usual  lens  formula;  1/a  +  1/6  =  1// 
becomes  l/{—  a)  +  l/b  =  1//,  since  a  refers  to  a  virtual  image, 
which  gives  —  1  +  a/b  =  a  //.  Here  a/6,  distance  of  image /object, 
as  always,  =  magnification,  which  becomes  M.P.  if  we  make  a 
25  cm.     Hence  M.P.  =  1  +  25  cm.|ycm.  =  l  +  J  D  as  above. 

What  of  the  Resolving  Power,  which  is  what  really  matters  ? 

If  you  were  a  small  fly  under  examination,  and  an  eye  came  down 
5  times  nearer  to  you,  it  would  merely  appear  to  you  to  have  stayed 
where  it  was,  but  grown  5  times  bigger  each  way,  25  times  in  area, 
25  pupils  to  report  information  instead  of  one.  That  is,  the  Re- 
solving Power  increases  proportionally  to  the  Magnifying  Power, 


§629]  OPTICAL   INSTRUMENTS  516 

and  nobody  could  suspect  that  there  might  be  any  distinction  between 
them.  You  simply  say  that  you  can  see  *  just  as  clearly  as  without 
the  glass/  only  bigger. 

Contrast  that  with  early  efforts  with  the  high  power  of  your 
compound  microscope  !  Contrast  it  also  with  a  little  lens  of  very 
short  focus,  like  Fig.  113.  A  1/32-in.  focus  lens,  even  if  spherical, 
can  be  only  1/16  in.  diameter,  and  more  than  half  this  must  be 
stopped  off  for  aberration,  so  that  only  a  third  of  the  diameter  of 
your  pupil,  a  ninth  of  its  area,  can  gain  any  information  at  all 
from  such  a  lens.  The  fly,  magnified  300,  looking  up  at  you, 
would  see  the  dark  inquisitive  pupil  so  obstructed  that  it  was  only 
100  times  wider,  and  he  would  be  perfectly  correct  in  calculating 
that  only  100^  as  much  was  being  learnt  from  him,  instead  of  the 
3002  the  microscopist  intended. 

You  would  say  the  lens  was  '  a  bit  fuzzy.'  It  is  so,  diffraction 
fuzz,  every  line  3  times  as  broadly  drawn. 

Robert  Brown,  §  367,  preferred  that  1/32-in.  microscope  to  a 
1/70-in.  which  DoUond  made  and  gave  him.  Rightly  so,  for  though 
this  magnified  twice  as  much,  it  cannot  have  given  information  to 
more  than  half  the  width  of  eye,  the  fly  would  not  have  seen  any 
change  in  the  size  of  the  inquiring  pupil.  The  user  would  see  twice 
as  big  a  picture,  but  twice  as  fuzzy. 

He  also  preferred  it  to  any  compound  microscope.  For  you 
will  find  that  old  pre-achromatic  microscopes  had  a  lot  of  little 
simple  convex  object-glasses,  masked  down,  to  reduce  aberration, 
smaller  than  your  pupillary  2  mm. ;  and  that  as  you  try  them  one 
after  another,  the  preserved  flea  or  the  butterfly  scale  gets  biffgcr 
and  bigger,  but  you  can  learn  no  more  about  it,  and  no  more  than 
you  could  by  using  any  one  of  them  as  a  simple  lens. 

For  it  is  open  as  wide  as  can  be,  and  even  then  is  not  as  big  as 
your  pupil,  i.e.  you  are  not  losing  any  scrap  of  information  it  can 
give,  and  no  process  of  squeezing  that  information  through  tubes  and 
lenses  higher  up  can  possibly  make  it  any  more  informative. 

It  is  the  way  in  which  the  modern  microscope-maker  has  been 
able  to  widen  out  the  size  of  his  lens,  without  losing  magnification, 
that  gives  its  value  to  the  compound  microscope. 

It  is  remarkable,  and  fortunate,  that  every  simple  macnifier, 
from  spectacles  up,  used  correctly,  is  achromatic,  in  mid-field. 
For  in  the  lower  half  of  Fig.  266,  the  greater  bending  of  the  blue 
(short  dots),  in  the  prismatic  edge  of  the  lens,  causes  the  blue  image 
to  lie  farther  off,  as  indicated  ;  but  since  all  the  images  lie  with  their 
ends  on  the  straight  central  rays  CX,  CY,  they  all  appear,  to  an  eye 
near  C,  to  cover  one  another  very  exactly,  their  sum  total  bemg 
a  colourless  image.     Contrast  Fig.  232. 

A  much  larger  flat  colourless  field  is,  however,  obtainable  with 
the  solid  *  triple  aplanats,'  of  which  a  couple  are  shown  in  section 
below  Fig.  266  ;  and  one  of  these,  costing  about  £1,  and  magnif>ing 
not  more  than  10,  is  the  only  high -power  pocket  lens  worth  investing 
in. 


616 


LIGHT 


[§630 


§  630.  The  Compound  Microscope.  Your  telescope  has  a  bigger 
eye  than  you  have,  perhaps  it  would  like  to  look  through  your 
pocket-lens,  tey  it,  and  the  two  magnifications  multiply  together, 
and  give  you  something  impressive,  and  that  is  all  the  calculation 
we  need  make  about  the  Magnifying  Power  of  the  Compound 
Microscope,  which  you  have  just  made.  Light  from  the  object 
at  the  principal  focus  of  the  pocket  lens  (focal  length  marked  in 
black)  is  made  '  parallel '  by  it,  enters  the  already  distance-focussed 
telescope,  Fig.  254,  and  that  is  how  it  works,  Fig.  267,  I. 

But  has  the  Resolving  Power  benefitted  ?  Well,  whatever  that 
of  the  pocket-lens  or  '  simple  microscope  '  may  be,  it  is  now  used 


Fig.  267. 


by  the  larger  eye  of  the  telescope  instead  of  your  little  one,  so  the 
R.P.  has  gone  up  in  that  ratio  of  sizes — provided  that  the  larger 
eye  is  kept  supplied  with  light  all  over,  and  that  is  a  big  and  often 
neglected  proviso  with  the  microscope. 

So  your  new  invention  really  is  worth  while. 

But  now  we  can  do  a  little  simplification,  as  they  do  in  wireless 
sets.  That  lens  G  is  only  a  weak  one,  why  not  actually  combine 
it  with  L,  making  L  a  little  fatter  instead,  as  shown  below,  in  II  ? 
(wherein  one  stream  of  light  has  been  shut  off,  for  clearness). 

What  has  become  of  focal  distance  F  ?  Lost.  Does  it  matter  ? 
TRY  IT,  alter  the  length  of  the  telescope  tube,  alter  the  object  distance 
until  you  see  it  clearly  every  time,  and  say  when  you  see  it  clearest. 
Different  magnifications,  that  is  all ;  details  always  about  the  same 
visibility. 

For  in  merging  the  two  lenses  you  not  only  lost  the  principal 
focal  length  of  the  back  one,  but  also  that,  marked  in  black,  of  the 


§630]  OPTICAL   INSTRUMENTS  517 

front  one.  Object  and  image  are  now  evidently  at  conjugate 
focal  distances  of  the  combined  object-glass,  object  lying  somewhere 
outside  the  new  principal  focus. 

So  that  essentially  a  Compound  Microscope  is  just  two  strong 
convex  lenses  a  good  way  apart,  nothing  more  definite  nor  more 
abstruse  than  that  ?  Look  at  your  own  microscope :  at  one  end 
a  choice  of  lenses  all  marked  with  short  focal  lengths,  2/3  in., 
4  mm.,  etc.,  at  the  other  an  eyepiece  of  little  fat  lenses;  between 
them  a  tube  and  a  drawtube  you  can  pull  out  how  you  like. 

Figs.  I  and  II  give  you  a  perfectly  sound  and  full  plan  of  the  action 
of  the  Microscope,  and  the  comparatively  unpleasant  and  un- 
informative  tangle  of  III,  where  all  distances  seem  to  have  come 
adrift,  is  put  in  mainly  to  meet  such  exam  questions  as  22 — 26, 
commented  on  later. 

In  it  I  have  changed  my  o.g.  for  a  stronger  one  (a  0-4  in.  on  actual 
scale),  have  planted  one  end  of  the  object  on  a  straight  ray  drawn 
through  both  lenses — which  therefore  assists  in  forming  the  tails  of 
all,  i.e.  they  must  all  touch  it — and  have  applied  the  Standard 
Construction  of  Fig.  203  to  find  the  Real  Image.  I  have  deliberately 
pushed  the  eyepiece  in  towards  this,  so  that  it  lies  inside  its  focal 
length,  and  have  then  started  afresh,  with  the  appropriate  modified 
construction  of  Fig.  204,  to  find  the  final  Virtual  Image  ;  and  to  save 
space  I  have  made  the  user  rather  short-sighted. 

That  skeleton  enables  you  to  meet  calculatory  requirements  : 
then  I  have  gone  on,  as  in  Fig.  210,  to  picture  how  the  gla.s.se8  are 
really  utilized ;  choosing  the  tail  as  object,  for  simplicity,  I  have 
filled  the  o.g.  with  light  from  it,  converged  this  down  to  form  the 
tail  of  the  image  and  carried  it  on  to  the  eyepiece,  and  this  then 
reduces  its  rapid  spreading  and  pours  it  into  the  eye,  fairly  filling  •<, 
as  if  it  emanated  from  the  point  we  call  the  tail  of  the  virtual  image. 

It  is  a  stock  laboratory  experiment  to  set  up  two  strong  lenses 
well  apart,  and  keeping  your  eye  a  couple  of  inches  from  one  of  them, 
focus  a  small  object  beyond  the  other.  It  does  not  matter  which 
way  you  look  through  ;  for,  unlike  a  telescope,  a  microscope  magni- 
fies equally  either  way.  The  lurid  colour  effects  you  get  with  this 
primitive  apparatus  soon  show  that  there  is  room  for  improvement, 
and  we  will  consider  it  briefly. 

We  have  said  already,  §629,  that  the  early  micro,  o.g.'s  were 
tiny  simple  convex  lenses,  stopped  down  very  heavily  just  as  the 
old  telescope  o.g.'s  were,  and  for  the  same  reason,  to  minimize 
their  aberrations.  In  1829  J.  J.  Lister  made  achromatic  object- 
glasses,  and  was  able  to  increase  their  size,  just  as  you  saw  ^lith 
achromatic  tele,  o.g.'s  :  up  went  the  Resolving  Power,  and  the 
Compound  Microscope  was  converted  from  a  beautiful  but  rather 
futile  elaboration  of  the  simple  one,  into  a  capable  weapon  of  re- 
search. His  son,  studying  '  germs  '  with  it,  grew  up  to  be  the 
apostle  of  antisepsis,  Lord  Lister. 

Your  low  power — 1-in.  or  2/3-in.  focus,  no  matter — consists  of 
one  small  achromatic  lens  :  or  better,  of  a  pair,  1/2  in.  or  so  apart, 


518  LIGHT  [§  630] 

larger,  and  with  correspondingly  more  resolving  value.  The  prin- 
cipal plane  of  the  lens  equivalent  to  the  pair,  §  542,  Fig.  214,  has 
a  position  somewhere  inside  the  brasswork,  so  the  actual  nozzle  of 
the  o.g.  is  much  nearer  the  object  than  2/3  in. 

Your  '  sixth-inch  '  has  a  thick  front  lens,  curved  on  the  back  to 
about  2  mm.  radius,  which  does  all  the  magnifying.  This  shape 
minimizes  its  spherical  aberration,  but  that,  and  all  its  chromatic, 
are  dealt  with  faithfully  and  wiped  out  by  two  pairs  of  correcting 
lenses  behind  it.  Your  '  twelfth,'  which  has  to  be  united  to  the 
object  by  a  drop  of  *  immersion  oil,'  has  a  hemispherical  front  lens, 
Fig.  230,  backed  by  a  crescentic  meniscus  in  which  aplanatic  re- 
fraction again  occurs,  and  then  the  two  pairs  of  correctors.  In  both 
these  o.g.'s  the  first  principal  plane  is  away  back,  which  leaves  very 
little  working  distance,  but  it  can't  be  helped. 

Microscope  tubes  are  shorter  than  they  used  to  be  purely  as  a 
matter  of  convenience  :  if  you  will  have  more  mere  magnification, 
you  can  pull  out  the  draw- tube. 

Eyepieces  have  been  described  in  §  623;  Again  it  is  the  Object 
Glass  which  is  the  elaborate  and  valuable  lens,  to  which  all  else  is 
subservient :  it  was  a  cracked  eyepiece  Ross  was  using  when  he 
discovered  the  malaria  parasite  in  the  4001st  mosquito's  stomach, 
and  transformed  life  in  the  tropics. 

If  you  want  to  dissect  under  the  microscope,  use,  as  Erecting 
Eyepiece,  the  small  joint  of  a  pocket  telescope,  Fig.  260,  adjusting 
by  draw -tube. 

Opaque  objects  you  illuminate  with  pocket-lamp,  or  any  bull's- 
eye  ;  for  transparent  ones  with  higher  powers  you  learn  to  manipulate 
the  substage  condenser,  or  your  microscope  remains  a  poor  peep- 
show. 

§631.  The  Magnifying  Power  of  a  microscope,  simple  or  com- 
pound, can  be  measured,  if  wanted  for  some  special  purpose,  as 
follows  :    Lay  the  microscope  horizontal  and  stand  it  on  a  steady 
pile  of  books  so  that  its  eyepiece  is  10  in.  above  a  paper  on  the 
table  ;    1-ay  a  weight  on  its  foot  if  necessary.     Stick 
yy\       a  cover-glass  on  the  top  of  its  eyepiece,  at  45°,  Fig. 
j^\^^      268,  with  a  speck  of  plasticine,  smoke  its  back  by 
passing  a  match  twice  under  it,  enough  to  dull  the 
second  reflection.      Then  an  eye  looking  vertically 
down  sees  paper  and  pencil  through  the  glass,  and 
also  the  whole  microscope  field  by  reflection,  as  if 
lying  on  the  paper. 
Fig.  268.  Focus   a   '  Stage  Micrometer  ' — a   very  fine-scale 

diamond-ruled  on  glass  in  tenths  mm. — under  the 
microscope,  and  trace  its  rulings  in  pencil  on  the  paper  :  the 
average  distance  between  two  marks,  divided  by  the  actual  tenth- 
mm.,  ==  M.P. 

This  method  does  just  as  well  for  Simple  Lenses,  for  which  a  mm. 
scale  serves  ;  retain  the  10  in.  from  the  paper. 


632J 


OPTICAL   INSTRUMENTS 


519 


This  45°  device,  which  facilitates  the  making  of  drawings  of 
microscopic  objects,  constitutes  a  Camera  Lucida.  The  pencil 
and  paper  should  be  brightly  lighted,  and  this  home-made  pattern 
does  very  well,  requiring  less  adjustment  than  most.  There  are 
many  others,  probably  the  best  is  the  Abbe,  with  large  mirror,  for 
which  see  makers'  lists. 

For  Eyepiece  Micrometers,  very  useful  for  measurements,  see 
§622. 

From  §  516  you  see  that  whatever  be  the  diametral  magnification, 
the  magnification  in  depth  is  the  square  of  it.  Hence  microscopes 
are  exceedingly  particular  about  exact  focussing,  and  a  steady  and 
true  Fine  Adjustment  is  absolutely  essential. 

Further,  it  makes  the  depth  of  view  of  high  powers  very  small, 
and  they  cut  several  '  optical  sections  *  out  of  a  5-micron  slice  of 
tissue.  This  saves  confusion,  but  it  means  that  you  have  to  keep 
the  fine  adjustment  in  action  all  the  time ;  and  it  makes  stereo- 
scopic binoculars,  which  are  delightful  for  low  powers,  almost 
unusable  for  high. 

§  632.  Let  us  now  obtain  the  all-important  Resolving  Power 
of  Telescope  and  Microscope  in  figures,  for  it  can  be  done  readily, 
and  the  result  accords  with  the  conclusions  of  two  generations  of 
observers  better  than  do  treatments  elaborated  upon  assumptions 
of  more  mathematical  convenience  than  accuracy. 


B 


uy 


i 


<  a 
h 


Fio.  269. 


In  Fig.  269  let  AC  and  BC  be  patches  cut  off  by  the  front-window 
frame,  width  w,  of  some  optical  instrument,  from  the  waves  of  light 
arriving  from  two  sources  so  distant  as  to  appear  mere  points  of 
light — a  twin  star.  *  i.  u 

The  instrument  presently  piles  these  up  mto  two  heaps  of  light, 
a  and  b,  the  images  of  the  distant  points  ;  heaps  of  some  size,  because 
energy  cannot  be  packed  into  mathematical  iwints  :  white  for  black, 
star  images  in  a  fine  telescope,  highly  magnifie<l.  look  very  like  my 
a  and  6. 


620  LIGHT  [§  632 

Let  the  two  stars,  originally  very  close  together,  gradually  move 
apart.  Their  images  are  at  first  bunched  into  one  heap,  which 
lengthens,  then  just  visibly  divides  into  two,  then,  continuing,  these 
are  separated  by  growing  width  of  darkness. 

Meanwhile,  as  the  angular  distance  apart  of  the  stars  (which  is 
the  angle  ACB)  increases,  AB,  from  being  a  fraction  of  a  wave- 
length of  light,  becomes  one  wave-length,  and  then  goes  on  increasing. 

There  is  one  unique  event  in  the  first  history,  and  that  is  when 
the  image  first  appears  distinctly  double  to  the  eye,  which  is  an 
optical  instrument,  just  like  every  other  optical  instrument  man 
ever  made  with  hands  and  used  his  eyes  to  perfect. 

There  is  one  unique  event  in  the  second  history,  and  that  is  when 
AB  is  one  wave-length. 

Now,  these  two  histories  are  indissolubly  connected  and  inter- 
dependent, and  if  these  two  unique  events  do  not  occur  simultane- 
ously, there  must  be  some  numerical  factor  which  settles  how  long 
after  the  one  event  the  other  is  delayed. 

No  such  factor  has  been  found,  nor  any  reason  for  one. 

Thus  the  image  is  just  resolved  into  two  when  AB  is  one  wave- 
length, and  the  angular  separation  of  the  two  points  is  Xjiju. 

[X,  lambda,  the  Greek  Z,  is  everybody's  symbol  for  wave-length 
of  light.] 

This  is  taking  it  for  granted  that  every  part  of  AB  and  AC  is 
as  good  as  every  other  part,  from  top  to  bottom,  i.e.  that  the  window 
is  sqvxire ;  but  most  optical  instruments  have  round  windows  ; 
what  must  we  allow  for  that  ? 

We  have  reckoned,  all  along,  the  Resolving  Power  as  so  many 
eye-pupils  packed  into  the  enlarged  eye  ;  it  cannot  matter  much 
whether  they  are  packed  into  square  or  circle,  so  long  as  there  are 
as  many  of  them.  That  is,  we  must  take  a  circle  of  equal  area 
to  the  square,  Tzd^ji:  =  w^,  which  gives  its  diameter  d  =  1-13  w. 

Therefore,  multiplying  top  and  bottom  by  1-13,  the  least  angular 
distance  apart,  or  minimum  angle  of  separation,  at  which  two  points 
may  lie,  to  be  resolvable  by  any  optical  instrument  with  a  limiting 
circular  aperture  of  diameter  d,  is 

Minimum  angle  of  separation  —r- 

White  light,  which  is  what  the  astronomer  mostly  gets  from 
stars,  and  what  most  microscopists  work  with,  and  our  eyes  mostly 
see  with,  is  polychromatic,  of  many  wave-lengths,  but  the  brightest 
part  may  be  reckoned  as  X  =  0-000555  mm. ;  so 

Minimum  angle  of  separation  in  White  Light  = 
1        ^  1 

imOd^r..  40,0006Z„ehes. 

This  is  in  natural  measure,  and  since  1  radian  =  207,500  sec. 
of   arc,  a   1-in.  telescope   should  resolve   double  stars  5"  apart ; 


§634] 


OPTICAL   INSTRUMENTS 


521 


a  6-5-in.,  0-8"  ;   while  the  best  effort  of  the  Lick  36-in.  will  be 
0-14". 

If  the  foregoing  argument  is  not  immediately  clear  and  acceptable 
to  you,  don't  worry  ;  it  has  taken  me  thirty-five  years  to  really 
tumble  to  its  essential  simplicity.  I  gather  that  it  has  met  with 
approval  at  the  Lick,  where  my  good  friend  Director  R.  G.  Aitken, 
working  under  the  finest  observing  conditions  in  the  world,  has 
built  up  year  by  year  a  monumental  mass  of  measurements  of 
Double  Stars — and  now,  knowing  more  about  them  than  any  man, 
confesses  himself  much  at  a  loss  to  know  why  such  things  exist 
at  all,  and  entirely  unwilling  to  attempt  to  discriminate  between 
competing  theories  of  their  origin. 

§  633.  The  minimum  angle  separable  by  your  own  keenly  focussed 
Eye,  2  mm.  diam.,  should  therefore  be  1/3200  radian,  1/320  in. 
at  10  in.  distance,  l/32nd  in.  at  100  in. ;  so  you  can  set  up  a  good 
plain  foot-rule  that  way  off,  in  the  best  light,  and  try  how  keen- 
sighted  you  are — or  better  with  wire  gauze,  as  below. 

Now  see  how  this  fits  in  with  its  internal  structure. 

I  have  measured  many  micro-sections,  and  find  the  retinal 
cones  are  very  uniformly  1/350  mm.  thick ;  frog  or  man,  it  looks 
as  if  Nature  has  standardized  production. 

Each  has  one  nerve  fibre,  and  is  either  stimulated  or  not :  plainly, 
if  all  are  stimulated,  no  structure  will  be  apparent ;  and  the  finest 
visible  structure  will  be  that  which  leaves  every  other  cone  in  the 
dark,  i.e.  which  has  a  spacing  1/175  mm.  on  the  retina. 

At  the  18-7-mm.  back  focal  length  of  the  eye  accommodated  for 
near  vision,  this  subtends  an  angle  1/(175  x  18-7)  =  1/3300  radian. 
Good  enough  ? 

The  experiment  I  have  set  you  is  quite  difficult,  and  if  you  try  it 
with  a  bit  of  wire-gauze,  set  up  in  front  of  a  diffused  light,  not  too 
glaring,  you  will  find  that  you  glimpse  the  structure  by  fits  and  starts: 
the  whole  fixed  pattern  can't  always  exactly  fit  the  retinal  carpet. 

§  634.  You  can  get  on  very  much  better  with  the  apparatua  of  Fig.  270, 
which  measures  the  Resolving  Power  of  a  Telescope,  effects  its  convorwon 
into  a  Microscope,  and  shows  the  action  of  that  instrument  and  its  CondeMer. 
It  consists  primarily  of  an  old  pocket-telescope  capped  with  an  iris  diaphragm. 


All  obstructions  to  the  left  of  the  iris  being  removed,  a  piece  of  1/64-m.  (or 
1/40  in.)  wire-gauze,  from  the  cheap  stores,  is  set  up  before  an  opal  lamp, 
or  the  window,  13  ft.  (or  21  ft.)  from  the  object-glass,  giving  a  structuro  mth 
an  angular  separation  of  1/10,000. 


622  LIGHT  [§634 

The  iris  is  opened  and  closed,  and  finally  set  at  the  diameter  at  which  the 
streaky  grey  blur  abruptly  resolves  into  dotted  gauze ;  this  is  d,  and  is  measured 
with  fme  dividers,  and  1-13  xjd  worked  out. 

The  erecting-glass  er  is  removed,  changing  the  telescope  into  an  inverting 
one  of  half  the  magnifying  power  :  the  iris  is  reset,  and  always  gives  the  same  d. 

A  quite  low-power  micro,  o.  g.  is  screwed  in  front  of  the  iris,  and  a  slide  of 
polycistina,  having  an  opal  lamp  close  behind,  is  focussed  by  the  microscope 
thus  constituted.  No  measurement  is  attempted,  but  the  effect  of  closing 
the  iris  on  the  details  visible  is  observed. 

The  tube  is  drastically  shortened,  or  lengthened,  and  the  slide  pushed  along 
into  focus  again ;  the  m.p.  can  be  varied  as  much  os  4  :  1,  but  the  same  details 
are  visible  at  the  same  iris  apertures,  as  near  as  can  be  judged. 

Finally,  the  microscope  is  directed  to  a  distant  lamp,  and  all  detail  disap- 
pears, for  the  cone  of  illumination  {i.e.  the  N.A.  in  use,  §  635),  and  therefore 
the  pip  of  light  which  enters  your  eye,  is  ridiculously  small.  Then  a  bull's- 
eye  is  stuck  behind  the  slide,  as  shown,  and  concentrates  the  distant  lamp's 
light  upon  it  in  a  cone  of  sufficient  angle,  and  N.A.  and  detail  are  restored. 

§  635.  The  best  pinhole  cameras,  §  472,  always  show  a  softness 
of  definition  due  to  inadequate  aperture,  but  all  photographers  use 
lenses  of  ample  aperture ;  and  simple  magnifying-glasses,  as  we 
have  seen,  look  after  themselves  quite  effectually.  But  you  have 
to  nurse  your  microscope  : 

Resolving  Power  of  the  Microscope.  Looking  at  Fig.  267,  you  see 
that  the  Minimum  Angle  of  Separation  now  takes  the  form  s/f, 
where  s  is  the  smallest  distance  apart  of  separate  details  in  the  object, 
and  /'  is  the  focus  of  the  strong  lens  which  converted  telescope 
into  microscope. 

s__  113X  _  1'13X 

'•    f-     d  "  '-   d/f 

(It  must  be  /'  on  the  slanting  line,  because  the  axial  one  is 
obstructed  by  thickness  of  glass.) 

Here  d/f  is  what  the  photographer  would  call  the  /  value  of  his 
aperture,  e.g.  if  d  were  l/5th/,  it  would  be//5,  see  §  613. 

The  microscopist  happened  to  choose  the  Radius  r  =  \d,  instead, 
and  calls  r/f  the  Numerical  Aperture  of  his  object-glass  ;  so  that  the 
photographer's //5  is  the  microscopist 's  N.A.  0-1 

_    113X         0-565X 
•  •  ^  -  2N:A.  """^  NAT 

Which  for  White  Light,  X  0-000555  mm.,  becomes 

Smallest  resolvable  distance  =  — ^^^-r —  =  „^^^  „  .    mm. 

N.A.  3200  N.A. 

so  that  3200  N.A.  spaces  or  dots  per  mm.  or  3-2  N.A.  per  micron, 
or  about  80,000  N.A.  per  in.,  is  the  finest  structure  a  microscope  can 
discern. 

You  see  that  the  focal  length  of  the  o.g.,  on  which  magnification 
depends,  has  disappeared.  It  does  come  in,  in  a  covert  way, 
because  the  maker  finds  it  difficult  to  get  a  large  N.A.  without 


§636]  OPTICAL   INSTRUMENTS  523 

making  it  short,  so  short  that  we  need  not  now  make  any  distinction 
between/'  and  the  nominal  focal  length /of  the  complete  objective  ; 
but,  apart  from  that,  mere  magnification  is  nothing  worth  ;  it  ia 
N.A.  that  determines  what  you  shall  see ;  and,  naturally,  what 
you  shall  pay.  A  1/6-in.  N.A.  0-85  costs  double  a  1/6  N.A.  ()-65, 
but  while  the  latter  might  tell  you  that  Pleurosigma  angulatum, 
§568,  was  a  clear  structureless  brownish  thing,  the  former  would 
insist  that  it  was  nothing  of  the  sort,  but  completely  colourlens,  and 
entu-ely  covered  with  a  fine  regular  structure  which  it  showed  you 
as  round  dots. 

§  636.  Illumination  of  the  Microscope.  But  whatever  money 
you  may  have  spent,  your  N.A.  is  valueless  unless  you  light  it  up, 
see  Fig.  271,  1. 

If  a  telescope  seems  to  be  performing  badly,  you  naturally  pull 
out  the  eyepiece  and  look  along  the  dark  tube,  to  see  if  the  round 
window  at  the  far  end  is  obstructed  in  any  way — by  something 
come  adrift  in  the  tube,  by  dew,  or  by  tree  branches — and  you  clear 
the  darkening  obstacle  away  as  a  matter  of  course. 

Fig.  267  shows  that  your  microscope  tube  is  a  telescope  tube,  so 
lift  out  your  eyepiece  and  look  at  the  round  window  at  the  bottom 
of  the  darkness  :  is  it  a  full  round  patch  of  light,  like  the  telescope 
showed  ? 

With  the  low-power,  quite  likely ;  but  with  the  *  sixth  *  ?  If 
not,  plainly  the  telescope  partner  in  the  combine  isn't  doing 
his  bit. 

Can  you,  with  a  glance  at  the  field,  tell  which  power  is  in  use  ? 
Is  the  high-power  picture  as  sharp  and  clear  and  finely  drawn  in 
every  detail  as  the  low  ? 

*  Oh,  but  something  must  be  allowed  for  high  magnification.' 
Must  it  ?  Did  you  buy  that  lens  just  for  the  plea.sure  of  looking 
at  a  poster  daub  instead  of  a  finely  painted  miniature  ;  or  because 
you  wanted  to  see  ten  times  as  much  of  the  innards  of  things  ? 
Would  that  sort  of  picture  please  you  in  a  telescope  that  was  any- 
thing more  than  a  family  heirloom  ? 

Light  up  that  window  :  and  you  try,  and  find  it  isn't  so  easy. 
Why  not  ?  In  Fig.  267,  draw  r  as  0-65/',  and  as  0-85/',  and  look 
at  the  big  angles  of  the  cone  of  light  that  must  be  poured  through 
your  translucent  object  to  fill  that  back  lens.     Or  observe  that 

r/f  =  N.A.  =  sine  {semi-angle  of  cone) 

and  look  up  the  angles  in  a  table  of  sines,  0-65  =  sin  41°  and  0-85  = 
sin  58°,  or  whole  angles  of  82°  and  116°  ;  or  see  §  638. 

Lying  on  your  back  on  the  roof,  and  holding  the  microscope  up 
to  the  broad  sky,  you  could  undoubtedly  get  these  cones  of  light, 
but  that  attitude  is  not  greatly  favoured  even  by  astronomen*. 
You  must  have  some  *  convergent '  svstem  to  *  condense  * 
'  parallel  light '  into  this  cone,  to  compel  plane  waves  of  a<lequatc 
width  to  cave  in  and  centre  down  on  to  the  object.    The  concave 


524  LIGHT  [§  636 

mirror  does  it  for  the  low  power,  after  a  nasty  astigmatic  fashion, 
but  even  one  6  in.  across  could  not  fill  the  cone  for  the  sixth. 

A  lens-system  is  indicated,  and  this  is  usually  the  two  fat  lenses 
of  the  Abbe  Condenser,  Fig.  274,  II,  shaped  so  as  to  reduce  spherical 
aberration  as  well  as  can  be  done  at  the  price,  i.e.  to  gather  in  and 
throw  light  at  wide  angles  truly  into  a  cone.  It  does  so  only  up 
to  N.A.  0-5,  i.e.  30°  half-angle  ;  beyond  that  the  focus  shortens 
like  Fig.  229  D  ;  and  by  N.A.  0-65  is  so  badly  out  that  one  has  to 
use  an  Iris  Diaphragm  to  cut  the  rest  of  it  off,  as  making  itself  a 
foggy  nuisance.  The  Condenser  is  always  used  with  the  Plane 
Mirror. 

Incidentally  that  means  that  when  you  have  focussed  the 
condenser-image  of  the  source  of  light  on  to  the  object  itself, 
under  the  low  power,  which  takes  in  only  a  narrow  cone,  and  have 
then  swung  round  the  sixth,  which  takes  in  the  whole,  it  makes 
out  iihat  the  focus  is  shorter  (nearer  D  than  F,  Fig.  229),  and  you 
may  have  to  bring  up  the  condenser  a  trifle  closer,  to  fill  the  back 
lens    of   the    o.g.   with  light,   Fig.  271   (2). 

O^^^^  Thus  although  you  must  focus  your  Abbe, 
.  ^^^^^  its  residual  aberration  soon  teaches  you  not 
I    ■  ■       to  fuss  with  it  overmuch,   as  e.g.  between 

J    ^^^^^       different  slide-thicknesses. 
'    ^^^^^  Fig.  274  II  shows  it  in  use  as  a  Dark- 

Ground  Illuminator,  very  close  up  and  im- 
mersed to  the  slide  with  water,  its  middle 
obstructed,  and  its  iris  opened  right  away. 
Ordinarily,  of  course,  you  reverse  all  this,  it 
Fig.  271.  jg  ^^^  g^  exceedingly  close  up,  it  is  used  dry, 

the  central  cone  is  in  use,  and  the  Iris  is 
cutting  off  all  the  edges  left  open  in  II. 

Fig.  271  (1)  shows  the  paltry  sort  of  lighting  you  see  on  the  back 
lens  of  your  0-85  N.A.  sixth  or  eighth,  with  Concave  Mirror.  No.  2 
shows  what  you  should  aim  at,  a  solid  cone  of  light  3/4  the  diameter, 
which  is  quite  the  best  an  Abbe  can  do  ;  don't  let  it  go  to  3  or  4, 
because  they  simply  cause  white  foggy  confusion.  No.  4  is  beginning 
to  show  the  action  in  use  in  Fig.  274,  except  that  the  central  spot 
is  stopped  out  then  :  it  is  too  close  to  the  slide  for  ordinary  working. 

§  637.  High  powers  of  the  Microscope.  So,  after  all,  you  are 
not  filling  the  '  telescope  '  lens  with  light,  and  you  are  not  getting 
your  money's  worth  ?  And  you  often  do  better  cutting  down 
quite  appreciably  with  the  iris  ?  In  fact,  you  can't  help  doing  so 
in  objects  of  insufficient  contrast,  unstained,  etc. ;  cf.  the  buff  hen's 
legs  in  §  618  ;  but  this  present  discussion  on  resolving  power  pre- 
supposes excellent  contrasts  in  the  object. 

Well,  a  high-power  micro,  o.g.  is  a  more  complex  problem  than 
a  tele,  o.g.,  and  above  0-65  N.A.  they  just  can't  be  made  good 
enough,  at  a  reasonable  price,  to  bear  the  full  blaze  of  light  up  to 
their  edges  ;  the  outer  zones  inevitably  have  some  aberrations  left. 


®9 


§638]  OPTICAL   INSTRUMENTS  525 

So  don't  be  greedy  and  flood  them  out,  but  leave  them  to  glean  the 
light  that  the  object  diffracts  all  around— you  see  them  doing  it 
as,  with  eyepiece  removed,  you  slide  the  section  in  and  out— and 
they  will  give  you  valuable  suggestions  that  a  smaller  N.A.  knows 
nothing  about.  The  maker,  so  to  speak,  sells  you  a  perfect  0-65 
and  gives  you  the  rest ;  just  as  the  camera-lens  maker  sells  you  a 
perfect //6  and  lets  you  open  it  to//4-5  if  you  must  have  speed. 

Your  '  twelfth,'  which  is  often  1/14  or  even  1/16  in.  focal  length 
and  is  marked  N.A.  1-30,  you  cannot  overload  with  light  from  the 
Abbe  ;  as  you  use  it,  it  is  mostly  gleaning,  its  immense  powers 
are  fully  brought  out  only  by  achromatic  condensers. 

No  sine  can  exceed  10  ;  r  cannot  be  greater  than/,  Fig.  267  :  what 
about  N.A.  1-30?  ^       e         , 

Keep  your  light  in  oil,  [i  =  1-5,  until  the  lens  has  dealt  faithfully 
with  it.  It  is  travelling  only  1/1-5  as  fast,  so  its  waves  are  only 
2/3  as  long,  Fig.  230,  and  can  search  out  details  only  2/3  aa  big, 
i.e.  its  Resolving  Power  is  increased  \l  times. 

So,  for  an  Immersion  Objective,  N.A.  =  jx  sin  {half  angle  of  cone). 

The  usual  immersion  medium  is  *  oil  of  cedar  wood,'  thickened 
with  Canada  balsam,  and  possessing  the  same  refraction  and  dis- 
persion as  crown  glass  for  most  colours,  so  that  it  is  •  homogeneous  * 
with  the  glass  of  cover-slip  and  front-lens.  Light  from  the  object 
therefore  goes  straight  through  to  the  hemispherical  back  of  the 
first  lens.  Fig.  230,  with  no  trouble  due  to  cover-glass  thickness 
(below),  and  leaves  the  hemisphere  strictly  as  if  it  came  from  a 
virtual  aplanatic  point.  This  simplifies  the  maker's  difficult ies 
enormously  ;  and,  above  all,  it  combs  out  the  details  of  the  object  with 
a  50%  finer  tooth.  '  Dry '  twelfths  have  followed  the  '  thirty- 
seconds  '  and  '  eightieths  '  of  the  dark  pre-N.A.  days,  into  oblivion. 

Achromatic  Substage-Condensers,  which  are  practically  big 
object-glasses  of  high  N.A.,  used  backwards,  far  excel  the  Abbe  in 
aplanatic  {i.e.  true  focussing)  aperture,  and  afford  very  perfect 
control  of  beautifully  colourless  illumination  ;  with  them  one  can 
verily  make  the  sun  shine  in  dark  places.  They  are  very  desirable 
in  photomicrography,  and  only  by  their  aid  can  the  full  power  of 
high-aperture  lenses  be  brought  out,  for  research  ;  but  no  nie<lical 
student's  mouth  need  water  for  one.  For,  knowing  what  fine 
effects  were  obtainable,  he  would  find  himself  spending  so  much  time 
always  adjusting  it  to  its  very  best,  that  more  necessary  work 
would  go  short.  Its  aberration  is  not  the  least  of  the  saving  graces 
of  the  serviceable  Abbe. 

§638.  Cover-glass  thickness  and  tube-length.— Your  object  is 
covered  with  glass,  \i  1-5,  and  is  O  in  Fig.  192.  Narrow-angle  low- 
power  o.g.'s  have  therefore  to  be  focussed  on  the  cusp  of  the  caustic, 
which  is  2/3  the  depth  of  O,  but  the  extreme  rays  collected  by 
wide-angle  o.g.'s  are  nearly  enough  the  45*^  for  0-7  N.A.,  the  60" 
for  0-85  N.A.,  and  the  76**  for  0-95  N.A.,  and    you   see   how 


526  LIGHT  •  [§  638 

successive  zones  of  these  lenses  have  to  focus  on  higher  and  higher 
points. 

This  aberration  is  countered  by  the  maker  increasing  the  air- 
space between  the  front  lens  of  the  o.g.  and  the  compensating  lenses 
which  follow  it ;  thereby,  of  course,  quite  unfitting  the  o.g.  for 
viewing  uncovered  objects. 

The  amount  of  it  is  proportional  to  the  thickness  of  the  glass, 
and  the  maker  assumes  that  you  will  use  No.  1  quality  cover-glass 
0-16  mm.  thick,  and  will  squeeze  out  and  remove  any  excess  of 
mountant,  a  layer  of  which  counts  as  so  much  more  glass. 

It  is  no  economy  to  use  thicker  glass,  for  you  get  fewer  pieces  for 
your  money,  and  you  can  soon  learn  the  knack  of  cleaning  thin 
ones  without  breakage  ;  lay  them  flat  on  wood  and  rub  pretty  hard 
with  a  rag,  with  a  trace  of  weak  acid  if  they  are  cloudy. 

Increased  cover-glass  thickness  can  be  compensated  for  by 
shortening  the  microscope  tube-length  :  correction  is  obtained  when 
a  minute  dot  in  the  shde  goes  out  of  focus  both  up  and  down  alike, 
not  forming  a  hard-ring  one  way  and  a  sudden  fuzzy  mist  the  other. 

Unfortunately,  this  is  seldom  available,  covers  are  never  too  thin, 
and  tubes  are  not  long  enough  to  shorten.  Therefore  let  your 
own  preparations,  at  all  events,  require  no  such  correction ;  and 
stick  to  the  Tube  Length,  from  large  flange  of  o.g.  to  top  of  eyepiece, 
that  is  marked  on  the  o.g. 

Objectives  of  N.A.  0-65  are  very  complaisant  as  to  cover-thickness 
and  tube-length  ;  with  N.A.  0-85  you  will  soon  find  it  pays  to  attend 
to  them  ;  N.A.  0-95  is  a  finicky  lady  of  high  degree,  with  a '  correction 
collar '  to  fiddle  with,  and  very  short-sighted,  and  from  her  it  is  a 
relief  to  turn  to  immersion  lenses. 

§  639.  Actual  measurement  of  N.A.  in  use.  First  measure  the 
Magnifying  Power,  as  in  §  631.  Then  measure  the  diameter  of  the 
sharp  bright  round  Eye -ring,  §  617,  by  a  micrometer  and  pocket - 
lens.  If  it  is  not  a  solid  round  patch  of  light  don't  waste  time 
on  it. 

Recollect,  §621,  that  this  diameter  is  inversely  proportional 
to  the  M.P. ;  it  would  be  double  at  half  the  M.P.,  it  would  be  the 
M.P.  times  as  much  at  magnification  1. 

Magnification  1  means  the  eye  itself,  working  at  a  distance  of 
250  mm. ;  and  (M.P.  x  eye-ring  diameter)  is  the  size  of  an  eye  which, 
working  at  that  distance,  could  resolve  just  as  much.  Half  this 
magnified  diameter,  divided  by  250  mm.,  is  thus  r/f,  the  N.A.  in  use. 

Sir  Almroth  Wright  described,  long  since,  his  Eikonometer, 
or  image  measurer,  for  measuring  M.P.  In  it  a  44-D  lens,  held 
over  the  eyepiece,  acts  as  the  object-glass  of  a  diminutive  telescope 
of  fixed  length  25  mm.,  and  forms  a  real  image,  on  a  scale  SS  divided 
into  tenths  of  a  mm.,  exactly  one-tenth  the  size  of  the  virtual  image 
you  would  otherwise  be  observing  at  the  conventional  250  mm. 
Image  and  scale  are  in  the  focus  of  a  Ramsden  eyepiece  X  10, 
and  the  effect  has  been  to  lay  a  millimetre  scale  on  the  usual  image. 


040J 


OPTICAL  INSTRUMENTS 


527 


The  Magnifying  Power  is  therefore  the  length,  read  in  mm.  on  this 
scale,  of  1  mm.  on  a  Stage  Micrometer  focus^  under  the  microacope. 

Between  us  we  have  developed  this  into  what        

may  be  called  an  '  Eikap,'  Fig.  272.  One  simply 
pulls  the  inch-long  telescope  apart  amidships,  and 
the  black  middle  sleeve  now  carries  the  tenth- 
mm.  scale  and  its  magnifier.  Apply  this  to  the 
Eye  Ring,  and  measure  its  diameter  :  then,  a.s 
above  (half  this,  in  mm.  x  M.P.),  divided  by 
250  mm.  =  N.A. 

Outer  dotted  sliding  steady -sleeves  are  a  con- 
venience in  use  to  rest  on  the  eyepiece. 

§640.  Colour  and  Resolution.  The  smallest 
separable  detail  being  spaced  at  0-565  X/N.A.  is 
evidently  proportional  to  the  wave-length,  which  is 
10%  less  for  F-line  blue,  and  20%  greater  for  C-line 
red.  Fig.  223. 

Try  setting  up  a  Pleurosigma  test-slide  under  a 
sixth,  and  stopping  down  the  iris  as  small  as  will  give  good  dot- 
resolution  ;    while  the  definition  remains  good  through  blue,  green 
or  amber  glass,  it  vanishes  completely,  beyond  call  of  refocuasing, 
in  red. 

It  should  also  be  an  advantage  to  use  full  blue  light,  but  this 
seldom  succeeds.  For  the  maker  has  perfected  the  corrections 
of  his  lens  for  the  brightest  part  of  the  visual  spectrum,  and  if  you 
will  go  far  from  that  into  the  darkening  blue,  you  must  expect  to 
get  into  the  rough. 

A  light  blue  (cobalt)  glass  is  often  used  to  kill  the  glare  and 
yellowness  of  lamplight,  and  it  makes  for  comfort.  Spectroscoping 
it,  however,  §  558,  one  finds  it  passes  a  conspicuous  band  of  red, 
blots  out  rather  a  lot  of  the  bright  middle  of  the  spectrum,  and  then 
passes  a  long  useless  tail  of  violet,  these  unruly  extremities  of  the 
spectrum  being  made  all  the  more  conspicuous  by  the  dimming  in 
the  middle.  Nothing  much  worse  could  be  contrived ;  and  the 
improvement  in  definition,  on  substituting  for  this  a  light  blue-green 
Giftard  or  similar  colour-filter,  which  cuts  out  the  violet  and  sub- 
dues the  red,  without  losing  any  stain-colour  worth  keeping,  is  veiy 
noticeable  indeed,  and  well  worth  the  cost. 

For  photomicrography  the  simplest  thing  is  to  cut  out  the  violet 
by  an  ordinary  orthochromatic  yellow  screen ;  and  the  method 
given  in  the  *  Hints  '  is  good  enough  to  start  with.  If  the  camera- 
lens  is  removed  it  involves  re-focussing  on  a  ground-glass,  and  then 
you  get  the  arrangement  of  Fig.  273. 

In  Apochromatic  Objectives  (apo  =  away  from),  by  the  use  of 
fluorite,  and  of  additional  glasses  (the  high  powers  contain  ton 
lenses),  it  has  been  possible  to  obtain  practically  complete  colour 
correction  throughout  the  spectrum,  and  also  to  eliminate  other 
aberrations  more  perfectly  and  tackle  minor  ones,  at  the  same 


628 


LIGHT 


[§640 


time  widening  out  the  N.A.  The  finishing  touches  are  given  in 
four-lens  compensating  eyepieces,  the  cost  of  which  adds  on  to  the 
already  five  times  greater  cost  of  the  objectives. 

For  photomicrography  in  short-"v\^ave  blue,  with  an  achromatic 
or  even  apochromatic  condenser,  and  also  for  dark-ground  illumina- 
tion, these  lenses  are  invaluable ;  but  for  ordinary  visual  work, 
as  compared  with  the  modern  achromatic  lenses  now  readily  avail- 


FiG.  273. 


able,  their  superiority  is  by  no  means  so  conspicuous.  To  make 
the  best  use  of  them,  one  would  prefer  also  a  modern  Binocular, 
with  cranked  eye-tubes.  A  1-4-N.A.  3-mm.  or  2-mm.  apo.  has 
long  been  the  best  lens  for  oil-  or  balsam-mounted  specimens  ;  a 
new  1-6  N.A.  demands  special  mountant,  a  flint-glass  cover,  and 
monobromide  of  naphthalene  immersion  fluid.  For  water  work, 
water-immersion  lenses  up  to  1-2  N.A.  are  convenient. 

§641.  The  Ultra-Violet  Microscope.  Since  5  =  0-565  X/N.A., 
halving  the  wave-length  would  halve  the  minimum  separable 
distance,  and  thereby  double  the  Resolving  Power,  enabling  four 
times  as  many  details  to  be  perceived  on  a  given  area. 

A  storm  of  short  electric  sparks,  fattened  up  by  Leyden  jars  in 
parallel  with  the  spark-gap,  between  magnesium  or  cadmium  poles, 
provides  a  choice  of  ultra-violet  radiations  in  the  neighbourhood  of 
X  0-275  micron — ^just  half  the  average  of  white  light — and  these 
radiations  are  passed  through  quartz  spectroscope  prisms,  and  one 
line  is  selected  to  provide  '  monochromatic  '  ultra-violet. 

To  this,  glass  is  opaque,  and  accordingly  the  condenser,  slide  and 
cover-slip,  and  '  eyepieces,'  must  be  worked  from  clear  fused 
silica,  or  crystalline  quartz,  while  the  object-glasses  are  '  monochro- 
mats  '  of  silica  and  fluorite.  Water,  normal  saline,  glycerine,  and 
castor  oil  are  transparent  to  the  radiation,  glycerine  being  used  as 
immersion  fluid  for  the  0-85  and  1-25  N.A.  lenses. 

The  radiation  is  invisible,  but  excites  green  fluorescence  in  a 
thin  slip  of  uranium  glass  in  the  focal  plane  of  an  eyepiece,  and 
preliminary  adjustments  are  made  by  aid  of  this,  but  fine  focussing 
must  be  done,  and  all  observations  made,  photographically. 

The  specimens  must  not  be  fixed,  hardened,  dried,  or  stained  : 
moving  bacteria,  spores,  yeasts,  etc.,  are  restrained  by  using  a 
very  thin  layer  of  fluid.  Differences  of  molecular  weight,  among 
the  substances  present  in  the  cells,  often  play  the  part  usually  taken 
by  differential  stains,  in  disclosing  internal  structure. 


642] 


OPTICAL   INSTRUMENTS 


529 


Subject  to  these  limitations  and  difficulties,  much  deUiled  in- 
formation  has  been  obtained  of  structures  which  to  ordinary  lieht 

are  a  perfect  blank.  * 

§  642.  The  Ultramicroscope  has  an  impressive  name,  but  if  you 
have  a  microscope  with  2/3  and  1/6  o.g.'s  and  ordinary  Abbe 
condenser,  the  further  expenditure  of  three-halfpence  on  a  stick  of 
plasticme  will  enable  you,  with  little  trouble,  to  rig  up  very  efficient 
ultramicroscopes  indeed. 

We  have  to  give  up  hope  of  seeing  microscopic  structure  smaller 
than  about  J  micron,  but  the  problem  now  remains  to  find  whether 
we  have  anything  there  or  not,  anything  sufficiently  big  and 
obstructive  to  knock  chips  enough  off  a  wave  of  light  for  us  to  sec. 

It  is  a  question  of  dark-ground  illumination;  that  chip  would 
never  be  missed  from  the  whole  wave,  but  we  look  for  it  where 


Fig.  274. 

no  waves  go.  You  do  not  look  for  dust  in  the  air  between  your  eye 
and  the  wide  window,  but  pull  the  blind  and  stand  aside  and  let 
the  sun  stream  in  through  a  chink,  and  you  see  the  dancing  motes 
in  its  beam,  against  the  background  of  the  darkened  ro<jm.  Only 
the  little  light  they  scatter  can  enter  your  eye,  the  main  stream  passes 
on  aside  ;  all  other  illumination  you  have  suppressed. 

Take  a  little  box  (a  match-box  casing  will  do),  cut -a  hole  as  in 
Fig.  274  I,  ink  it  all  over  inside.  Take  the  lower  lens  of  your 
Abbe  condenser  and  stick  it  on  the  end  of  the  box.  Puff  smoke 
into  the  box,  and  reflect  a  strong  light — sunlight  is  best,  or  a  harsh 
bare-wire  lamp,  or  a  gas-mantle — through  the  condenser  lens,  to 
form  a  brilliant,  rather  'aberrated,'  neck  of  light  in  the  smoke. 
Bring  down  the  2/3  and  focus  on  this  neck ;  stop  up  draughts,  of 
course,  and  there  is  Ultramicroscope  No  1. 

Wet  smoke  will  show  you  wheeling  battalions  of  brilliant  droplets, 
with  dry  smoke  the  particles  are  very  much  smaller. 

By  making  up  a  tin  resonance  tube  the  right  length,  and  exciting 
it  with  the  fork,  the  smoke  particles  streak  out  lengthwise,  showing 
the  longitudinal  motion  of  the  air,  Fig.  143. 

In  liquids,  the  problem  is  to  study  nutrient  or  antiseptic  emulsions. 
to  see  living  and  unstained  bacteria,  to  explore  '  filterable  vinis«/ 
colloidal  solutions,  etc. 


630  LIGHT  [§  642 

Rebuild  your  Abbe,  stick  a  shilling,  or  a  disc  that  size,  centrally 
on  the  lower  lens,  and  put  a  big  drop  of  water  on  the  top  one. 
Bring  it  up  into  *  optical  contact '  with  the  under  side  of  an  ordinary 
1/20-in.  slide,  on  which  you  can  then  put  a  drop  of  water  tinged 
with  the  least  suspicion  of  burnt  umber,  carmine,  etc.  Turn  on 
your  strong  light,  and  under  the  2/3  focus  the  condenser  until  the 
black  spot  shrinks  away,  leaving  a  bright  one ;  focus  the  l/6th  on 
this,  and  there  is  Ultramicroscope  No.  2  showing  the  Brownian 
movement  in  full  swing. 

Fig.  274  II  shows  how  the  central  illumination  is  stopped  out, 
while  the  very  aberrant  wide-angle  marginal  rays,  which  you  cut 
off  in  ordinary  practice,  now  light  up  the  object  from  all  sides 
(only  a  vague  outside  edge  is  suggested  for  them  in  the  figure)  and 
provide  enough  for  small  particles  to  scatter,  and  so  declare  their 
existence,  all  other  light  missing  the  lens. 

Fig.  274  III  shows  the  usual  Cassegrain  dark-ground  illuminator. 
Light  falls  on  the  silver  ball  in  the  middle  of  the  glass  block, 
is  reflected  to  the  silvered  concave  surface,  and  focussed  by  it  in 
a  hollow  cone,  as  before  (it  is  shown  working  into  oil,  and  so  missing 
the  wide -spreading  aerial  refraction  seen  in  II)  ;  it  is  a  miniature 
Cassegrain  telescope,  Fig.  253,  working  backwards.  It  focusses 
the  light  better,  and  keeps  it  colourless,  but  for  all  ordinary  use 
the  patch- stopped  Abbe  does  just  as  well. 

If  the  immersion  water  trickles  out,  use  glycerine  or  treacle ; 
never  use  immersion-oil  under  a  slide,  it  is  a  pest. 

Ultramicroscopic  particles,  far  too  small  for  any  shape  to  be 
*  resolvable,'  disclose  their  presence  as  minute  '  diffraction-discs  ' 
of  light,  of  size  dependent  on  the  optical  system  of  the  microscope, 
and  of  brightness  dependent  on  the  size  and  obstructiveness  of  the 
particle,  and  of  course  on  the  intensity  of  the  source  of  light.  With 
the  electric  arc  particles  0-015  micron  in  diameter  have  been  glimpsed, 
and  with  sunlight  0-005  micron,  one  trillionth  of  a  gram  of  gold. 

§  643.  Microscope  Hints.  Keep  locked  up,  and  don't  lose  the 
key. 

Lubricate  sliding  parts  with  vaseUne,  and  axles  with  thin  machine 
oil. 

Tighten  the  coarse  adjustment  by  the  screws  on  the  pinion 
bearing. 

Never  let  alcohol  touch  finished  brasswork  or  achromatic  lenses. 

Spots  on  the  field  of  view  may  be  on  lower  lens  of  eyepiece,  or  on 
condenser  lenses,  or  on  mirror  ;  rotate  the  suspected  part. 

Clean  lenses  with  a  clean  handkerchief  ;  the  sixth's  nose  may  get 
stuck  over  with  balsam  ;  wet  the  rag  with  one  drop  of  xylol.  This 
also  cleans  oil  off  the  twelfth  ;  don't  leave  this  oil  to  dry  on,  and 
never  use  it  as  a  lubricant. 

An  eyepiece  kept  in  place  keeps  out  the  dust,  but  eyepieces 
drop  out  of  inverted  tubes.  Use  the  lower  eyepiece  mostly,  re- 
serving the  higher  for  fine  detail  in  mid-field. 


§  643]  OPTICAL  INSTRUMENTS  53  i 

Dust  on  back  lens  of  o.g.  comes  out  on  a  clean  camel-hair  bruah. 
If  the  2/3  consists  of  two  lenses,  the  front  can  be  screwed  off  to 
leave  a  very  low  power,  sometimes  useful  in  searching. 
Don't  attempt  to  take  high  powers  to  pieces. 

Drawtube.— Low  powers  tolerate  any  length  of  tul>e,  any  eye- 
piece,  and  any  thickness  of  cover-glass,  but  higher  object-gUnM 
are  optically  corrected  for  a  tube-length  of  170  mm.,  and  any  con- 
siderable departure  from  this  impairs  their  definition.  (Tbot.— An 
isolated  speck  in  the  plane  of  the  object  must  go  out  of  focus  exactly 
the  same  up  and  down,  not  hard  ring-speck-fuzz.)  With  the  sixth, 
an  unduly  thick  cover-glass  calls  for  a  shortened  tube,  which  is 
usually  impracticable  ;  therefore  use  only  best  thin  cover-glaaii,  and 
squeeze  out  excess  of  mountant. 

Illumination  is  all-important.  Pull  out  the  eyepiece  and  look 
at  the  back  lens  of  the  o.g.  ;  the  2/3  should  be  full  of  light,  higher 
powers  should  show  a  solid  patch  of  light  1/2  to  2,3  the  lens- 
diameter.  If  not,  it  is  as  futile  as  using  a  telescope  with  your 
fingers  held  in  front. 

The  smallest  perceptible  angle  between  entering  light -waves  is, 
as  in  all  optical  instruments,  0-56  (wave-length)  (radius  of  thi« 
window).  This  angle  is  the  closest  distance  of  perceptible  details 
in  object  divided  by  the  focal  length  of  the  objective.  Therefore 
closest  details  resolvable  =  0-56  (w.l.)/(radius/focus),  or,  inverting, 
Maximum  number  of  details  resolvable  per  unit  distance  is  (Numerical 
Aperture  of  objective  in  use)/(0-56  wave-length  of  light),  which, 
with  white  or  green-tinted  light,  comes  to  3200  N.A.  dots  per  mm. 
or  just  over  80,000  per  inch. 

With  broad  skylight,  the  mirrors  alone  do  fairly  well ;  with 
lamplight  the  concave  serves,  but  is  apt  to  throw  skew  shadows ; 
a  glance  at  the  back  lens  will  show  why. 

The  Substage  Condenser  is  used  with  the  plane  mirror ;  focus  a 
slide  under  the  low  power  and  screw  condenser  up  until  a  picture  of 
the  source  of  light  itself  is  focussed  on  the  object,  swing  round  the 
high  power,  fine-focus  it,  and  open  or  close  the  iris  to  *  best  seeing.* 

Too  wide  causes  white  fog ;  too  narrow,  heavy  outlines  and  lost 
detail. 

(If  the  condenser  has  centring  adjustments,  you  searched  for  the 
little  aperture  of  the  closed  iris  with  vour  lowest  power,  and  centred 
it.) 

If  in  doubt,  always  look  at  that  back  lens. 

Now  let  the  condenser  be  :  the  na«ty  view  of  the  source  of  light 
that  shows  under  the  2/3  can  be  blurred  out  by  turning  up  the 
concave  temporarily. 

Dim  a  dazzling  light  by  tinted  glasses,  or  by  using  its  reflection 
in  unsilvered  glass,  and  not  by  closing  the  iris. 

If  only  that  eyesore,  a  bare-wire  lamp,  is  available,  lay  a  slip  of 
ground-glass  under  the  slide. 


63^  LIGHT  [§  643 

For  continued  low-power  work,  use  the  lower  lens  of  the  condenser 
alone. 

For  really  fine  seeing,  you  must  be  content  with  a  very  small 
illuminated  area  (edge  of  lamp  flame).  A  light-green  filter  always 
helps. 

Although  at  the  expense  of  fine  definition,  a  narrow  or  even 
oblique  illuminating  cone  may  be  necessary  to  produce  sufficient 
contrasts  in  living  and  unstained  subjects  to  make  them  visible 
at  all. 

Get  Dark-ground  Illumination  from  your  Abbe  by  sticking  a 
central  shilling  beneath  it,  bringing  closer  up,  and  '  immersing  ' 
to  slide  with  a  drop  of  water. 

For  Photography,  lay  horizontal  and  focus,  relaxing  your  eye  to 
'  distance,'  bring  up  camera  with  lens  at  infinity  focus  and  open 
wide,  use  orthochromatic  filter  and  film,  have  everything  very 
steady,  and  give  several  seconds  exposure. 


EXAM   QUESTIONS,   CHAPTER  XL 

That  is  a  Treatise  on  Optical  Instruments  ample  enough  for  most  of  you 
for  a  good  few  years  to  come,  and  a  more  liberal  treatment  than  the  exam 
questions  have  yet  arrived  at.  They  want  you  to  stalk  through  22 — 26, 
computations  that  in  practice  nobody  dreams  of  making ;  slide-rule  pushing. 
But  who  can  tackle  27 — as  yet  only  a  '  possible  ' — has  got  the  alpha  and  omega 
of  it,  for  what  happens  at  the  two  ends  of  the  instrument  is  what  he  uses  it  for. 

The  perpetual  inquiries  about  magnifying  power — like  asking  for  the  mmabers 
of  teeth  on  the  wheels  in  a  clock,  or  like  the  '  magnetic  islands  '  put  in  a 
mediaeval  map  because  customers  would  complain  if  they  were  left  out — 
are  excusable,  for  the  really  fundamental  Resolving  Power,  introduced  in 
§  620,  and  tackled  in  §§  632  to  the  end,  has  hitherto  made  its  appearance 
only  in  an  obscruing  web  of  '  higher  maths,'  %vith  which  in  essence  it  has 
nothing  whatever  to  do.  My  treatment  contains  no  weak  points,  and  yet 
approximates  rather  to  Tennyson's  dictum,  that  anything  worth  proving 
doesn't  want  it. 

Descriptive  questions  are  not  intended  to  be  particularly  searching,  but 
when  you  draw  us  a  diagram,  do  let  us  see  shortly  that  you  understand  it, 
and  can  explain  away  possible  slips ;  or  your  lot  is  likely  to  be  the  comment 
'  Seen  it  in  a  book.'     As  always,  handle  everything  you  can  get  hold  of. 

The  latter  end  of  the  chapter  is  the  distillate  of  forty  years  of  experience, 
blended  for  yom*  use;  if  you  want  more  than  semi -efficiency  from  your 
costliest  investment,  §  643  is  worth  the  price  of  the  book. 

1 .  How  would  you  attempt  to  produce  a  very  intense  beam  of  '  parallel ' 
light,  and  how  test  its  parallelism  ? 

Explain  the  action  of  a  pinhole  camera,  and  the  advantages  and  difficulties 
of  using  a  lens. 

2.  Figure  the  essential  parts  of  a  lantern  for  projecting.  With  an  8-in. 
lens  15  ft.  from  the  screen,  find  size  of  pictiu-e  of  slide  3  X  3  in.  Compare 
the  illmninations  of  slide  and  picture. 

Why  is  a  very  good  lens  necessary  in  projecting  opaque  objects  ? 

3.  Describe,  with  diagram,  the  construction  of  a  simple  telescope,  and  state 
modifications  in  practice  to  obtain  a  clear,  coloiu-less  and  extensive  field  of 
view.     (  X  3) 


OPTICAL  INSTRUMENTS  633 

4.  Describe,  with  diagram,  the  construction  and  action  of  an  a«tronomic«l 
telescope,  and  find  an  expression  for  its  magnifying  power.  How  can  iu 
image  be  made  erect  ?     (  x  3)  ^     o  r  «*.  «• 

5.  What  is  meant  by  the  magnifying  power  (1)  of  a  teI«»cope,  (2)  of  a 
microscope?      How    can    it    be    determined    experimentally    or    calculated 

theoretically  ? 

6.  Where  are  the  cross-wires  put  in  a  teleecope,  and  why  in  any  particular 

position  ?  </  J  I 

7.  Why  is  a  telescope  with  cross- wires  in  the  eyepiece  umhI  for  observing 
directions  of  objects  which  can  bo  quite  easily  soon  directly  ?  How  could 
you  ascertain  whether  its  cross-wires  intersect  on  its  axis  ? 

8.  Lenses  of  9  cm.  and  3  cm.  are  used  to  form  an  erect-image  telescope; 
draw  a  diagram  showing  the  magnification  of  a  distant  object.     (  X  2) 

9.  Describe  opera  glass,  and  find  focal  lengths  of  lenses  of  one  which  is 
3  in.  long  and  magnifies  3  times. 

10.  Describe  with  diagram  a  telescope  or  a  microscope;  how  ought  the 
eyepiece  to  be  shifted  for  a  long-sighted  person  ? 

11 .  Describe  some  form  of  telescope  to  produce  erect  virtual  images.  Draw 
a  diagram,  and  indicate  the  relative  focal  lengths. 

How  do  you  re-focus  (a)  for  short  sight,  (6)  to  produce  a  real  image  on  a 
photographic  plate  ?     (  X  2) 

12.  Explain  visual  accommodation.  A  4-cm.  lens  is  used  as  a  magnifier 
by  a  person  with  Dn  25  cm.     Draw  a  diagram  and  de<luco  M.P. 

13.  Explain  how  a  convex  lens  close  to  the  eye  acts  as  a  magnify ing-glass. 
and  calculate  the  focal  length  of  a  lens  with  M.P.  10.     (  x  3) 

14.  What  is  meant  by  the  '  magnifying  power,'  for  a  normal  eye,  of  an 
optical  instrument  ? 

What  is  that  of  a  lens  which  permits  an  object  o  cm.  away  to  bo  soon  clearly  ? 
Find  its  longest  permissible  focal  length,  and  draw  a  diagram.     (  x  2) 

15.  A  2-in.  focus  magnifier  is  held  1  in.  from  eye  with  D,  9  in.  MTiero 
must  object  be  ? 

16.  Show  that  two  convex  lenses,  1  in.  and  2  in.,  can  be  used  as  simple 
microscope,  telescope,  or  compoimd  microscope. 

17.  Draw  diagrams  showing  the  paths  of  several  rays  through  two  2-cm. 
convex  lenses  combined  into  (o)  a  telescope,  (h)  a  compound  microscope. 

With  a  number  of  lenses  to  choose  from,  what  magnitudes  of  focal  length 
would  you  select  for  these  two  instruments  ? 

18.  If  the  Mt.  Wilson  telescope  mirror  has  a  focal  length  of  60  ft.,  what  ia 
its  magnifying  power  without  any  accessories  ? 

19.  24  ft.  in  front  of  a  telescope  mirror  of  30  ft.  focus  is  a  convex  Casaegnun 
mirror  which  reflects  the  light  to  form  an  image  1  ft.  behind  iho  (perforated) 
concave.     What  is  its  focal  length,  and  what  is  the  magnification  7 

20.  Draw  a  diagram  showing  the  path  of  the  rays  through  a  compound 
microscope. 

How  would  you  define,  and  measure,  the  magnifying  power  ?     (  x  3) 

21.  How  can  a  single  lens  be  usetl  as  a  microscope  ?  What  condition  sets 
a  limit  to  its  magnification  in  practice  ?  Show  how  a  compound  miorosoope 
can  obtain  greater  magnification.  How  is  the  colour-dispersion  of  ita  leoMi 
corrected  ?     (  X  2) 

22.  Give  a  diagram  of  a  compound  microscope.  Can  it  be  rolated  in  any 
way  to  the  telescope  ? 

Where  must  an  object  be  placed  to  be  seen,  through  two  3-cm.  focus  leoM 
16  cm.  apart,  by  an  eye  focussed  for  distance?  What  is  the  magnifioaiioa 
of  the  intermediate  real  image,  and  the  total  M.P.  ? 

[Internal  image  3  cm.  from  eye-lens,  of  which  mfn.  is  then  25/3,  without 
any  +  1,  §  629.  Calculate  conjugate  focal  distances  for  o.g.  from  1/fnmt 
focus  4-  1/13  =  1/3,  then  13/3-9  is  mfn.  of  internal  image  itself;   final  27-8.) 


534  LIGHT 

23.  Ditto,  two  1-in.  lenses  8  in.  apart  ? 

24.  Lenses  of  focal  length  4  and  5  cm.  are  used  as  objective  and  eyepiece 
of  a  microscope.  Where  must  an  object  be  placed  to  produce  a  final  image 
26  cm.  from  the  eyepiece  ?     The  distance  between  the  lenses  is  20  cm. 

26.  An  object  is  0-55  in.  from  a  A-in.  o.g.,  and  the  1-in.  eye-lens  is  6  in. 
beyond.     Make  a  diagram,  and  find  the  final  image. 

26.  Arrange  two  convex  lenses  of  2  and  10  cm.  focal  lengths  to  give  maxi- 
mum M.P.  when  the  nearest  allowable  approach  to  the  object  is  3  cm.  Make 
a  diagram,  and  find  M.P.  and  distance  between  lenses. 

27.  Is  there  advantage,  or  disadvantage,  in  compoTuiding,  with  tube  and 
eyepiece,  a  simple  microscope  consisting  of  one  lens  of  short  focus  ? 


PRACTICAL  QUESTIONS 

Find  the  magnifying  power  of  a  simple  lens. 

Choose  lenses  and  construct  telescopes  of  two  kinds,  measure  their  magnify- 
ing powers  and  deduce  either  the  ratio  of  the  strengths  of  the  lenses,  or  the 
focal  length  of  the  concave  lens  (measuring  that  of  the  convex  in  the  very 
simplest  way). 

Choose  lenses  to  set  up  a  microscope,  and  measure  its  magnifying  power. 


CHAPTER  XLI 
POLARIZED  LIGHT 


§  651 .  We  come  to  a  property  of  light  waves  that  compels  a  sharp 
distinction  to  be  drawn  between  the  motion  of  the  particles  in  them 
and  in  sound-waves.  Of  two  beams  of  Hght,  perfectly  in- 
distinguishable to  the  eye,  one  may  pass  unhinderwl  through 
certain  pieces  of  clear  colourless  spar  which  quite  stop  the  other. 
A  glossy  surface  will  always  reflect  one  but  may  blot  out  the  other, 
or  reflect  it  feebly,  or  fully,  according  to  position. 

Such  light  is  said  to  be  polarized.  Nothing  like  it  occurs  in 
Sound.     It  is  possible  because 

Light  vibrations  are  transverse. 

Imagine  a  stick  and  some  vertical  palings.  Held  lengthways  it 
can  always  be  pushed  through  the  fence,  but  held  crossways  iii  the 
middle  it  will  go  through  when  parallel  to  the  palings  but  be 
bounced  back  when  horizontal :  make  your  own  diagram. 

Let  the  stick  represent  the  to-and-fro  track  of  a  particle  taking 
part  in  a  travelling  wave  motion.  In  the  first  case  the  vibration 
is  longitudinal,  a  *  push-wave,'  as  we  know  it  in  sound  waves.  In 
both  the  other  cases  it  is  transverse  (Fig.  126,  T),  a  '  shake-wave/ 
and  what  happens  resembles  the  effects  of  polarization  described 
above.  It  is  concluded  that  the  vibrations  in  light  waves  are 
transverse,  each  particle  being  confined  to  its  own  plane,  per- 
pendicular to  the  direction  of  travel  of  the  light. 

In  ordinary  light  it  can  vibrate  in  that  plane  in  lines  and  ellipses 
wandering  in  all  directions  in  turn  ;  anywhere  in  planes  cutting 
the  paper  perpendicularly  in  the  upright  diameters  of  Fig.  126. 

In  plane  polarized  light  it  is  confined  to  one  fixed  direction 
in  that  plane,  e.g.  in  the  upright  diameters  of  the  figure. 

Or  in  Fig.  121,  the  vibrations  in  ordinary  light  coming  towanls 
you  can  be  in  any  of  the  forms  at  the  bottom  of  the  figure,  or  inter- 
mediate ones  ;  but  in  plane-polarized  they  are  confine<i  to  a  line. 

In  walking,  the  vibrations  of  your  legs  are  polarized  in  a  vertical 
plane  ;  in  playing  tennis,  they  are  unpolarized. 

Radio  waves,  sent  out  from  a  vertical  aerial,  are  polarized  in  a 
vertical  plane.  Fig.  382.  To  receive  them  you  set  up  another 
vertical  pole  ;  if,  instead,  you  stretched  out  your  aerial  all  on  a  level 
with  your  set,  and  squarely  fticing  the  transmitting  station,  like 
a  donkey's  ears,  you  would  hear  little. 

Light-waves  are  tiny  radio-waves  sent  out  from  atoms,  and  as 
there  are  billions  of  atoms  in  a  flame,  arranged  just  anyhow,  onlinary 
light  vibrates  equally  in  all  directions.    One  way  of  sorting  them 

535 


536 


LIGHT 


[§651 


out  into  ups  and  downs,  and  rights  and  left,  i.e.  of  getting  Polarized 
Light,  is  by  Reflection ;  another  is  by  passing  them  through  a 
Crystal. 

§  652.  Polarization  by  reflection.  Light  reflected  obliquely  from 
any  glossy  surface  (but  not  from  metals)  is  more  or  less  polarized, 
and  at  a  particular  Polarizing  Angle  {of  reflection)  light  reflected 
from  a  perfectly  clean  surface  is  wholly  plane  polarized,  and  is  vibrating 
parallel  to  the  surface.  Meanwhile,  most  of  the  light  plunges  into 
the  surface,  and  contains  a  corresponding  excess  of  perpendicular 
vibration,  and  the  transmitted  light  is  therefore  partially  polarized. 
It  is  just  exactly  '  ducks  and  drakes.' 

Tan  (polarizing  angle)  =  y.  gives  a  means  of  finding  the  refrac- 
tive index  of  pitch,  ebonite,  etc.     For  water  it  is  53°,  glass  57^°. 

Water  made  inky  to  hide  the  bottom,  glass  laid  on  black  velveteen, 
or  a  shiny  black  book  laid  on  the  window-sill,  and  looked  at,  at 
about  these  angles  from  the  vertical,  make  splendid  polarizers. 
So  does  a  stack  of  glass  plates  ;  while  a  dozen  microscope  slides 
cleaned  up,  stuck  in  a  bundle  at  nearly  60°  in  a  square  card  tube, 
and  looked  through,  does  as  well  as  a  nicol.  The  virtue  of  a  Bundle 
is  that  one  surface  polarizes  only  a  little  of  the  great  bulk  of 
transmitted  light,  therefore  continue  the  treatment  ad  lib. 

§  653.  Passage  of  light  through  a  crystal.  Suppose  some  shot 
set  bouncing  to  and  fro  across  a  circular  pipe.  Fig.  275  (upper). 
Each  can  continue  to  bounce  along  the  diameter  it  starts  in,  because 
it  hits  the  wall  perpendicularly  at  each  end.  In  an  elliptical  pipe, 
Fig.  275  (lower),  however,  it  is  usually  flung  back  a 
different  way  at  each  bounce,  and  the  only  two 
directions  in  which  vibration  can  continue  permanently 
are  the  long  and  short  axes  of  the  ellipse  which  are 
perpendicular  to  the  walls  at  their  ends,  and  are  at 
right  angles  to  each  other. 

If  the  pipe  were  circular,  but  its  walls  of  indiarub- 
ber,  much  softer  top  and  bottom  than  at  the  sides, 
this  would  permit  a  longer  swing  up  and  down,  just 
as  in  the  elliptical  pipe,  with  the  same  result. 

Finally,  if  the  pipe  were  filled  in  with  rubber, 
embedding  the  shot,  but  controlling  it  more  softly 
up  and  down  than  right  and  left,  it  would  have  the 
same  effect. 

Now,  in  a  crystal  (except  cubic)  the  Elasticity 
differs  in  different  directions,  e.g.  a  crystal  is  actually 
stronger  and  harder  one  way  than  another. 

The  vibrations  carrying  light  through  a  crystal 
are  therefore  not  elliptical,  or  anyhow,  but  are  confined  to  two 
directions  at  right  angles  to  each  other  (and  necessarily  to  the 
direction  of  travel). 

That  constitutes  two  polarized  '  beams  '  completely  intermixed , 
and  the  problem  is  to  separate  them.    If  the  original  light  were 


Fig.  275. 


§654] 


POLARIZED   LIGHT 


ft37 


Fio.  276. 


'  ordinary,'  containing  just  as  much  up  and  down  as  right  and 
left,  they  will  of  course  be  equally  strong. 

From  §  396,  the  speed  of  travel  V  =  \/(E/D),  therefore  if  the 
elasticities  differ  in  these  two  directions  at  right  angles  (and  possibly 
the  densities  of  packing  of  the  atoms  also,  as  in  sardines ;  and  not 
proportionally),  these  two  different  beams  will  travel  at  different 
speeds. 

Now,  since  y.  =  Vair/Vmedium»  and  they  travel  at  unequal  speeds 
in  the  crystal,  they  are  unequally  refracted,  and  usually  follow 
different  tracks.    This  double  refraction 
is  best  seen  in  Iceland  spar  (calcite),  a 
cleavage  piece  of  which  lying  over  Fig. 
275  produced  Fig.  276. 

As  the  flat  piece  is  rotated  on  the  page, 
one  image — the  '  ordinary  ' — remains 
fixed,  but  the  '  extra-ordinary  '  moves 
round  it,  and  does  not  obey  the  first  law 
of  refraction,  §  485. 

[Little  bits  of  calcite  can  often  be  cleft 
out  of  chemical  laboratory  stuff.] 

A  prismatic  rock-crystal  (quartz),  held 
to  the  eye,  shows  two  little  overlapping  spectra  of  a  lamp. 

8  654.  To  get  a  single  beam  of  plane  polarized  light.    From  a 

large  piece  of  spar  the  two  polarizwi 
beams,  each  half  as  bright  as  the 
original  beam,  will  emerge  quite 
separated ;  but  large  spar  is  hard  to 
get. 

The  Nicol  Prism  is  our  best  means. 
It  is  shown  in  section  in  Fig.  277.  A 
long  '  rhomb  '  of  clear  Iceland  spar 
is  sawn  across  very  obliquely. 
poHshed,  and  re-cemented  with 
Canada  balsam.  Now,  [i  balsam  = 
1-53  and  (x  calcite,  ordinary  ray, 
=  1-66,  therefore  when  this  tries  to 
pass  very  obliquely  into  the  optical  I  v 
lighter  balsam,  it  is  toUlly  reflectwi. 
and  thrown  aside,  to  be  ultimately 
absorbed  in  black  varnish  on  the  side 
of  the  prism.  But  the  extraordinarv 
ray,  jx  calcite  1-49,  passes  through 
unaffected,  for  the  balsam  is  optically 
the  denser  now.  The  vibration 
transmitted  is  along  the  abort  dia- 
gonal of  the  rhombic  end,  as  shown. 
Schorl,  or  Tourmaline,  is  a  dark 
mineral  which  absorbs  one  of  the  vibrations  much  more  than  the 
Xr     Therefore,  if  ordinary  light  falls  on  a  J-m.  slice  cut  length- 


Fig.  277. 


Fio.  278. 


638  LIGHT  [§  654 

wise  from  a  schorl  crystal  the  dim  brown  or  green  light  that  does 
get  through  is  plane  polarized,  Fig.  278.  It  is  cheap,  but  dismal ; 
uncoloured  tourmaline  is  useless. 

Most  other  available  crystals  are  but  feebly  doubly  refracting ; 
they  polarize  the  two  beams,  but  fail  to  separate  them  to  any  extent. 

In  all  crystals  there  is  an  optic  axial  direction  in  which  no  double 
refraction  occurs.  It  is  parallel  to  the  length  of  a  rock  crystal ; 
it  enters  the  blunt  corner  of  a  calcite  rhomb  symmetrically  to  the 
three  faces.  Evidently  quartz  for  making  lenses  must  be  sliced 
up  straight  across  the  crystal.  Gypsum,  sugar,  etc.,  have  two 
optic  axial  directions. 

Rock-salt  and  fluor-spar  are  cubic,  and  do  not  doubly  refract. 
So  is  diamond,  but  it  is  useless  for  optical  purposes,  being  always 
irregularly  doubly  refractive  from  internal  strain. 

§  655.  Having  now  got  a  supply  of  plane-polarized  light,  let  us 
study  it. 

[Some  eyes,  among  them  mine,  can  detect  its  polarization.  Right 
in  front,  as  big  as  a  halfpenny  at  arm's  length,  appear  Haidinger's 
Brushes,  a  pair  of  blue  quadrants  crossed  by  a  brownish  pair. 
Clear  skylight,  being  produced  by  scattered  reflection  in  the  atmos- 
phere, is  partially  polarized,  vibrating  across  the  direction  of  the  sun. 
Looking  up,  and  looking  quickly  from  place  to  place  (or  the  ap- 
parition soon  fades),  the  brown  brushes  always  point  to  the  sun, 
so  one  can  tell  where  he  is  quite  a  time  after  sunset :  on  the  re- 
flecting surface  of  the  sea  around  the  ship  they  stand  upright.] 

If  this  polarized  light  meets  a  second  polarizing  arrangement 
of  any  sort  (called  the  '  Analyser  ')  it  will 

(a)  continue  unchecked  if  the  direction  of  possible  vibration  in 
the  analyser  is  the  same  as  its  own, 

(6)  be  dimmed  if  they  are  inclined,  and 

(c)  be  stopped  if  they  are  at  right  angles,  for  a  motion  has  no 
component  at  right  angles  to  itself,  §  15. 

For  instance,  light  is  reflected  again  from  a  second  plate  when 
parallel  to  that  which  polarized  it,  but  not  when  turned  through 
90°  on  the  ray  as  axis.  It  passes  through  a  bundle  when  per- 
pendicular, but  not  when  parallel,  to  the  first  reflector.  '  Crossing  ' 
tourmalines  or  nicols  blackens  the  field  of  view.  Fig.  278.  In 
Fig.  279  at  the  top  is  a  pair  of  nicols  '  crossed  '  ;  below  is  a 
polarizing  plate  with  '  bundle  '  extinguishing  reflected  light. 

Mostly  one  makes  use  of  the  Dark  Field,  where  it  is  a  question 
of  some  light  or  none  at  all. 

§  656.  *  Depolarizing  '  effect  of  thin  crystalline  plates.  A  piece 
of  a  crystal  held  between  '  crossed  nicols  '  appears  bright  in  the 
dark  field.  (Try  chips  of  mica  between  your  inky  water  and 
analysing  bundle,  as  in  Fig.  279.) 

The  crystal  has  to  split  the  incident  vibration,  by  the  usual 
rectangular  parallelogram  law,   §  15,  into  two  components  in  its 


657] 


POLARIZED   LIGHT 


639 


own  possible  directions  of  vibration.  One  component  travel  faster  • 
keeping  step  but  taking  longer  steps,  it  reaches  the  other  side 
a  fraction  of  a  wave  ahead  of  its  companion,  there  is  a  phase 
difference,  and  now  when  these  rays  meet  the  analyser  they  do  leave 
a  component  in  its  direction,  i.e.  light  passes  through. 

[Cf.  Fig.  121 ;  the  straight  line  in  the  comer  has  opened  out  into 
one  of  the  elhpses,  which  has  breadth  parallel  to  the  line  in  the 
opposite  comer,  which  is  the  analyser,  perpendicular  to  the  polarizer  ] 


Fig.  279. 


Fio.  280. 


Thin  crystal  plates  show  soap-bubble  colours,  when  one  colour 
has  gained  a  whole  wave  and  is  extinct  again,  but  thick  plates, 
exactly  like  thick  films,  are  not  coloured  (cf.  §  564). 

§657.  Various  uses  of  polarized  light.  When  the  sheen  of  a 
polished  surface  becomes  troublesome,  recollect  that  it  is  sure  to 
be  pretty  largely  polarized,  and  can  therefore  be  cut  down  by  a 
Nicol  Prism.  Skyed  pictures  become  visible,  the  trout  angler 
wipes  the  skylight  off  the  stream  and  sees  his  fish  below,  the 
navigator  clears  up  his  skyline  in  the  sextant,  for  sea  surface  and 
sky-haze  polarize  at  angles. 

As  Analyser  is  turned  from  the  dark  cross-position,  to  parallel 
to  Polarizer,  the  light  transmitted  increases  from  zero  exactly 
proportionally  to  (sine)*  of  angle  turned  through  :  this  finds  ail 
sorts  of  applications  in  Photometry. 

Nicol  prisms  applied  to  the  microscope  are  invaluable  to  the 
mineralogist,  and  they  enable  the  biologist  to  pick  out  cr>'staU. 
starch  grains  (which  are  *  sphere-crystals  ),  hard  tissues,  etc.,  in  a 
smother  of  tissue  (Fig.  280  left,  ordinary;  right.  i)olarize<i  light). 
The  pathologist  uses  them  in  detecting  certain  crystalline  fat 
deposits. 

Crystallizations  in  progress  on  a  microecope  slide  present  daudixig 
spectacles.    Some  melted  mixtures,  such  as  chole«terin  and  stearic 


640  LIGHT  [§  657 

acid,  produce  little  round  moving  '  liquid  crystals  '  before  they 
solidify. 

All  transparent  solids  become  doubly-refracting  under  strain. 
Light  will  gradually  reappear  in  a  bit  of  glass  as  you  squeeze  it  in 
pincers  in  your  dark  field.  Hence  glass  intended  for  optical  purposes 
is  carefully  examined  in  polarized  light  to  detect  any  temperature 
strains  not  annealed  out,  which  would  warp  the  images. 

So,  by  countless  thousands,  are  glass  jam-pots,  by  the  plate 
arrangement  of  Fig.  279  :  if  they  light  up  noticeably  they  go  back 
to  the  melting-pot,  for  the  hot  jam  would  crack  them. 

The  engineer  makes  model  sections  in  soft  celluloid  of  masonry 
dams,  bridge  structures,  ships'  frames,  steel  railway-carriage  sides, 
etc.,  loads  them  and  computes  their  strains  from  the  colours  they 
develop.  He  takes  a  narrow  strip  of  the  same  celluloid,  lays  it 
over  the  model,  hitches  on  a  spring -balance,  and  alters  pull  and 
direction  until  the  colour  vanishes ;  then  the  strain  at  that  point 
in  the  model  is  the  exact  reverse  of  that  in  the  simple  strip. 

§  658.  Polarimetry.  Certain  substances,  e.g.  solutions  of  the 
sugars  or  tartaric  acids,  the  terpenes,  and  all  that  contain  an 
'  asymmetric  carbon  atom,'  possess  the  property  of  rotating  the  plane 
of  polarization,  i.e.  as  the  polarized  light  travels  through  them,  the 
direction  of  vibration  in  it  slews  round,  through  an  angle  to  right 
or  left  characteristic  of  the  (substance  X  quantity  passed  through)  ; 
and  this  affords  a  convenient  means  of  analysing  them. 


--JT^E^B-SS 


Fig.  281. 

The  Rotation  is  right-handed,  or  dextro-rotatory,  if  you  see  it 
turning  clockwise  as  it  comes  towards  you,  so  that  the  top  of  the 
analyser  must  be  moved  to  your  right. 

Specific  Rotations  are  tabulated,  meaning  the  rotation  produced 
by  10  cm.  length  of  a  solution,  in  inactive  solvent,  containing  1  gm. 
of  active  substance  per  c.c.  of  solution.  Such  concentration  is 
seldom  attainable,  but  rotation  is  proportional  to  concentration, 
reckoned  in  this  way. 

Some  Specific  Rotations  are,  for  sodium  light,  at  20°  C.  : 

Cane-sugar     R    66-7°  Rochelle  salt  R    29'8° 

Invert-sugar  L     19-7°  Nicotine  L  162° 

Dextrose         R  105°,  falling  to  half  in  Camphor         R    54-5° 

6  hours 

Laevulose        L  104°,  falling  to  92°  in  Quartz     R  or  L,  21-7°  per  ram. 

^  hour 


§  658]  POLARIZED   LIGHT  541 

In  Polarimeters  the  clarified  solution  is  put  in  a  glass-ended 
tube  20  cm.  long,  between  crossed  nicols,  Fig.  281  ;  light  entem 
the  polarizing  nicol  on  the  left,  and  the  angle  the  analyser,  mounted 
in  a  graduated  circle,  has  to  be  turned  through  to  restore  darkne«8, 
gives  a  means  of  estimating  the  containe<l  iSugar,  say  (for  which 
they  are  largely  used).  Since  the  deepest  darkness  is  hanl  to  decide 
on,  they  contain  also  a  crystal  device,  and  the  adjustment  is  to 
equate  the  dimness  of  two  semicircular  halves  of  the  little  field  of 
view,  one  of  which  is  still  getting  darker  as  the  other  has  begun  to 
get  lighter. 

This  is  often  a  Laurent  half-wave-plate  of  quartz,  which  confines 
the  user  to  sodium  light  in  a  dark  room  :  it  is  the  standard  instni- 
ment  in  sugar  factories.  For  the  weak  concentrations  occurring 
in  diabetes,  however,  sufficient  accuracy  is  attainable  with  daylight 
passed  through  a  gelatine  filter-film  dyed  a  special  yellow,  and 
measurements  can  be  made  at  the  bedside  on  a  direct -reading 
*  Diabetometer ' :  this  is  much  easier  than  a  Fehling  cuprous-oxide 
sugar  estimation  (which,  besides,  is  vitiated  by  creatinine  present). 

Or  accessory  prisms,  of  nicol  type,  are  put  to  cover  the  sides  of 
the  field,  again  so  as  to  pass  light  polarized  at  a  small  adjustable 
angle  to  the  central  beam,  getting  the  same  equality  effect  with  light 
of  any  wave-length. 

The  *  biquartz  '  plate,  used  in  daylight  to  bring  the  halves  of  the 
field  from  red  and  blue  forget-me-not  hues  to  the  same  lavender 
'  sensitive  tint,'  is  affected  by  the  yellow  or  brown  so  difficult  to 
remove  completely  from  organic  solutions,  and  is  less  accurate. 

Knowing,  as  in  sugar  refineries,  what  particular  sugar  is  present, 
you  can  determine  its  concentration.  Conversely,  if  it  is  changing 
into  another  form,  as  Cane-sugar  into  Invert-sugar,  dextrose  -f- 
laevulose,  under  hydrolysis  by  hot  acid  or  alkali,  or  by  invertase 

CigHaaOn  +  HgO  =  C.H^jO.  dextro.  -f  C,H„0,  laevo. 

the  change  can  be  watched  and  measured. 

Polarimeters  are  used  largely  in  the  essential-oil  industry,  wher© 
any  chemical  interference  is  apt  to  cause  very  unwante<l  ^'^^^JH^ 
among  the  labile  terpenes,  diflFerent  *  optical  isomers '  of  which 

often  differ  widely  in  commercial  or  therapeutic  value. 

.               ,          ,         observed  rotation  ^  U)  em. 

Concentrahan.gms.lcc.  soln.  =  ^^^^^^ X  J^j^^  ^^d 


642  LIGHT 


EXAM   QUESTIONS,   CHAPTER   XLI 

1.  How  does  polarized  light  differ  from  ordinary  light?  Describe  experi- 
ments to  show  this  difference  and  sketch  the  arrangement  of  apparatus 
used.     (  X  5) 

2.  Given  a  number  of  glass  plates,  such  as  microscope  slides,  how  would 
you  produce  polarized  light,  and  detect  the  polarization  ?  Can  the  unaided 
eye  do  so  ?     (  X  2) 

3.  When  a  piece  of  calc-spar  is  laid  on  the  page,  the  print  beneath  it  appears 
doubled.  How  do  you  account  for  this,  what  inference  has  been  drawn 
from  it  as  to  the  nature  of  light,  and  to  what  practical  use  is  it  put  ?     (  X  3) 

4.  Describe  the  construction  of  a  Nicol's  prism.  Contrast  the  properties 
of  light  before  and  after  it  has  passed  through  such  a  prism. 

5.  Describe  either  a  direct-vision  spectroscope,  or  a  saccharimeter  [a 
polarimeter  graduated  for  cane-  or  beet-sugar]. 

6.  What  is  meant  by  plane  polarized  light,  and  how  may  it  be  produced  ? 
Describe  the  property  which  a  solution  of  sugar  has  with  respect  to  it.     (  X  4) 

7.  Give  an  account  of  the  rotation  of  the  plane  of  polarization  of  light 
produced  by  '  optically  active '  substances.  Describe  an  instrument  by 
which  this  effect  is  measured. 

A  2%  solution  of  such  a  substance  is  placed  in  a  tube  2  dm.  long.  If  the 
specific  rotatory  power  of  the  substance  is  30°,  show  that  the  plane  of  polariza- 
tion is  rotated  1*2°.     (  X  5) 


i 


MAGNETISM 

CHAPTER  XLU 
MAGNETS  AND  MAGNETIC  MATERIALS 


§  661.  It  must  have  been  a  Palajolithie  discovery  that  there  was 
a  sort  of  black  stone  that  had  the  power  of  attracting  and  holding 
little  fragments  of  itself.  The  name  by  which  we  know  this  rich 
ore  of  iron,  Fe^O^ — Magnetite — appears  to  be  derived  from  a  locality 
so  prolific  in  minerals  as  to  have  conferred  its  name  also  on  two 
others,  magnesia  and  manganese. 

In  Britain  it  was  the  Lodestone,  because  it  led  fragments  to  itself. 
or  perhaps  because  a  bar  of  it,  hung  by  a  hair,  would  turn  and  point 
northwards,  towards  the  steadfast  leading-  or  Lode-star  of  the 
mariner. 

Instances  of  the  fables  that  gathered  round  it  you  have  met  with 
already,  in  Hans  Andersen,  and  in  the  Story  of  the  Third  Royal 
Mendicant  in  the  '  Arabian  Nights.' 

Steel  rubbed  by  the  lodestone  acquired  its  powers,  and  the 
Levantine  sailors,  magnetizing  a  needle  of  it  by  their  stone,  stuck  it 
through  a  reed,  floated  it  in  a  brazen  bowl,  and  steered  by  *  the  little 
frog  '  in  cloudy  weather. 

A  Chinese  compass  reached  Europe  in  1260,  and  by  1269  Peter 
the  Pilgrim  was  making  very  much  better  ones,  divided  into  360^. 

If  you  examine  the  attractive  power  of  a  Magnet — the  lodestone, 
or  a  piece  of  steel  rubbed  by  it — you  find  it  concentrated  in  parts 
called  Poles,  whereto  iron  nails  and  filings  cling  thickly,  and  toy 
compasses  frantically  point.  There  is  usually  a  strong  jkiIc  near 
each  end,  but  there  may  be  others,  called  consequent  poles,  anywhorr, 
often  where  the  steel  has  been  casually  touched  by  the  magnet . 

§662.  North  and  South  poles.  Taking  hencefon»anl  the  usual 
steel  magnet  with  a  pole  near  each  end,  it  will  1k»  found  that  while 
both  attract  iron,  and  if  movable  are  attracte<l  towards  it  (by  the 
third  law  of  motion),  one  will  attract,  and  the  other  rvpel,  one  end 
of  another  magnet.  They  are  evidently  of  opposite  'polarity,* 
and  are  distinguished  as  North  and  South. 

The  bar  shows  no  signs  of  magnetism  at  its  middle,  and  one 
might  expect  that  breaking  it  there  would  leave  the  one  original 
pole  on  each  half.  But  if  you  will  magnetize  an  old  hack*8aw,  by 
stroking  it  from  end  to  end  with  a  magnet,  and  break  it  in  pieces, 

543 


544 


MAGNETISM 


[§662 


you  will  find  that  new  poles  develop  instantly  at  the  broken  ends — 
opposite  poles,  for  the  ends  cling  together — and  each  part  is  now 
a  complete  magnet.  In  fact,  a  piece  of  lodestone,  or  of  steel  hardened 
to  brittleness  by  plunging  red-hot  into  water,  and  magnetized,  can 


Fig.  282. 


Fig.  283. 


Fig.  284. 


Fig.  285. 


be  powdered  up,  and  every  particle  will  of  itself  chng  to  iron,  being 
a  complete  little  magnet. 

Apparently  the  stream  of  magnetism  runs  right  through  the 
bar,  but  only  shows  itself  where  it  enters  and  leaves. 

In  agreement  with  this  is  the  fact  that  the  amounts  of  North  and 
South  magnetism  in  a  magnet  are  always  equal  to  each  other,  whether 
collected  into  two  poles  or  scattered  among  several  consequent 


§  666]  MAGNETS   AND   MAGNETIC   MATERIALS  545 

poles.  For  a  Magnet  set  afloat  on  a  board,  in  a  wash-basin,  turns 
and  sets  itself  N.  and  S,,  and  makes  no  further  movement,  whereas 
if  one  of  its  magnetic  charges  were  stronger  than  the  other,  the  action 
of  the  great  magnet,  Earth,  on  that  charge,  would  be  greater,  and 
would  drag  the  magnet  bodily  along,  north  or  south. 

§  663.  We  can  follow  the  stream  of  magnetism  as  it  spreads  out 
from  the  North  pole  into  the  surrounding  space — the  magnet's 
'  field  ' — where  it  gives  rise  to  all  the  various  magnetic  actions. 
Fine  iron  filings  are  sprinkled  on  a  card  laid  on  the  magnet,  the 
card  is  gently  tapped,  and  the  filings  arrange  themselves  in  lines 
which  are  stream-lines  of  the  magnetic  flow  in  that  particular 
plane  in  which  the  card  cuts  the  magnetic  field.  In  the  photo- 
graph Fig.  282  the  card  is  lying  on  a  bar  magnet,  and  in  Fig.  283 
on  the  poles  of  a  vertical  '  horseshoe  '  electro- magnet. 

The  stream  that  flows  in  at  the  S.  pole  is  just  the  stream  that 
left  the  N.  pole  ;  we  have  already  found  it  convenient  to  think 
of  the  stream  as  continuous  right  through  the  steel ;  it  follows 
then  that  each  magnetic  line  is  a  closed  endless  loop. 

BY  CONVENTION  all  N.  polcs  and  N.  polarity  drift  downstream. 
S.  poles  and  S.  polarity  travel  up  against  stream. 

Fig.  284  shows  two  N.  poles,  each  sending  out  its  own  streams 
of  lines  ;  the  two  sets  never  mix. 

In  Fig.  285  an  iron  nut  has  been  placed  in  the  field ;  notice  how 
the  lines  bend  round  and  crowd  into  it ;  evidently  they  find  it 
easier  to  run  through  iron  than  through  air,  so  much  easier  that 
the  filings  show  hardly  any  flow  on  the  card  (in  the  air)  just  above 
the  nut. 

Bits  of  iron  always  prefer  clinging  to  the  edges  rather  than  to 
the  flat  face  of  a  magnet  pole.  For  the  lines  crowd  into  the 
corners  so  as  to  have  as  great  a  proportion  of  their  course  in  the 
easily  permeable  metal  as  they  can,  hence  the  attractive  force  is 
greater  there. 

§  664.  Magnetic  shielding.  Few  lines  emerge  into  the  hollow 
middle  of  the  nut  of  Fig.  285  ;  it  is  easier  to  run  round  in  the  iron 
than  to  jump  across,  the  thick  iron  shell  shields  the  space  inside  it 
from  outside  magnetic  influence.  n    •      k 

This  plan  of  surrounding  a  space  with  thick  iron  walls  is  the 
only  known  means  of  keeping  out  external  magnetic  influence. 
Delicate  galvanometers  have  sometimes  to  be  protected  from 
electrical  machinery  by  a  close-fitting  jacket  of  '  soft '  iron.  The 
conning-tower  of  a  battleship  is  a  poor  place  for  a  binnacle,  but  a 
quarter  or  more  of  the  earth's  magnetic  force  still  pervades  it ;  which 
was  lucky  for  H.M.S.  Vindictive  as  she  steamed  from  Zeebrugge, 
with  a  few  stray  steel  compliments  even  inside  the  tower. 

§  665.  Magnetization  by  Induction.    The  lines  crowding  in  and 
out  of  the  iron  give  it  the  appearance  of  a  magnet ;    this  is  still 
T 


646 


MAGNETISM 


[§665 


better  seen  in  Fig.  286,  where  a  wrought-iron  bar  has  been  placed 
in  the  magnetic  field.  And  by  trial  we  find  that  for  the  time  being 
it  is  a  magnet.  An  iron  nail,  for  instance,  held  with  one  end  near 
a  magnet  pole,  will  pick  up  pen-nibs,  etc.,  on  its  far  end,  though 
they  fall  when  the  magnet  is  removed.  A  yard  of  wire  rope  will 
carry  the  magnetic  stream  round  from  a  magnet  to  a  compass 
which  previously  was  but  little  affected. 

These  things  are  magnetized  by  *  induction,'  magnetization  is 
induced  in  them. 

This  explains  how  a  magnet  pole  (N.,  say)  which  attracts  a 
S.  and  repels  a  N.  pole,  and  should  presumably  have  no  effect  on 
a  neutral  body,  yet  always  attracts  ordinary  unmagnetized  iron. 
Lines  from  the  pole  crowd  into  the  iron,  inducing  an  opposite  pole, 
and  these  two  poles  attract  each  other. 


Fig.  286. 


Fig.  287. 


Every  little  filing  in  the  field  becomes  an  '  induced  magnet ' 
and  sets  itself  head  to  tail  with  its  neighbours,  hence  the  con- 
tinuous dark  lines  of  them. 

The  Earth  is  a  magnet,  with  its  Northern  Magnetic  Pole  north 
of  Canada,  and  magnetically  a  south  pole,  and  all  iron  upon  it  is  more 
or  less  magnetized  by  its  induction.  In  this  country  its  magnetic 
lines  run  steeply  downwards  slightly  west  of  N. ;  holding  a  poker 
N.  and  S.,  or  vertical,  or  best  in  the  described  direction,  and 
hammering  it,  it  will  acquire  and  retain  magnetism  enough  to 
affect  a  compass  quickly,  and  perhaps  to  pick  up  filings. 

Hammering  always  seems  to  help,  one  says  vibration  '  shakes 
up  the  molecules,'  as  tapping  shook  up  the  fiUngs,  easing  their 
friction  on  the  card  and  enabling  them  to  turn  into  line.  Vibration 
in  a  contrary  field  reduces  magnetization.  Steel  ships  built 
N.  and  S.  are  a  magnetic  nuisance  to  their  navigators  until  a 


§  667]  MAGNETS   AND  MAGNETIC  MATERIALS  547 

year's  roundabout  voyaging  has  shaken  out  most  of  their  acquired 
magnetism. 

But  iron  placed  right  across  the  lines  as  in  Fig.  287  (e.g.  a  girder 
east  and  west)  does  not  get  magnetized.  For  the  lines  would 
gain  nothing  by  turning  quite  at  right  angles  to  run  along  it,  and 
the  extra  facility  of  traversing  its  small  thickness  is  not  sufficient 
inducement  to  bring  many  lines  out  of  their  short  direct  couniefl, 
i.e.  the  lines  do  not  crowd  in,  but  pay  little  more  heed  to  it  than 
to  a  bit  of  wood  ;  it  shows  no  magnetic  difference  from  it«  surround- 
ings, it  is  not  perceptibly  magnetized.  It  is  impracticable,  for 
instance,  to  permanently  magnetize  a  thin  steel  plate  to  have 
one  face  all  N.  and  the  other  all  S.  pole  ;  it  is  difficult  to  magnetize 
a  bicycle  ball  strongly ;  it  is  many  hundred  times  easier  to 
magnetize  a  rod  lengthwise  than  crosswise  ;  whatever  direction  the 
poker  be  held  in  when  hammered,  its  poles  will  be  near  its  end« 
and  its  magnetization  parallel  to  its  length.  A  rough  bit  of  lode- 
stone  always  has  poles  at  its  ends. 

§  666.  Methods  of  magnetizing.  All  magnetization  is  effected 
by  induction.  The  commonest  process,  that  of  stroking  the  steel 
from  end  to  end,  always  one  way,  with  a  magnet  pole,  is  simplv 
exposing  every  particle  of  it  in  succession  to  a  strong  field.  If 
instead  of  the  solid,  a  tube  full  of  steel  fiUngs  be  used,  they  can  be 
seen  turning  to  point  to  the  pole  as  it  passes  :  it  coaxes  them  all 
down  one  way,  and  the  tube  (or  rod)  showing  the  sum  total  of  all 
these  little  (molecular)  magnets,  exhibits,  on  the  end  at  which  the 
magnet  left,  opposite  polarity  to  the  stroking  pole. 

[Subsequently  shaking  up  the  tube  jumbles  the  steel  filings 
together  and  obliterates  their  united  effect,  though  each  may 
remain  magnetized.] 

In  the  extra  work  you  do,  in  separating  the  inducing  from  this 
induced  pole,  is  the  source  of  the  magnetized  steel  s  store  of 
Potential  Energy,  which  enables  it  to  turn  compass  cards,  or  move 
iron,  or  drag  itself  towards  it. 

The  objection  to  this  way  of  propagating  magnetization  is  that 
the  outer  layers  act  as  magnetic  shields  to  the  inner,  and  only  thin 
strips  are  '  done  all  through.' 

With  a  large  electro-magnet  available,  one  places  the  bar  to  join 
its  poles,  packing  in  any  gap  with  lumps  of  iron,  and  taps  it.  The 
intense  magnetic  stream  flows  through  the  bar  and  thoroughly 
magnetizes  it. 

Magnetization  by  circulating  electric  currents  must  be  deferred 

k     until  later,  we  may  only  mention  here  that  strong  local  magnetiia- 
tion  in  native   magnetite   (constituting   it   lodestone)   is   usually 
'       ascribed  to  lightning  having  struck  near  it,  for  the  great  bulk  of 
the  ore  is  not  naturally  magnetized. 

§667.  Magnetic  Permeability.  The  ratio  of  the  numbo-  of 
lines  which  flow  through  1  sq.  cm.  cross-section  of  a  long  rod  of 


548  MAGNETISM  t§  667 

iron,  etc.,  placed  along  their  natural  course,  to  the  number  flowing 
if  the  iron  were  not  there,  is  called  the  permeability,  P,  of  the 
material. 

The  lines  referred  to  are  '  unit '  lines,  to  be  defined  in  §  686. 
Their  number  per  square  centimetre  in  the  iron  is  B,  the  '  density 
of  induction,'  and  in  air,  is  the  magnetizing  '  field  strength,'  both 
reckoned  in  '  gauss.' 

Magnetizable  substances  therefore  possess  a  Permeability  greater 
than  1,  for  if  lines  don't  crowd,  there  is  no  magnetization. 

Some  average  values  are,  for  B  about  5000  : 

Soft  cast  iron      .........  500 

Mild  steel 600 

Hard  magnet  steels  (retentive)     ......  80-200 

Laminated  soft  iron  and  steel  for  machinery  .  .  .  3000-6000 

Ditto  at  B  20,000 200 

Permalloy  Nii'Fe 12,000 

Electrolytic  iron,  melted  in  vacuo  and  annealed  (Mumetal,  etc.) 

for  small  fields 25,000-50,000 

Nickel  (retentive)         .........       50 

Cobalt 80 

Heusler  alloy  (Mn,Al,2Cu)  annealed       ......       40 

With  the  following,  B  can  only  be  small 

Nickel  steel,  special  .  .  .  .  .  .  .  .1*2 

Manganese  steel  (hard  white,  tramway  points)         ....         1-05 

Magnetite  .  .  .  .  .  .  .  .  .  .4 

Iron  in  the  field  not  only  gathers  together  the  magnetic  stream 
but  actually  increases  the  total  flow,  having  made  the  circulation 
so  much  easier.  Electrical  engineers  therefore  build  their  machines 
of  massive  soft  iron  with  as  Httle  air  gap  as  possible. 

It  can  be  shown  that  the  force  of  magnetic  adhesion  to  iron  is 
0-04  (lines  per  sq.  cm.)2,  in  dynes  per  sq.  cm. 

§  668.  Temporary  and  permanent  magnetization.  In  soft  malleable 
iron  magnetization  is  easily  induced,  P  being  over  1000,  but  it 
vanishes  when  the  magnetizing  influence  is  removed.  This  power 
of  quickly  acquiring  and  losing  magnetism  is  made  great  use  of  in 
electro-magnets . 

In  hard  steel ;  better,  tungsten-steel ;  and  best,  chrome-cobalt- 
steel  ;  particularly  when  very  hard,  magnetization  is  far  less  easily 
induced,  P  averaging  80-160 ;  but  now  a  large  proportion  of  it 
is  retained  '  permanently ' ;  though  warming,  knocking  about, 
the  proximity  of  contrary  magnets,  etc.,  gradually  enfeeble  this 
permanent  residue,  greatly  in  carbon-steel,  far  less  in  the  modern 
alloys. 

§  669.  Temperature.  At  a  red  heat  iron  is  more  easily  mag- 
netizable, but  at  780°,  its  cherry  red  '  temperature  of  recalescence,' 
at  which  it  interrupts  its  cooling  to  glow  out  red  again,  it  suddenly 
loses  all  magnetic  properties.  Permanent  magnetization  diminishes 
as  the  temperature  rises  and  disappears  at  the  same  temperature. 


670] 


MAGNETS   AND   MAGNETIC  MATERIALS 


549 


Heating  to  redness  is  therefore  sometimes  used  to  demagnetize 
specimens,  but  they  must  afterwards  be  placwl  magnetic  ea«t 
and  west,  or  they  will  pick  up  no  little  magnetization  from  the 
earth  as  they  cool  through  the  temperatures  oi  high  permeal)ility. 

Magnetite  likewise  demagnetizes  at  bright  redness,  nickel  at  only 
320°  C,  as  you  can  easily  try.  A  curious  nickel  steel  demagnetizes 
at  600°  and  has  to  be  frozen  before  again  becoming  magnetic  at  all. 

§  670.  Magnetization  Curves.    If  a  bar  of  soft  iron  is  subjected 

to  a  gradually  increasing  field,  its  magnetization  increases  in  three 


B/ooo 


distinct    stages.    Fig.    288,    where    B  =  induction    density,    H  = 
magnetizing  field. 

From  O  to  A  the  magnetization  is  proportional  to  the  field 
strength,  P  is  constant,  but  small,  and  when  the  field  is  removed 
the  specimen  immediately  and  perfectly  demagnetizes. 

From  A  to  K  the  magnetization  increases  enormously,  the 
apparent  permeability  increasing  from  its  steady  value  AU/OU 
to  a  maximum  KV/OV. 

Further  increase  in  field-strength  evokes  only  slight  responuo 
from  the  specimen,  which  presently  becomes 
practically  saturated. 

The  magnetization  of  hard  steel  rises  far  less 
rapidly,  and  the  A'  and  K'  bends  are  smoothly 
rounded.  Saturation  demands  a  field  far  be- 
yond the  diagram,  and  yet  means  much  less 
magnetization  than  in  iron  ;  notice  the  marked 
figures. 

Returnmg,  as  the  field  dimmishes,  the  mag- 
netization falls  only  slowly,  so  that  at  R. 
where  the  field  is  zero,  there  is  still  left  the 
Permanent  Magnetization  OR,  and  it  takes  a 
reversed  field  strength  OX,  called  the  '  coercive 
force,'  to  remove  this.  .         ...      ,^i;j 

This  'Hysteresis'  or  *  sticking  '  of  the  magnetism,  like  soUd 
friction,  causes  a  loss  of  energy  which  goes  to  heat  up  tiie  8P«^;^on. 
In  alternating-current  machinery  it  absorbs  about  2-5  h.p.  per  ton  . 
steel  would  soon  get  red  hot. 


Fio.  289. 


550  MAGNETISM  [§  670 

Demagnetization  cannot  be  effected  in  practice  by  the  reversed 
field  OX,  because  the  curve  is  so  steep  at  X  that  the  least  over- 
running puts  in  an  appreciable  reversed  magnetization.  Demag- 
netization is  effected  by  reversing  the  field  again  and  again,  mean- 
while gradually  weakening  it,  like  Fig.  289,  which  gradually  shrinks 
up  to  nothing.  Thus  a  watch  that  has  become  magnetized,  and 
either  gains  or  sticks,  can  be  cured  by  putting  it  inside  a  coil  in 
which  flows  a  current  frequently  alternated  while  gradually  weakened. 

§671.  On  the  molecular  magnet  theory  it  can  be  supposed  that 
the  atoms  are  all  individually  magnetized  to  start  with,  but  that 
they  are  arranged  haphazard,  and  the  total  effect  is  nil.  As  the 
magnetizing  force  increases  they  swing  round,  more  and  more, 
until  finally  all  point  one  way  and  the  iron  is  magnetically  saturated. 

Ewing  imitated  these  actions  by  a  swarm  of  little  compass 
needles,  of  which  it  suffices  to  consider  four.  At  first,  these  are 
settled  as  in  Fig.  290  (i),  under  their  mutual  attraction,  head  to 
tail  in  a  closed  ring,  which  produces  no  magnetic  effects  outside. 


/    I  ^       /  n  V,  — ^m 

/  /       5'      ^ ^'V  ^'^ 


n: 


Fig.  290. 

A  weak  field  in  the  direction  of  the  arrow  persuades  them  to 
move  apart  a  little  to  follow  it,  as  in  (ii)  ;  removed,  they  return  at 
once  to  the  original  '  unmagnetized  '  position. 

A  stronger  field  presently  draws  the  poles  Ti^Sg'  ^3*4  ^^  ^^^  apart, 
enfeebling  their  mutual  attraction,  that  the  4  needles  become 
unstable  and  swing  round,  as  in  (iii)  ;  all  point  the  same  way, 
imitating  a  magnetized  specimen,  K,  Fig.  288. 

Further  increase  of  field  can  now  only  separate  s^n^,  s^n^  a 
little,  as  it  pulls  the  needles  into  line  with  itself  :  the  specimen 
is  saturated. 

Removing  the  field,  the  needles  remain  almost  as  in  (iii)  under 
their  mutual  attraction :  the  specimen  remains  permanently 
magnetized. 

A  reverse  field  of  some  strength  will  be  necessary  to  upset  this 
stable  arrangement,  and  when  it  does  so  the  needles  are  not  likely 
to  return  to  the  half-and-half  position  (i),  but  to  all  swing  round 
through  X,  Fig.  288,  into  (iv). 

The  best  way  to  get  them  back  to  (i)  is  to  stir  them  round 
violently  with  a  magnet,  then  remove  it  and  leave  them  to  quiet 
down  {a  diminishing  alternating  field). 

The  magnetization  of  an  atom  may  be  considered  as  due  to  the 
orbital  revolution  of  an  electron  inside  it :  the  orbit  tilts,  without 
the  atom  moving  as  a  whole,  cf .  §  767. 


§  673]  MAGNETS   AND   MAGNETIC  MATERIALS  661 

§  672.  Now,  the  portions  of  the  magnetic  lines  outside  the  iron 
inay  be  regarded  in  another  way,  and  that  is  as  lines  of  Force, 
elastic  lines  on  the  stretch,  trying  to  shorten,  pulling  together 
the  pieces  of  iron  they  connect. 

Thus  in  Fig.  285  the  iron  nut  is  being  pulled  by  two  dense 
bundles  of  lines,  and  in  Fig.  286  the  bar  is  being  pulled  round  into 
line  with  the  two  poles.  In  the  actual  experiments  these  had  to 
be  fastened  in  position. 


Fig.  291. 


Each  line  in  Fig.  291  (drawn  in  an  exam)  represents  the  track  of 
a  little  compass  stepped  along  in  the  direction  it  pointed.  The 
needle  of  course  set  itself  along  the  line  of  greatest  pull,  just  as  if 
it  had  threads  attached  to  each  end  and  pulled  opposite  ways. 
Being  pivoted,  it  turned  more  easily  than  the  iron  filings  of  ¥\a. 
282  and  the  field  is  traced  farther  out,  past  the  Neutral  Points  X\, 
into  regions  where  the  earth's  influence  preponderates.  The 
earth's  lines  when  undisturbed  are  of  course  straight  lines  running 
magnetic  N.  and  S. 

§  673.  Para-   and   dia-magnetic   substances.    The   very   intense 

magnetic  fields  (from  10,000  to  50,000  gauss)  obtainable  between 
the  pointed  pole-pieces  of  a  great  electromagnet,  disclose  a  feeble 
magnetic  activity  in  almost  all  substances.  Those  that  behave 
like  iron,  having  a  permeability  greater  than  1,  are  called  para- 
magnetic, some  are  : 


Air  (compared  with  vacuum)  .... 

Liquid  oxygen  (runs  up  tube  to  polee) 

Ferric  salt  solutions  (01  gm.  Fe  per  c.c.)  . 

Ferrous  ,,  ,,  ,,        ,, 

Igneous  rocks  (containing  disseminated  magnetite) 

Ferro-  and  ferri-cyanides  .... 


1-0000004 
1-0026 
I-O00S3 
I-000S6 
1-0001-I-036 
1-00000 


A  threepenny  piece  of  old  English  standard  silver  seta  strongly 
to  the  edge  of  the  pole  pieces,  so  does  a  little  stick  of  that  very  useful 


552 


MAGNETISM 


[§673 


electrical  cement,  Chatterton.  Copper  wire  to  be  used  in  sensitive 
galvanometer  moving-coils,  where  any  trace  of  magnetism  is 
objectionable,  is  washed  free  of  iron  by  HCl,  as  is  also  its  insulating 
silk ;  and  the  coils  must  not  be  touched  by  iron  tools,  or  by  damp 
fingers,  or  be  allowed  to  gather  dust. 

Diamagnetic  substances  have  a  permeability  less  than  1,  refusing 
to  pass  the  magnetic  stream  as  readily  as  air,  so  they  get  pushed  into 
positions  in  which  they  cause  least  hindrance  to  it ;  a  rod  of  bismuth, 
1 — 0-0002,  or  one  of  its  tin  alloys,  1 — 0-002  (the  greatest  known), 
therefore  sets  across  (dia-)  the  field,  placing  its  ends  in  the  weaker 
outer  parts.  So  does  a  thin  slice  of  ordinary  red  rubber  pressure- 
tubing,  at  right  angles  to  the  position  assumed  by  the  threepenny- 
bit.     Water  is  diamagnetic,  with  permeability  1 — 0-00001. 

Para-  and  dia-magnetism  form  the  basis  of  modern  magnetic 
theory,  but  this  book  cannot  venture  into  that. 


EXAM   QUESTIONS,   CHAPTER   XLII 


1.  How  would  you  show  experimentally  that  the  two  poles  of  a  magnet 
are  equal  and  opposite  ?  How  is  their  equality  accounted  for  ?  How  can 
you  show  magnetization  extends  throughout  the  whole  length  ? 

2.  Why  is  it  difficult  to  magnetize  a  short  piece  of  steel  ?  How  can  you 
magnetize  iron  rings  ? 

3.  Describe  how  the  intensity  of  magnetization  of  iron  depends  on  the 
magnetizing  force,  and  explain  magnetic  saturation. 

Show  the  distribution  of  the  lines  of  force  in  a  horseshoe  magnet,  and  the 
effect  of  putting  a  piece  of  iron  in  the  space  between  the  poles. 

4.  In  what  respects  do  the  magnetic  properties  of  hard  steel  and  soft  iron 
differ  from  one  another  ?  How  would  you  experimentally  show  these 
differences  ? 

Mention  two  or  three  physical  instruments  in  which  these  magnetic  properties 
of  steel  or  of  soft  iron  are  put  to  use.     (  X  2) 

5.  How  do  soft  iron  and  hard  tungsten  steel  differ  magnetically  ?  Give 
a  brief  account  of  the  molecular  theory  of  magnetism.     (  X  2) 

6.  Explain  the  molecular  theory  of  magnetism  and  describe  experiments 
in  support  of  it.     (  X  2) 

[In  5  and  6,  Ewing  details  are  not  expected.] 

7.  Show  how  it  is  possible  to  have  an  iron  rod  magnetized  differently  at 
different  times,  though  under  the  influence  of  the  same  magnetizing  force. 


CHAPTER  XLIII 
MAGNETIC  FIELDS 


§  681.  Our  few  magnetic  calculations  can  be  done  most  simply  by 
laying  aside  the  stream-line  idea,  and  regarding  magnetic  action  a« 
direct  attraction  or  repulsion  between  point-poles  at  a  distance. 

The  point-poles  of  a  magnet  can  be  regarded  as  the  *  centres 
of  gravity '  of  two  magnetic  charges.  To  find  them,  bring  up 
one  end  of  the  magnet  close  to  a  charm  compass,  in  such  a  way  that 
the  needle  is  not  deflected  at  all :  ink  on  the  magnet  the  line  of  pointincr 
of  the  needle.  Slew  the  magnet  round  to  some  other  position,  still 
so  as  not  to  deflect  the  needle,  again  ink  in  its  line  of  pointing. 
The  ink  lines  cross  at  the  *  pole.' 

In  a  bar  magnet  the  poles  are  usually  0-85  its  length  apart  (one- 
fourteenth  from  either  end).  The  line  joining  them  is  the  magnet's 
magnetic  axis;  it  is  this  line,  of  course,  which  sets  in  the  Magnetic 
Meridian,  when  the  magnet  is  free  to  turn  in  the  earth's  field. 

To  find  it,  suspend  the  magnet  in  a  paper  stirrup  by  a  length 
of  the  finest  plaited  silk  fishing-line,  without  twist,  and  let  it  settle. 
Lay  a  book  on  the  bench  with  its  edge  parallel  to  the  bar,  turn  the 
bar  over  sidewise,  top  for  bottom,  let  settle  again.  Its  magnetic 
axis,  always  in  the  magnetic  meridian,  bisects  the  angle  (if  any) 
between  book -edge  and  present  bar-edge. 

§682.  Strength  of  Pole. 

I.  The  unit  (N.)  pole  repels  with  a  force  of  1  dyne  another  unit 
(N.)  pole  placed  1  cm.  away  from  it. 

II.  A  pole  of  m  units  repels  unit  pole  with  m  dynes,  and  further, 
it  repels  pole  of  strength  m'  with  their  product  mm'  dynes.  [South 
poles  are  given  a  —  sign,  a  —  force  means  attraction.] 

This  can  be  demonstrated  by  a  *  Magnetic  Balance  '  as  follows ; 
though  with  limited  accuracy. 

Several  steel  knitting-needles  AA',  BB',  etc.,  are  magnetissed 
(or,  50  times  better,  procure  long  3/16-in. -thick  cobalt-steel  magnets 
from  Darwin's  of  Sheffield) ;  let  their  north-pole  strengths  be  A,  B. 
C,  etc.  AA'  is  laid  on  the  pan  of  a  balance  with  no  iron  about  it 
and  counterpoised.  Above  it  is  fixed  BB'  with  its  N.  pole  B 
vertically  above  A  at  a  distance  d.  Fig.  292,  and  repelling  it  down. 
The  extra  weight  necessary  in  the  other  pan  to  restore  equilibrium 
may  be  called  AB,  the  repulsion  between  poles  A  and  B.  It  should 
be  in  dvnes,  but  as  the  experimental  accuracy  is  vitiated  by  cross 
attractions  with  the  distant  S.  poles  A'  and  B',  milUgram  weights 
serve  well  enough  as  units  of  force. 

663 


554 


MAGNETISM 


[§682 


BB'  is  now  exactly  replaced  by  CC,  etc.  :  weights  AC,  etc., 
restore  equilibrium.  Since  the  repelled  pole  has  remained  the  same, 
the 

ratio  of  pole  B  to  pole  C,  etc.  =  force  AB  :  AC  :  etc. 

This  gives  a  relative  measure  of  the  poles  :   now,  removing  AA' 

and  taking  any  pairs,  placing  one 
on  the  pan  and  the  other  at  d 
above  it,  the  repulsion  between 
them  will  be  found  proportional  to 
their  product. 

III.  The  force  varies  inversely 
as  the  square  of  the  distance. 

Put  magnet  BB'  at  other  dis- 
tances, \'2d,  d,  0-Sd,  etc.,  and  the 
repulsions  will  be  found  to  be  AB/1-44,  AB,  AB/0'64,  etc. 
Hence  the  complete  law  of  magnetic  action  can  be  put : 

The  repulsion,  measured  in  dynes,  between  two  point-poles,  is 
equal  to  the  product  of  their  strengths  divided  by  the  square  of 
their  distance  apart  in  centimetres. 


Fig.  292. 


mm 


We  can  calculate  the  Absolute  Values  of  the  poles. 
B  and  C,  their  repulsions  on  A  give 


For  taking 


Pole  ratio 


—  =  repulsive  force  ratio  -^ 


Then,  replacing  AA'  by  BB',  their  mutual  repulsion  at  distance 
d  cm. 

^      =  BC  dynes. 

Multiplying  the  two  equations  together 

1  X  5-^  -  ^  X  BC 
C  ^      d2      "  AC  ^ 


.*.  pole  B  =  (Z  X  a/ Tp units,  and  C  follows  easily. 

§  683.  Strength  of  field.  The  strength  of  the  magnetic  field  (often 
briefly  referred  to  as  *  the  field  ')  at  a  place,  is  defined  as  being  equal 
to  the  force  in  dynes  that  would  be  exerted  on  a  unit  N.  pole  placed 
there.     It  is  commonly  (mis)quoted  in  *  gauss  '  (gowss). 

It  is  sometimes  miscalled  the  magnetic  force,  but  it  should  be 
carefully  distinguished  from  actual  Mechanical  Force,  to  obtain 
which  it  must  evidently  be  multiplied  by  pole  strength. 


§685]  MAGNETIC   FIELDS  655 

Taking  one  particular  instance,  the/  of  last  section, 

/  =  -T2~  ~'^^  ^   =  ^^^^  ^^  P^^®  ><  exploring  pole  strength, 

thus  strength  of  field  at  distance  d  due  to  a  single  pole 
=  ±  pole  -^  d^,  directly  away  from  a  N.  pole  or  towards  a  S.  pole. 

But  the  value  of  the  conception  of  Fiela  is  that  it  takes  you  away 
from  the  particular  things  that  are  causing  it. 

§  684.  Earth-induced  and  permanent  poles  on  a  vertical  iron  rod. 

If  an  iron  or  mild  steel  rod,  perhaps  J  yd.  x  i  in.,  be  held  upright 
in  a  wooden  stand,  with  its  lower  end  resting  on  the  bencn,  the 
Earth's  magnetic  lines,  plunging  down  towards  the  arctic,  stream 
down  through  it  and  spread  out  all  round  its  lower  end,  which  is 
therefore  an  induced  N.  pole ;  let  its  strength  be  E.  But  the  rod 
may  be  feebly  permanently  magnetized  to  start  with,  with  poles 
i  P,  so  that  one  way  up  the  lower  pole  is  E  +  P,  and  the  other 
way  up  E  —  P. 

The  efflux  of  lines  at  the  bottom  is  soon  swept  away  towards 
the  north  on  the  bench  by  the  *  horizontal  field  H  of  the  earth. 
§  692,  with  the  result  that  at  distance  d  or  d'  south  of  the  rod  a  toy 
compass,  plotting  the  resultant  lines,  finds  a  Neutral  Point,  like  those 
at  the  side  of  Fig.  291,  where  (E  +  P)/<^,  or  (E  -  P)/rf'«,  due  to 
the  pole,  just  =  H. 

Measurement  of  d  and  d'  therefore  enables  you  to  find  the  ratio 
E/P,  the  proportions  of  pole  strength  due  to  earth -induct  ion  and 
to  permanent  magnetization.  And  if,  in  a  non-ferrous  building, 
you  assume  H  =  0-185,  you  can  deduce  the  pole  strengths  in  absolute 
value. 

§685.  Moment  of  a  magnet.    Suppose  a  magnet  held  at  right 
angles  to  the  lines  of  a  field  H  as  in  Fig.  293.     A  force  Hm  acts  at 
right  angles  on  the  N.  pole  and  an  equal  force  in  the  opposite  direction 
on  the  S.  pole,  both  combining  to  turn  it  one 
way,  with  a  turning  moment  about  any  point  O. 
=  Hw  X  NO  +  Hm  X  SO  =  Hm  x  NS 
=  field  X  [pole  X  straight    length  of   magnet 
between  poles]. 

Thus  if  the  magnet  lies  with  its  axis  at  right 
angles  to  a  field  of  unit  strength,  the  couple  or 
turning  moment  acting  on  it 
=  strength  of  one  pole  x  length  between  its  poles 
and  this  product  is  called  the  Magnetic  Moment,  Fia.  IW. 

M,  of  the  magnet.  i_      •♦ 

Neither  the  strength  nor  the  position  of  a  pole  can  De  quii© 
satisfactorUy  ascertained,  but  this  product  can  be  metumred  tnih 
accuracy  (as  below),  and  is  used  to  express  the  magnetic  value  ol 
the  magnet. 


Hmt 

I     . 


;  i  ; '  i  i    ! 


556 


MAGNETISM 


§  686.  Magnetic  lines  in  relation  to  field  strength,  pole  strength,  etc. 
Here  we  may  bring  together  the  geometrical  theory  and  the 
magnetic  stream-lines.  Taking  a  square  centimetre  at  right 
angles  to  the  lines,  A  field  of  unit  strength  is  represented  by  one 
'unit  magnetic  line'  passing  perpendicularly  through  that  square 
centimetre  :  a  field  strength  H  by  H  unit  lines  per  sq.  cm. 

Actually,  of  course,  the  stream  pervades  the  whole  square 
centimetre,  there  is  no  striated  structure  in  the  field — as  the  card 
is  tapped  some  of  the  lines  of  filings  will  probably  move  sideways 
and  settle  down  where  blank  spaces  were — but  it  is  convenient  to 
think  of  unit  lines,  each  the  axis  of  a  tube  of  flow,  as  one  might 
count  wicks  in  a  box  of  candles  that  had  accidentally  softened 
into  a  solid  lump.  For  instance,  the  earth's  '  total  field  '  0-33  is 
represented  by  1  unit  line  to  each  3  sq.  cm.  of  an  area  perpendicular 
to  it. 


Measurement  of  the  strength  of  a  magnetic  field.  Theoretically, 
of  course,  one  puts  a  unit  N.  pole  at  the  place  and  feels  the  force 
on  it  in  dynes.  Practically,  the  problem  has  to  be  tackled  very 
indirectly.  There  are  two  distinct  ways  of  comparing  field- strengths, 
and,  fortunately,  a  little  calculation  can  so  combine  the  two  as  to 
give  us  field-strengths  and  magnet-strengths  in  absolute  value  with 
accuracy  : — 

§687.  Method  I.  Comparison  of  horizontal  magnetic  fields  by 
Deflection.  The  Earth's  field  H  cannot  be  got  rid  of,  ergo  the  best 
thing  to  do  is  to  make  use  of  it,  as  a  standard.  Its  value  may  be 
altered  by  the  proximity  of  girders,  gas-pipes,  etc.,  but  it  will  keep 
sufficiently  constant  at  any  one  place,  provided  one  guards  against 
movable  iron  anywhere  near — stray  magnets,  pocket-knives,  etc. 

Arrange  the  field  under  test  so  as  to  act  at 
right  angles  to  the  Earth's  field  on  a  compass. 
This,  which  is  called  in  this  connection  a  Deflec- 
tion Magnetometer,  should  have  a  short  stout 
magnetic  '  needle  '  and  a  long  light  pointer, 
stuck  on  usually  at  right  angles  to  the  magnet 
(§  767),  moving  over  a  scale  of  degrees.  There 
should  be  a  mirror  bottom  to  the  box,  and 
pointer  must  cover  its  reflection  as  you  read 
both  ends,  to  eliminate  error  of  centring. 

Then  (Fig.  294)  the  N.  pole  m  of  the  com- 
pass needle  is  being  pulled  magnetic  north- 
wards by  the  earth's  field  H  with  a  force  m  x  H  dynes,  and 
magnetic  east  (or  west)  by  the  test  field  F  with  force  m  X  F  dynes. 
The  needle  turns  and  settles  down  so  that  the  resultant  pull  acts 
along  the  line  joining  pole  to  pivot,  when,  of  course,  it  has  no 
further  turning  moment  either  way.  Meanwhile  the  S.  pole  is 
undergoing  exactly  similar  but  opposite  actions. 

Then  by  the  rectangular  parallelogram  of  forces  wABC 


Fig.  294. 


688]  MAGNETIC   FIELDS  667 

w  X  F       AC       F       ,      ^ 
=  -i —  or  TT  =  tan  D. 


m  X  H      Am      H 


Where  D  is  the  angle  the  needle  is  deflected  from  it8  natural  poMition. 
Other  test  fields  then  replace  the  first  and  produce  deflect iona 
D',  D",  etc.,  when  evidently 

F:F':F"  .  .  .  :  H  =  tan  D  :  tan  D' :  tan  D"  .  .  .  :1 

You  are  usually  asked,  in  practical  exams,  to  compare  the  fields 
produced  by  magnets  at  various  distances  along  the  directions 
of  their  axes. 

§  688.  Calculation  o!  field  due  to  a  magnet  at  a  point  on  Its  axis 
produced — *  End  on.' 

^ d >j 

k---l--~>\  " 

Fio.  295. 

Let  magnet  have  poles  ±  ^  separated  length  / ;  to  find  field  at 
d  cm.  from  its  centre  along  axis  produced.     Fig,  295. 

Field  strength  at  P  due  to  N.  pole  =  ^       ^^  to  right. 


S.     ..    =^^^^o^eH. 


m 


/.  Resultant  field  strength  F  at  P  =  ^  _  ,^>,  -  ^^  ^  |^^,  to  right. 
_  m{d^W-m{d-\X)^  _     2rM_ 

=  ^J^\m2  to  right  along  axis 

whence,  if  d  is  much  greater  than  I  (say  5  times),  very  nearly 

F  =  2M/£. 

so  that  the  field  strength  of  a  small  magnet  falls  off  ver>'  rapidly. 
being  inversely  as  the  cube  of  the  distance. 

Example.     Calculate  field  on  axis  at  25  cm^  and  85  cm.  beyond  Um»  N. 
pole  of  a  10-cm.  bar  magnet  of  pole  strength  260. 

Here  d  =  30  or  90  cm..  I  =  8-6  cm.  (S  681)    ,^  «  -  MO  X  8-5. 
Hence  at  30  cm.,  accurately  0164,  approx.  0;157  (loo  ciom). 
And  at  90  cm.,  accurately  000583,  approx.  000682. 


558 


MAGNETISM 


[§689 


§  689.  These  two  paragraphs  taken  together  enable  us  to  compare 
the  Moments  of  Magnets. 

Lay  the  magnet  magnetic  E.  or  W.  of  the  compass,  pointing 
towards  it,  centres  d  cm.  apart,  then 


F 
H 


2M 

dm 


tan  D 


M       d\      ^ 


and  replacing  by  another  magnet  at  the  same  d 

M/M'  =  tan  D/tan  D' 

or  opposing  by  another  magnet,  at  d'  the  other  side,  until  the  two 
deflections  cancel  each  other,  a  quick  rough  test 

M/M'  =  d^/d'^ 

This  is  an  experiment  the  details  of  which  you  must  learn  in  the 
laboratory,  and  unless  I  is  less  than  d/5  you  must  use  the  long  formula. 


§  690.  At  d  cm.  from  the  magnet,  but  at  right  angles  to  its  axis, 
in  an  equatorial  plane,  '  broadside-on,'  the  field  strength  is  only 
M/d^,  and  is  parallel  to  the  magnet,  but  backwards  :  in  inclined 
directions  it  has  intermediate  values  and  is  oblique,   Fig.   296. 

These  results  follow  from  rather  trickier 
calculations  with  which  your  examiners 
have  never  wished  you  to  concern  your- 
selves. 

The  fact  that  the  one  field- strength  is 
exactly  half  the  other  was  made  by  Gauss 
the  basis  of  the  most  accurate  proof  of 
the  Inverse  Square  Law  of  magnetic  ac- 
tion ;  but,  with  the  simplifying  approxi- 
mations we  are  making,  we  shall  do 
just    as    well   to    accept    the    results    of 


Fig.  296. 


§  682   III,   using  cobalt-steel  magnets. 


§691.  Method  II.  Comparison  of  horizontal  magnetic  fields  by 
Vibration.  Let  now  a  magnet,  pivoted  or  suspended  by  a  torsionless 
fibre  and  balanced  level,  be  turned  through  a  sw^lU  horizontal 
angle  from  its  position  of  rest  (straight  down  the  field),  and  then 
let  go.  It  oscillates  under  the  steady  horizontal  pull  of  the  field 
H  just  exactly  as  a  compound  pendulum  does  under  the  steady 
vertical  pull  of  gravity,  and  according  to  §  90  its  time  of  vibration 
T  sec.  =  2:1  X  V  (moment  of  inertia,  I  ~  turning  moment  acting  on 
it  when  held  out  at  right  angles  to  the  force).  By  the  argument 
of  §  685  this  turning  moment  =  field  H  X  moment  M,  of 
magnet. 


§  692]  MAGNETIC   FIEUOS  559 

.-.  T  =  27c^/III 
V   HM 

provided  that  the  arc  of  swing  is  small,  §  85, 
or  MH  =  -^ 

To  use  this  to  compare  two  field-strengths,  time  the  swings  of 
the  magnet  first  in  one  field,  then  in  the  other  ;  M  and  I  do  not  alter, 
and  hence  H  oc  l/T*,  the  strengths  of  the  fields  are  inversely  as 
the  squares  of  the  periods  of  vibration.  A  swing  twice  as  fast 
means  a  field  4  times  the  strength.  If  n  is  the  number  of  swings 
per  minute,  say,  w  oc  1/T,  and  hence  H  oc  n* ;  the  field  strength 
is  proportional  to  the  square  of  the  number  of  swings  per  minute. 

The  vibration  method  is  used  in  magnetic  survey.s,  on  all  scales 
of  magnitude.  Thus  in  Fig.  291  the  values  of  field  strength 
marked  would  be  obtained  from  the  squares  of  the  numbers  of 
vibrations  per  minute  of  a  magnetometer  (a  fibre-suspended  brass 
bob  with  J  in.  of  knitting-needle  stuck  through  ;  vibrates  longer 
and  steadier  than  the  charm  compass).  In  the  absence  of  the 
magnet  the  earth's  field  H  =  0-185  gives  a  standard  N*,  hence  the 
actual  values  of  the  fields  =  squares  of  vibrations -J- N*  X  0*185 
c.g.s.  units. 

[In  point  of  fact  the  pole  strengths  and  field  values  marked 
on  Fig.  291  were  obtained,  three  years  after  it  was  drawn,  by 
graphic  calculation.] 

§  692.  To  calculate  M  of  a  magnet,  and  H  the  horizontal  magnetic 
field  of  the  Earth,  in  absolute  value. 

§  689  gave  us  M/H,  and  applying  §  691  to  the  same  magnet 
gives  MH ;  multiplying  these  together  gives  M*,  and  dividing,  H*, 
hence  M  and  H  absolutely. 

In  algebra  it  looks  appalling,  so  let  us  take  an  actual  example  : 

Deflection  experiment.  At  30  cm.  E.  or  W.  of  a  compass  a  magnet 
caused  a  mean  deflection  25-5° 

2M  2M  .         or:  ro        a  A'7'7 

5SH  =  3OTI  =  *""  ^-^  =  "•*" 
.       ^^?L^^  0-411  =  6420. 

Vibration  experiment.  This  magnet  weighed  43-5  gm.  and  was 
7-8  cm.  long  and  1  cm.  diameter,  /.  by  §  89,  I  =  43-5  X  (S-S*;^  + 
0-52/4)  =  224.  ,         .  , 

Suspended  by  a  foot  of  finest  plaited  silk  fishing-hne,  it  made 
50  horizontal  swings  to-and-fro  in  300  sec.  .*.  T  =^  6-0  sec. 


660 


MAGNETISM 


„[§  692 


[But  the  thread  itself  takes  a  hand  in  controlHng  these  swings, 
adding  its  own  resistance  to  twist  to  the  couple  MH.  To  find  what 
this  amounted  to,  a  brass  bar  of  the  same  length  and  mass,  quite 
insusceptible  to  magnetism,  was  suspended,  and  made  1  complete 
swing  in  30  sec. 


For  this,  t 


'^V^ 


I 


stiffness  of  thread 


30  =  27r 


V. 


224 


stiffness 
stiffness =8] 


Therefore,  for  the  magnet 

.   / I 

VmH  +  stiffness 


27r 


•0  =  2.^^ 


224 


MH  +  8 

.-.  MH  +  8  =  224  X  39-5  ^  36     /.  MH=238 
M  =:V(M/H  X  MH)  =^(6420  X  238)  =  1235 
and  H  =  MH  -i-  M  =  238  -^  1235  =  0193 


THE   EARTH'S  MAGNETISM 


DEC 


§  693.  We  have  seen  that  a  magnet  freely  poised,  under  the 
influence  of  magnetic  force  alone,  sets  itself  along  a  magnetic 
line  with  its  N.  pole  '  down  stream.'  Such  a  magnet  controlled  by 
the  Earth's  Magnetic  Field  sets  itself  (in  this  country)  with 
its  N.  pole  pointing  between  N.W.  and  N.  and 
dipping  steeply  downwards,  i.e.  the  field  runs  in  this 
direction,  Fig.  297,  T. 

To  get  an  accurate  description  of  the  direction, 
the  vertical  plane  in  which  the  magnet's  axis  lies  is 
first  drawn.  This  is  called  the  plane  of  the  Magnetic 
Meridian.  The  horizontal  angle  ('  bearing ')  between 
this  plane  and  the  Geographical  (astronomical)  Meri- 
dian is  called  the  Declination  (*  Dec.')  or,  nautically, 
the  Variation  of  the  Compass,  from  true  N.  The 
angle  in  the  magnetic  meridian  plane  at  which  the 
magnetic  axis  dips  down  below  the  horizontal  is  the 
Dip. 

These    angles    are    actually    studied    separately, 

the   Dec.   in   a  compass  the   S.   end   of    which  is 

overweighted  to  prevent  the  N.  dipping,  the   Dip 

with  a  '  dip  needle  '  which  is  placed  in  the  meridian  and  allowed 

to  roll  on  a  horizontal  cross-axis. 


Fia.  297. 


§  697]  MAGNETIC   FIELDS  5el 

§694.  Declination.  This  Variation  of  the  Compaq  from  true 
N.,  which  you  and  a  limited  number  of  other  people  know  about, 
was  noticed  in  Europe  in  1269,  and  Columbus  in  1492  obaerved 
how  the  angle  changed  on  his  voyage  across  the  Atlantic. 

You  can  measure  it  on  your  own  pocket  compass  by  sighting 
the  pole  star,  and  allowing  that  the  celestial  pole  is  really  1°,  two 
moon's  diameters,  down  the  line  from  the  pole  star  to  the  middle 
star  of  the  Great  Bear's  tail  (the  handle  of  the  Plough).  Or  the 
shadow  of  the  door-post  in  London  at  1  p.m.  Summer  Time  will 
put  you  right  within  1°. 

§  695.  Dip  was  discovered  by  Norman,  a  London  instniment- 
maker,  in  1576.  '  Always  finishing  and  ending  his  neo<llc*s  l>efore 
touching  them  with  the  stone,  he  continually  found  himiielf 
constrained  to  put  some  small  piece  of  ware  on  the  S.  point  to 
make  them  level  again.  And  having  once  spoiled  a  large  needle 
by  cutting  too  much  ofif  the  north  end,  in  some  choler  he  applied 
himself  to  seek  further  into  this  effect, *  and  he  straightaway  made 
the  first  Dip  Circle  and  found  a  dip  of  71°  50'. 

Dip  circles  as  good  as  that  are  costly,  and  tedious  in  u«e,  and 
poorer  ones  are  emphatically  not  worth  using.  A  shorter  and  more 
modern  way  will  be  described  in  §§  753,  774. 

§696.  The  Earth's  magnetic  field  at  a  place  runs,  of  course, 
in  the  direction  of  the  Dip  needle.  It  is  awkward  to  measure  its 
'  Total '  field-strength  T  in  that  inclined  direction,  and  more  usually 
one  notices  these  relations  between  it  and  the  Vertical  component 
field  V  and  Horizontal  component  field  H,  which  we  measured  in 
§692 

V/H  =  tan  Dip 
V  =  T  sine  Dip 

§  697.  The  Magnetic  State  of  the  Earth.  In  considering  maps  of 
the  magnetic  state  of  the  earth  as  a  whole,  avoid  Mercator  s  pro- 
jection,  unless  you  are  a  navigator.  The  right  hemisphere  of  Fig. 
298  will  tell  you  all  you  can  possibly  want,  and  is  not  difficult  to 
get  the  hang  of.  . 

Dip.  The  needle  sets  nearly  horizonUl  in  equatoruU  region*, 
the  N.  end  dips  more  and  more  as  it  is  carried  N..  while  m  the 
southern  hemisphere  the  S.  end  dips  increasingly.  That  is.  the 
lines  of  force  run  out  of  the  southern  hemisphere  of  the  earth  mto 
space  and  return  into  the  northern  hemisphere,  ven;  much  m 
the  same  way  as  they  run  out  of  and  into  the  circle  in  tig.  2H3. 

The  great  circle  of  the  earth  on  which  there  is  no  dip  is  the 
Magnetic  Equator.  N.  and  S.  of  it  are  the  successive  -small 
circles '  of  magneUc  latitude  (called  also  Uocttnili)  on  which  the 
dip  is  r,  2°  .  .  .  89°.  Those  for  every  15°  are  shown  m  rig.  W5. 
The  two  places  at  which  the  needle  stands  always  vortical  are  called 
the  Magnetic  Poles,  both  on  the  left  hemisphere  of  the  figure. 


662 


MAGNETISM 


■[§  697 


This  system  of  circles  is  inclined  to  the  geographical  system, 
the  magnetic  equator  rising  to  10°  N.  latitude  in  the  Indian  Ocean 
and  sinking  to  16°  S.  latitude  in  Brazil.  The  circles  are  more  or 
less  distorted. 

Declination.  Dividing  the  magnetic  equator  into  360°,  starting 
off  from  each  and  following  the  direction  of  the  compass  needle 
N.  or  S.  there  are  traced  out  meridians  of  magnetic  longitude  all 
converging  towards  the  magnetic  poles.  Fig.  298.  The  indispensable 
arrows,  indicating  magnetic  N.S.,  drawn  at  various  parts  of  a  chart, 
are  really  short  pieces  of  these  lines,  which  are  the  '  lines  of  H 
component  force.'     They  are  all  great  circles,  slightly  distorted ; 


Fig.  298. 


one  of  them  practically  coincides  with  the  geographical  meridian 
90°  E.W.  (the  outside  rim  of  the  figure),  the  others  are  all  more  or 
less  inclined,  this,  of  course,  reaching  a  maximum  90°  away  from 
the  foregoing,  or  practically  on  the  0°  180°  meridian  of  Greenwich 
(centre  lines  of  figure). 

That  is,  the  line  of  '  no  variation  '  should  be  the  rim  of  the 
hemisphere ;  actually  it  runs  through  Superior,  Florida,  and  the 
River  Plate,  and  returns  through  W.  Australia  and  Malaya,  makes 
a  great  wobble  across  India  to  the  Red  Sea  and  thence  to  the  North 
Cape,  with  the  result  that  over  most  of  the  Near  and  Far  East  the 
compass  points  very  few  degrees  wide  of  true  North.  On  the  con- 
trary, on  the  N.  Atlantic  run  it  swings  out  as  far  as  33°  W. 

Nothing  can  be  put  in  here  about  the  troubles  of  the  Compass  on 
shipboard,  and  how  they  are  overcome,  but  your  pocket  .compass 
can  soon  give  you  an  idea  of  their  magnitude. 

§  698.  Changes  in  the  Earth's  magnetic  state.  The  earth's 
magnetism  continuously  alters,  the  magnetic  elements — Dec, 
Dip,  and  Force — at  any  place,  undergoing  a  slow  but  considerable 
Secular  Change,  and  even  the  poles  wandering. 


MAGNETIC   FIELDS 


56S 


The  recorded  motion  of  the  N.  pole  of  the  free  needle  at  Green- 
wich, as  seen  from  its  centre,  is  shown  in  Fig.  299. 

There  is  also  a  diurnal  variation  which  is  a  sort  of  miniature 
of  the  secular,  the  pole  of  the  needle  travelling  round  in  a  cycle  of, 
roughly,  8'  angular  diameter,  every  day. 

Breaking  in  on  this  quiet  daily  march  come  Magnetic  Storms 
which  may  fling  the  compass  needle  to  and  fro  more  than  a  degree 
from  its  mean  position. 

It  is  known  that  these  daily  variations  are  due  to  some  caune 
exterior  to  the  earth's  surface,  and  magnetic  storms  are  often 
accompanied  by  notable  displays  of  the  Aurora  Polaris,  mani- 
festing electrical  disturbance  of  the  upper  atmosphere.  That 
both  these  are  connected  with  that  variation  of  .solar  activity 
evidenced  by  the  prevalence  and  extent  of  sunspots,  is  de<lucihle 
from  the  occasional  violent  ter- 
restrial disturbance  accompany-  W  ^V  y  ^ 
ing  a  particularly  large  and  active 
sunspot,  and  with  more  certainty 
from  the  occurrence  of  an  eleven- 
year  period  in  the  frequency  of 
all  three,  their  maxima  coinciding 
within  a  year  or  two. 

At  the  Magnetic  Observatory,  at  Abinger  in  Surrey,  the  Declina- 
tion in  1935  was  11°  30'  W. ;  it  is  decreasing  11'  per  annum,  and 
usually  swings  during  the  day  from  5'  W.  to  3'  E.  of  its  mean  value. 
The  Dip  was  66°  42',  increasing  1'  per  annum.  H  =  0-1851, 
decreasing  0-0001. 

Ross  in  1831  obtained  a  dip  of  practically  90°  m  Boothia  at 
70°  N.  lat.,  97°  W.  The  Antarctic  pole  has  been  located  thrice 
much  more  recently ;  it  seems  to  be  wandering  about  somewhere 
between  72°  and  73°  S.  lat.,  155°  and  156°  E. 


EXAM   QUESTIONS.  CHAPTER  XLIII 

The  calculations  of  §§  682.  692  have  "ot  been  ajikedJTor.  othem  you  w»U 
use  in  the  lab.     Not  much  inquiry  i8  made  about  the  Earth. 

1  Describe  an  experiment  showing  that  force  between  pole,  is  in^-ewely 
^2';Xrtht  ;ltr^Tor  a  ma.net,  and  show  how  the  poeition.  of  the 
-^'^^jAronTS^'TL^^^^^  how  that  or  a  .hort  b.. 

magnet  can  be  determined. 


664  MAGNETISM 

3.  A  compass  needle  of  magnetic  moment  M  is  forcibly  deflected  from  the 
meridian  through  an  angle  A;  what  is  the  turning  moment  (of  the  couple) 
acting  on  it,  if  the  field  strength  is  H  ?  What  quantities  control  its  rate  of 
vibration  if  released  ? 

4.  A  25-cm.  2-gm.  knitting-needle  balanced  at  its  centre  until  magnetized, 
and  then  at  2  nma.  from  its  centre,  in  a  vertical  field  0*44.  Calculate  moment 
and  pole-strength,  assuming  the  poles  1  cm.  from  ends. 

5.  Define  field,  pole,  moment.  How  would  you  observe  the  effect  of  a 
single  pole,  and  compare  it  with  H  ? 

6.  A  bar  magnet  24  cm.  long  stands  upright  on  a  sheet  of  cardboard,  north 
pole  downwards.     Sketch  the  lines  of  force  on  the  card. 

Assuming  that  there  is  a  neutral  point  8  cm.  from  the  magnet,  calculate 
its  pole-strength.     (H,  0-20  gauss.)     (  X  2) 

7.  An  upright  iron  column  stands  in  the  earth's  magnetic  field,  in  London. 
Sketch  and  explain  the  resultant  distribution  of  lines  of  force  in  horizontal 
planes  near  the  top,  middle,  and  foot  of  the  column. 

By  the  compass  a  neutral  point  is  found  1  m.  from  the  foot  of  the  column ; 
calculate  approximately  the  magnetic  moment  of  the  column  if  it  is  8  m. 
high  and  H  is  0-18. 

8.  Describe  how,  with  the  aid  of  a  small  comipass,  you  would  distinguish 
between  slight  permanent  magnetization  of  an  iron  rod,  such  as  a  poker, 
and  temporary  magnetization  due  to  the  earth's  induction. 

How  would  you 

(a)  demagnetize  the  rod ; 

(b)  obtain  maximum  magnetization  from  the  action  of  the  earth; 

(c)  make  a  rough  estimate  of  the  dip  ? 

9.  Draw  the  lines  of  force  of  a  horse-shoe  magnet ;  and  of  a  pair  of  bar- 
magnets  crossing  in  the  middle  at  right  angles.     Disregard  the  earth's  field. 

10.  What  do  you  understand  by  a  line  of  magnetic  force  ? 

Give  two  methods  of  plotting  lines,  and  sketch  them  for  a  bar  magnet 
(a)  if  the  earth's  force  is  neutralized,  (6)  with  N  pole  N. 

11.  What  do  you  understand  by  the  intensity  of  a  magnetic  field  ?  Explain 
why  there  are  points  where  the  intensity  is  zero  near  a  small  magnet  placed 
in  the  earth's  field. 

12.  Explain  the  meaning  of:  Magnetic  Axis  of  a  magnet.  Resultant  Field. 
A  magnet  placed  with  its  axis  along  the  magnetic  meridian  produces  at 

a  point  along  the  axis  a  resultant  field  zero.  Diagram  this  resultant  field, 
indicating  the  point  at  which  it  is  zero,  and  also  that  at  which  it  has  double 
value. 

13.  A  short  magnet  lies  in  the  meridian  with  N.  pole  south,  and  there  is 
a  neutral  point  25  cm.  from  its  centre.     H  =  0-2,  calculate  M. 

14.  Draw  a  diagram  to  show  the  lines  of  force  near  a  bar  magnet  with  its 
axis  in  the  magnetic  meridian  and  its  north  pole  towards  the  south. 

If  there  is  a  neutral  point  15  cm.  from  the  north  pole,  and  the  distance 
between  the  poles  of  the  magnet  is  20  cm.,  what  is  the  pole  strength  (H  = 
0*18),  and  where  is  the  other  neutral  point  ?     (  x  2) 

15.  What  do  you  understand  by  intensity  of  magnetization  ?  Give  some 
method  of  comparison  for  different  specimens. 

[It  is  pole-strength  per  sq.  cm.  cross-section  of  bar,  compare  poles  and 
measure  areas.] 

16.  Describe  an  instrument  for  comparing  the  strengths  of  two  magnetic 
fields,  and  explain  how  it  could  be  used  for  comparing  the  field  strengths 
at  different  distances  along  the  axis  produced  of  a  bar  magnet.     (  X  2) 

17.  Define  magnetic  momients;  and  give  a  method  of  comparing  them. 

18.  How  would  you  compare  the  field  10  cm.  away  from  the  centre  of  a 
small  bar  magnet  (a)  along  its  axis,  (6)  at  right  angles  to  it ;  and  how  would 
you  expect  the  results  to  come  out  ? 


MAGNETIC   FIELDS  565 

19.  How  would  you  define  the  direction  and  magnitude  of  the  field  on  the 
bench,  and  find  possible  variations  ? 

20.  A  magnet  can  swing  in  a  horizontal  plane.  Upon  what  does  ite  time 
of  oscillation  depend  ?  How  would  you  increase  or  decreaae  it  without 
touching  the  magnet  ?  How  would  you  distinguish  between  a  change  in 
the  intensity  of  the  earth's  field,  and  in  its  direction  ? 

21.  What  is  meant  by  the  terms  :  unit  nmgnetic  pole,  magnetic  moment, 
magnetic  substance,  magnetic  permeability  7 

A  magnet  is  oscillating  freely  in  the  earth's  field ;  to  what  extent  would 
the  period  be  affected  if  the  field  were  reduced  by  20%  ? 

22.  What  do  you  mean  by  H  =  0-18  gauss?  A  little  suspended  magnet 
has  made  20  vibrations  in  184  sec,  and  when  a  short  magnet  of  moment  M 
is  placed  10  cm.  south  of  it,  makes  30  in  159  sec.     Calculate  M. 

23.  Define  the  Magnetic  Meridian  an<l  describe  carefully  how  you  would 
find  its  direction.     How  is  it  related  to  the  geographical  meridian  ?     (  X  2) 

24.  What  is  the  meaning  of  the  North  Magnetic  Pole  of  the  Karth  ?  Uliat 
instruments  would  you  use  to  detect  variations  of  the  Earth's  magnetism  ? 

25.  Explain  the  meaning  of  (o)  horizontal  field  or  intensity.  (6)  total 
intensity,  (c)  dip,  as  applied  to  the  eculh's  magnetic  field.  What  is  the  relation 
between  them  ? 

How  would  you  determine  one  of  them,  and  in  what  unit  would  you  exprsM 
your  result?     (  x  2) 

26.  Give  some  experimental  method  of  comparing  H  and  V. 

27.  How  would  you  determine  the  Dip,  and  how  does  it  vary  from  place 
to  place  on  the  earth  ?     (  X  2) 

28.  Describe  briefly  the  magnetism  of  the  Earth  and  its  variations.  If 
H  =  018  and  Dip  is  60°,  find  Total  Force.     (  x  2^ 

29.  Describe  an  experiment  to  determine  H,  and  outline  the  calculation. 


PRACTICAL  QUESTIONS 

Plot  the  equipotential  lines  near  a  single  pole,  and  from  the  neutral  point 

deduce  M.  .  .  ^    •  i.*         i      *    i-..— 

[Magnet  will  be  long  and  vertical,  equipotentials  are  at  right  angles  to  lines 

offorce,  cf.  Fig.  313.] 

Plot  lines  round  base  of  a  vertical  iron  rod,  replot  reversed,  and  find  what 
part  of  pole-strength  is  earth -induced,  and  what  is  quasi-permanent. 

Plot  the  field  between  lower  end  of  a  vertical  magnet,  of  known  pole-strongth, 
and  a  short  horizontal  magnet  pointing  towanis  it,  deduce  its  M. 

Find  the  magnetic  axis  of  a  plate,  set  it  in  the  meridian,  plot,  and  find  M. 

Plot  the  field  on  a  drawing  board,  and  mark  the  position  and  polarity  of 
concealed  magnets. 

Use  a  deflection  magnetometer  to  find  the  effective  length  of  a  magne*. 

Find  how  strength  of  field  varies  from  centre,  (o)  along  axis,  (6)  at  right 
angles  to  axis  of  magnet. 

Compare  pole  strengths  of  two  magnets;  also  momenU. 

Compare  moments  by  oscillation ;   ditto  fields. 

Find  how  M  of  an  electromagnet  depends  on  the  current. 


ELECTROSTATICS 


CHAPTER   XLIV 


FRICTIONAL  ELECTRICITY 


§701.  The  crackle  and  sparkle  in  dry  hair  and  fur,  and  its 
lifting  towards  the  hand  that  stroked  it,  must  have  been  abnost 
as  well  known  to  the  hunters  of  the  cave  bear  and  the  mammoth 
as  to  their  descendants  of  the  present  day.  And  to  the  Egyptians 
of  the  later  dynasties,  with  their  employment  of  resins,  their 
veneration  for  cats,  the  guardians  of  their  granaries,  and  their 
torrid  climate,  the  active  attraction  and  adhesion  of  dust  and  light 
stuff  was  doubtless  an  occurrence  too  familiar  to  be  placed  on  record. 

The  first  discourses  on  the  attractions  of  the  lodestone,  and  of 
the  fossil  resin  YjXeKTpov  (amber),  are  ascribed  to  Thales  of  Miletus, 
traveller  in  Egypt,  first  predictor  of  eclipses  (585  B.C.),  one  of 
the  Ionian  thinkers  renowned  as  the  Seven  Wise  Men  of  Greece ; 
men  who,  abandoning  the  old  myths,  took  the  natural  world  out 
of  the  hands  of  the  gods,  drove  the  eagle  of  Zeus  from  the  sky,  and, 
striving  always  to  discern  the  natural  laws  governing  all  things, 
laid,  by  the  greatest  achievement  of  the  human  intellect  of  all  time, 
the  foundations  of  the  new  world  of  thought  that  we  call  science 
and  philosophy. 

The  '  Electron  '  is  now  the  atom — the  uncut  unit — of  negative 
'  Electricity.' 

Many  writers  invented  and  elaborated  ludicrous  legends ;  but 
the  first  real  advance  was  made  by  William  Gilbert,  born  in  Colchester 
1544,  and  ultimately  established  in  1573  as  a  physician  on  St.  Peter's 
Hill,  near  the  Royal  College  of  Physicians,  and  within  bowshot  of 
St.  Paul's,  the  great  spire  of  which  had  been  pulled  down  in  1561, 
after  repeated  damage  by  lightning.  President  of  the  College  in 
1599,  he  was  appointed  Physician  to  Queen  Elizabeth  ;  died  not 
long  after  her  in  1603,  and  lies  buried  in  the  Saxon- towered  church 
of  Trinity,  Colchester. 

He  spent  £5000  on  his  experiments,  but  all  his  apparatus  and 
MSS.,  bequeathed  to  the  College,  were  burnt  in  the  Great  Eire 
of  1665.  He  was  the  first  English  exponent  of  the  Copernican 
theory,  and  one  of  the  chief  contributors  to  the  original  London 
Pharmacopoeia. 

He  was  a  man  who  in  his  writings  absolutely  disregarded  authority, 
and  accepted  nothing  at  second  hand :   the  value  of  his  electrical 

566 


§702]  FRICTIONAL   ELECTRICITY  667 

work  is  based  even  more  largely  on  the  scientific  method  which  he 
was  the  first  to  inculcate  and  practise,  than  upon  the  importance 
of  his  actual  discoveries.  *  Careful  experiment  and  observation. 
and  not  the  inner  consciousness,  are,'  he  writes,  *  the  only  foundation 
of  true  science.  .  .  .     Nothing  hath  been  set  down   in   my  book 

which  hath  not  been  explored  and  many  times  repeated  by  myself 

It  is  very  easy  for  men  of  acute  intellect,  apart  from*  cxjieriment 
and  practice,  to  slip  and  err.' 

Altogether  an  outstanding  physician  ;  a  vast  contrast  to  others, 
whose  mentality,  if  such  it  be,  seems  more  akin  to  that  of  a  certain 
dean  ;  of  whom  it  is  related  that,  when  a  few  students  of  natural 
philosophy  ventured  to  suggest  that  they  would  much  like  to 
actually  see  some  of  those  things  they  had  been  diligently  learning 
about,  he  froze  in  shivering  dignity,  intimated  that  their  requeiit 
was  a  singularly  ill-conceived  and  reprehensible  reflection  upon 
their  instructor,  who  was  a  graduate  of  high  culture,  and  shortly 
about  to  take  Orders,  and  warned  them  that  any  further  instance 
of  such  inconsiderate  and  distressing  temerity  would  be  met  with 
a  strong  hand — and  so  on,  ad  nauseam  :  perhaps  you  have  met  him 
too. 

Born  in  the  sister  town,  next  door  to  that  Thomas  Wolsey 
whose  name  and  fame  must  have  been  fresh  in  memory  in  the 
Gilbert  household,  and  bred  in  the  same  countryside.  I  verilv 
hope  that  something  of  the  true  Gilbert  spirit  may  breathe  forth 
to  you  from  these  pages,  printed  in  mine  own  county,  of  this  my 
book. 

§  702.  In  his  epoch-making  '  De  magnete  .  .  .  etc.,  jAurimis 
experimentis  demonstrata'  Gilbert  noted  that  without  Friction 
few  bodies  gave  out  their  natural  '  emanation  and  effluvium,'  and 
he  made  up  lists  of  things  in  which  friction  excited  this  attractive 
effluvium,  and  gave  them  the  name  of  Electrics.  Such  are  amber, 
resin,  lac,  wax,  sulphur,  paper,  dry  wood,  silk,  glass,  etc. 

Substances  from  which  friction  drew  no  effluvium — metaU, 
stone,  etc. — were  Non-electrics. 

'  Electrics  attract  all  things  save  flame  and  objects  aflame  and 
thinnest  air,  the  effluvia  are  consumed  by  flame  and  igneous  heat, 
vet  they  draw  to  themselves  the  smoke  of  an  extinguishetl  candle.* 
*  He  found  the  necessity  of  getting  rid  of  the  damp  usually  adherent 
to  everything.  *  Moisture  suppresses  the  effluvium,  but  olive  oil 
does  not,'  and  as  a  practical  point  in  frictional  electrical  ex- 
periments this  is  all-important.  Everything  should  ^J*  »".>'*"" 
warm,  short  of  melting  wax  or  cracking  glass ;  and  so  should  the 
atmosphere.  The  presence  of  a  number  of  people  may  moisten 
the  air  too  much  ;  frictional  electrical  experimenU  are  among  tho«» 
things  that  will  go  wrong  in  public.  , 

Amber  is  always  reliable,  but  unobtainable  m  any  size.  HticKs 
and  plates  of  Ebonite  (black  hard -vulcanized  rubber)  are  m«it 
generally  useful :    it  should  be  tough  and  of  good  quahty,  cheap 


568  ELECTROSTATICS  [§  702 

brittle  varieties  are  not  much  use.  Preferably  its  polish  should  be 
removed  with  sand-paper,  and  thereafter  it  should  be  kept  in  the 
dark,  otherwise  its  surface  is  apt  to  oxidize  to  sulphuric  acid. 
Sulphur,  Shellac,  and  fine  Sealing-wax  are  good,  but  brittle  ;  brown 
paper  is  excellent  when  dry,  but  is  hygroscopic,  and  must  be  scorched 
before  the  fire  every  minute  or  two.  Celluloid  electrifies  easily, 
but  leaks.  All  these  are  electrified  by  rubbing  with  dry  fur  or  flannel 
or  the  coat-sleeve.  Glass  is  apt  to  collect  a  surface  film  of  moist 
dirt,  it  should  be  washed  in  hot  soap-and-water,  rinsed  in  hot  water, 
and  wiped  dry,  and  is  then  freely  electrified  by  warm  silk. 

The  property  of  becoming  electrified  by  friction  is,  however,  not 
confined  to  '  electrics,'  as  Gilbert  supposed.  No  amount  of  drying 
can  make  metals  '  electrics,'  but  if  a  tube  or  plate  of  metal  be 
mounted  on  a  handle  made  of  an  *  electric,'  and  be  whacked  with 
dry  fur  or  silk,  it  will  be  found  electrified. 

Tests  of  electrification.  The  picking  up  of  light  stuff — paper, 
feathers,  hair,  dust,  etc. — is  a  rough  test.  A  curious  woolly  tickling 
is  felt  on  the  nose  and  face  when  an  electrified  plate  is  held  close 
to  it  :  much  more  delicate  is  the  attraction  of  a  little  pill  of  elder- 
pith,  or  of  cork,  suspended  by  a  fine  thread,  and  still  more  sensitive 
is  the  gold-leaf  electroscope,  to  be  described  later. 

§  703.  It  soon  appears  that  there  are  two  opposite  kinds  of 
electricity  obtained  by  rubbing  different  substances  ;  much  as  there 
are  two  opposite  polarities  of  magnetism.  The  mutual  repulsion  of 
bodies  charged  with  the  same  kind  (same  '  sign  ')  of  electricity 
is  shown  by  rubbing  two  sticks  of  sealing-wax,  placing  one  in  a 
stirrup  of  wire  or  card  suspended  by  a  plaited  thread  or  very 
narrow  ribbon,  and  bringing  the  other  near  it.  The  same  repul- 
sion occurs  with  glass  rods  ;  but  glass  and  sealing-wax  attract 
each  other.  The  repulsion  is  very  easily  shown  by  stripping 
a  doubled  silk  ribbon  through  the  fingers ;  the  two  halves  straddle 
wide  apart.  And  on  occasions  one's  own  hair,  dried  after  a  wash, 
becomes  electrified  and  quite  unmanageable. 

Indirectly,  the  repulsion  can  be  shown  by  use  of  a  pith  ball 
hung  by  a  thread  or  fibre  of  silk.  It  comes  up  to  touch  the 
electrified  glass  and  then  flies  away.  That  this  repulsion  is  due  to 
its  having  picked  up  electrification  from  the  rod  is  proved  by  the 
now  increased  strength  of  its  attraction  towards  rubbed  sealing- 
wax  :  the  ball  jumps  rapidly  to  and  fro  between  the  two  opposite 
rods. 

The  electrification  developed  on  glass  was  called  '  vitreous,' 
and  now  positive  (  +  ) ;  that  upon  resin,  sealing-wax,  sulphur, 
ebonite,  etc.,  '  resinous,'  now  negative  (  — ). 

§  704.  If  the  pith  ball  is  suspended  by  cotton  from  a  glass  or  sealing- 
wax  holder,  it  will  be  found  to  gain  an  electrical  charge  from  the 
electrified  rod  drawn  across  the  thread.  The  electricity  has  travelled, 
or  has  been  conducted,  along  the  '  non-electric  '  material. 


§705]  FRICTIONAL  ELECTRICITY  669 

This  immediately  explains  why  non-electrics  do  not  ordinarily 
show  electrification  after  friction ;  they  conduct  the  developed 
electricity  away  to  the  hand,  and  it  passes  through  the  experi- 
menter's body  down  to  the  Earth,  the  great  receptacle  for  all  stray 
electric  charges. 

Consequently  *  non-electrics  '  are  nowadays  called  Condactors, 
while  '  electrics  '  are  non-conductors  or  Insulators,  for  on  them  the 
electricity  cannot  travel  about,  but  remains  isolated  in  patches, 
often  difficult  to  scrape  off.  (Wiping  only  makes  more  :  clasp  in 
a  damp  hand,  or,  best,  pass  over  a  flame,  to  diselectrify.) 

§  705.  The  attraction  exerted  by  electrified  on  unelectrified 
objects  leads  by  the  argument  of  §665  to  the  idea  of  Bleetrie 
Induction.  The  mere  proximity  of  an  electric  charge  induces  a 
separation  of  -|-  and  —  electricity  in  the  uncharged  body  ;  which- 
ever charge  is  of  opposite  sign  is  drawn  nearer  to  the  inducing 
charge,  and  the  attraction  between  these  overbalances  the 
repulsion  between  the  inducing  charge  and  the  more  distant 
residuum  of  the  same  sign. 

That  this  separation  does  occur  is  shown  by  an  experiment 
like  Fig.  300.  A  '  conductor  '  is  made  up  of  two  separable  halvee» 
e.g.  two  apples,  hung  by  silk  threads  and 
touching  each  other ;  a  charged  rod  is 
brought  near  one,  and  they  are  separated. 
Both  will  now  affect  an  electroscope,  but 
oppositely,  and  the  effect  of  the  one  that 
was  nearer  the  rod  is  opposite  to  the  rod's 
own  effect. 

Thus  as  in  Magnetism  opposite  charges 
have  been  induced  to  separate  :  quite 
unlike  Magnetism  they  can  be  isolated  on 
separated  halves  of  the  conductor.  They 
reunite  if  the  two  halves  are  touched 
together  again,  in  the  absence  of  neighbour- 
ing charge  (iii). 

On  a  Non-conductor  these  charges  cannot 
move    apart,    and    if  this   explanation   of 

attraction  of  uncharged  bodies  is  true,  a  non-conductor  ought  not 
to  be  appreciably  attracted  towards  a  charged  rod. 

A  very  simple  and  striking  experiment  shows  this,  cotton  and 
silk  threads  hang  side  by  side  over  the  finger,  a  rubbed  rod  of  glass 
or  sealing-wax  is  brought  near,  the  conducting  cotton  rises  high  to 
meet  it,  the  non-conducting  silk  hangs  indifferent. 

The  third  law  of  motion  applies,  of  course,  to  force*  of  electrical 
origin  just  as  to  any  others  :  that  there  is  attraction  between 
two  bodies  leaves  it  an  open  question  as  to  which  carries  the  '  in- 
ducing charge.'  Other  circumstances  sometimes  tell  us,  e.g.  if  a 
pith  ball  spontaneously  moves  up  to  meet  the  hand  we  know  that 
it  was  the  ball  that  was  electrified. 

Mutual  repulsion  necessarily  means  electrification  oj  both. 


570  ELECTROSTATICS  [§  706 

§  706.  Since  the  two  opposite  induced  charges  sprang  into 
being  without  the  conductor  being  touched  in  any  way  (and 
without  any  conduction  through  the  air  by  spark),  and  subse- 
quently neutralized  each  other  without  leaving  any  residue,  they 
must  have  been  exactly  equal.  There  was  a  temporary  separation 
of  electricities,  but  no  creation. 

There  never  is  a  creation.  Mount  on  sealing-wax  handles  a 
disc  of  ebonite  as  big  as  a  penny  and  a  similar  disc  of  card 
covered  with  cloth.  Rub  them  together,  the  handles  prevent 
either  of  them  losing  any  of  the  charge  developed  on  it  by  the 
friction.  Hold  them  together  near  a  pith  ball — no  effect — 
separate  them,  and  the  ball  dances  from  one  to  the  other,  showing 
that  they  are  oppositely  charged. 

Ordinarily  the  cloth,  etc.,  used  as  rubber,  is  held  in  the  hand, 
and  as  it  is  not  a  good  insulator  its  electrification  soon  travels 
down  through  the  experimenter  to  earth  and  is  lost  sight  of,  and 
there  appears  to  have  been  a  production  of  one  sort  of  electricity 
only  (and  rubbing  a  metal  plate  both  charges  travel  away, 
leaving  no  signs  of  electrification  at  all). 

Why  friction  should  cause  this  separation  of  positive  and 
negative  electricities  (this  driving  of  electrons  down,  or  bringing 
them  nearer  the  surface),  we  do  not  know.  Conceivably,  however,  it 
produces  local  heating  and  increases  the  natural  tendency  to 
oxidation  of  the  sulphur,  resin,  insulated  metal,  etc.,  and  this 
may  be  the  obscure  beginning  of  an  electro-chemical  process. 
This  makes  it  depend  on  the  presence  of  an  atmosphere,  but  of 
that  the  merest  clinging  traces  would  suffice,  and  the  question  is 
not  to  be  tested  by  merely  pumping  out  '  a  vacuum  '  over  the 
surfaces. 

§  707.  Just  as  with  Magnetism,  the  most  graphic  way  of 
explaining  electrical  actions  is  by  filling  the  field  with  Lines  of 
Electric  Force. 

Each  line  links  together  a  -\-  and  a  —  charge  ;  it  is  said  to  originate 
on  the  +  charge  and  run  from  it  until  it  ends  on  the  —  charge.  The 
shapes  of  the  lines  in  a  few  cases  are  shown  in  the  tracings,  taken 
from  photographs  which  do  not  reproduce  well  on  this  paper. 

In  Fig.  301  they  are  radiating  from  a  +  charged  body  to  end 
on  an  equal  —  charge  induced  on  a  surrounding  wall.  Fig.  302 
shows  their  path  from  a  +  to  an  equal  —  charged  body  (such  as 
two  conductors  in  a  cable  ;  or,  half  of  it,  from  thundercloud  to 
ground)  ;  below  it  Fig.  304  shows  two  equal  charges  of  the  same 
sign  (the  powder  has  been  repelled  right  clear  of  the  conductors), 
notice  the  straight  impassable  barrier  which  forms.  Fig.  303  shows 
the  action  of  a  gold-leaf  electroscope ;  it  also  illustrates  §  708  ; 
*  there  is  no  force  inside  a  closed  conductor.' 

Notice  how  the  '  repulsion  of  similar  charges  '  appears  rather 
as  a  pulling  apart,  by  the  lines  joining  them  to  opposite  charges 
which  they  have  induced  on  surrounding  conductors,  e.g.  the  metal 
case  of  the  electroscope. 


708] 


FRICTIONAL   ELECTRICITY 


571 


Notice  that  these  electric  lines  are  not  supposed  to  be  continued 
through  the  substance  of  the  conductor,  as  magnetic  lines  were 
through  the  magnetized  iron.     The  existence  of  a  line  presuppoaes 


Fio.  301. 


Fig.  303. 


Fio.  304. 


a  +  and  —  charge  at  its  ends,  and  if  these  are  situated  in  the  same 
conductor  the  line  joining  them  immediately  pulls  them  together, 
and  line  and  charges  disappear.  Conductors  are  blanks  on  an 
electric -line  diagram. 

§  708.  Nor  is  there  any  need  for  the  conductor  to  be  aolid 
throughout.  For  suppose  there  were  -f  and  —  charges  on  a 
hollow  conducting  shell,  Fig.  305,  and  a  line  joined  them  ;  under 
its  pull  the  charges  ran  round  the  shell  to  meet 
each  other  and  coalesce,  and  the  line  dis- 
appeared. 

But  might  there  not  be  a  line,  such  as  XY, 
crossing  the  hollow  on  its  way  from  a  -}-  charge 
on  the  conductor  to  some  remote  —  charge 
elsewhere  ?  This  would  have  to  cut  through 
the  conductor  at  Im,  the  piece  Im  disappears 
since  it  is  in  a  conducting  material,  therefore 
/  is  the  end  of  a  line  XI  and  must  be  a  —  charge, 
XI  shrinks  up  as  before,  and  all  that  is  left  is 
the  beginning  of  a  line,  i.e.  a  +  charge,  at  m.  In  other  words,  X 
has  travelled  round  to  m  under  the  pull  of  the  line  to  the  distant- 
charge. 


Fto.  905. 


572  ELECTROSTATICS  [§  708 

Hence  a  charge  on  a  conductor  produces  no  lines  inside  it,  i.e. 
no  electric  force  inside  it,  whether  it  is  solid  or  hollow. 

Faraday  tested  this  thoroughly.  He  built  a  large  box,  sus- 
pended it  by  silken  ropes,  and  connected  it  with  an  electrical  machine 
so  that  sparks  several  inches  long  could  be  obtained  from  all  over  it : 
meanwhile  he,  inside,  with  delicate  electroscopes,  tried,  and  failed, 
to  detect  any  sign  of  electric  force  there. 

§  7(39.  It  has  an  important  practical  application  in  Electric 
Sliielding.  Any  instrument  entirely  enclosed  in  a  conducting 
envelope  connected  to  earth  is  perfectly  shielded  from  all  external 
electrical  disturbance.  All  that  the  latter  can  do  is  to  induce 
various  charges  on  the  sheath.  All  parts  of  a  radio  set  should  be 
individually  shielded  like  this. 

It  is  very  strikingly  shown  by  an  experiment  in  which  a  pith  ball 
hangs  inside  a  soap  bubble ;  an  electrified  rod  is  brought  near,  and 
the  soap  bubble  bulges  out  to  meet  it,  but  the  pith  ball  hangs  quite 
unaffected.  Bringing  the  rod  too  near,  the  bubble  bursts,  and  in- 
stantly the  ball  flies  up  towards  the  rod.  Coarse  wire  gauze  makes 
a  sufficient  shield.  Carefully  paraffin-waxed  paper  is  almost 
the  only  insulator  perfect  enough  to  have  no  screening  action. 

§  710.  Since  the  charge  on  a  hollow  conductor  is  unable  to 
produce  lines  inside  it,  no  part  of  the  charge  is  on  the  inner 
surface.  For  if  it  were,  lines  would  arise  from  it  and  must  pass 
across  the  cavity. 

And  none  of  the  charge  remains  at  rest  in  the  body  of  the  metal 
by  the  argument  of  §  708. 

All  is  on  the  outer  surface,  brought  there  by  the  pull  of  the  lines 
joining  it  to  the  equal  and  opposite  charges  on  other  conductors 
elsewhere.  And  the  lines  it  emits  leave  the  surface  perpendicularly, 
otherwise  their  '  resolved  component '  parallel  to  the  conducting 
surface  would  tow  the  charges  along  it,  until  the  pull  became 
entirely  at  right  angles  to  it. 

This  Absence  of  Charge  inside  a  Closed  hollow  Conductor  is 
easily  demonstrated.  The  hollow  conductor  may  be  a  tin  can, 
with  a  If -in.  hole  cut  in  its  lid,  insulated  by  standing  on  wax 
or  ebonite,  and  charged.  A  small  insulated  conductor,  called  a 
'  Proof  Plane,'  say  a  halfpenny  on  the  end  of  a  stick  of  sealing-wax, 
is  lowered  into  the  can  and  touched  on  its  inside,  then  taken  out 
and  touched  on  a  gold-leaf  electroscope.  No  effect.  But  if 
touched  on  the  outside  of  the  can  and  then  tested,  the  leaves,  of 
course,  diverge. 

[Notice  particularly  that  if  a  wire  attached  to  the  electroscope 
and  twisted  round  a  sealing-wax  handle  is  lowered  in  to  touch  the 
inside  of  the  can,  the  leaves  do  diverge  just  as  much  as  if  the  wire 
touched  the  outside.  For  now  can,  wire,  and  gold  leaves  combine 
to  form  one  conductor,  and  this  is  not  a  hollow  or  nearly  closed 
one.] 


§  713]  FRICTIONAL  ELECTRICITY  57S 

It  is  easy  to  show  that  an  insulated  charged  *  Faraday  butterfly- 
net '  gives  up  no  charge  from  its  inside  to  a  proof  plane,  and  that 
when  pulled  inside  out,  by  a  silk  thread  attached  to  its  bottom,  the 
charge  travels  through  so  as  still  to  be  on  the  outaide  only. 

§711.  Nothing  that  has  been  said  precludes  the  existence  of 
lines  inside  a  conductor  provided  that  they  emanate  from  separate* 
charged  bodies  inside  and  insulated  from  it,  and  these  lines  then 
do  induce  opposite  charges  on  the  inner  surface  of  the  cavity.  For 
instance,  an  electrified  rod  inside  a  room. 

But  it  does  follow  that  if  any  of  these  charged  bodies  Is  touched 
on  the  wall  it  gives  up  the  whole  of  its  charge  to  the  hollow  con- 
ductor, instead  of  merely  sharing  it.  Thus  we  can  transfer  the  whole 
of  the  charge  on  anything — a  proof  plane,  for  instance — to  an  electro- 
scope,  by  standing  a  deep  narrow  can  on  the  plate  of  the  electro«cope 
and  lowering  the  proof  plane  to  touch  the  can  inside  near  the  bottom. 
Nearly  enough,  it  is  then  *  inside  a  closed  conductor.* 

As  the  charged  proof  plane  is  lowered  into  the  deep  cavity  the 
leaves  spread  out,  and  it  will  be  noticed  that  the  final  touching 
has  no  sudden  effect.  This  leads  on  to  the  whole  question  of 
Charging  by  Induction,  and  its  explanation. 

§  712.  Charging  by  Induction.  We  have  seen  that  when  a  charge 
is  produced  by  friction  there  is  an  equal  and  opposite  charge  on 
the  rubber.  As  the  two  things  are  separated,  the  a uaai -elastic  line« 
of  electric  force  draw  out,  and  spread  out,  so  as  to  fill  the  surrounding 
space,  but  each  trying  to  remain  as  short  as  it  can,  consistently 
with  the  sideways  pressure  of  its  neighbours. 

If  a  magnetic  line  ran  near  iron  it  bent,  so  as  to  run  a«  much 
of  its  course  as  possible  in  the  iron.  When  an  electric  line  comes 
near  a  conductor,  it  bends  towards  it,  and  may  break  in  halvM, 
the  broken  ends  on  the  conductor — meaning  that  equal  and  opposite 
charges  are  induced  on  it — and  these  broken  ends  (charges) 
separate  without  difficulty.  This  happens  when  electric  wave 
meets  radio-receiving  aerial. 

The  weakening  of  the  magnetic  line  in  iron  is  superse<ied  bv  the 
total  obliteration  of  the  electric  line  in  the  conductor,  and  the 
pieces  left  at  the  ends  are  together  shorter  than  the  original  line. 

With  the  aid  of  these  lines  one  can  solve  electrostatic  problems 
wholesale,  but  for  the  little  we  want,  a  very  simple  conception  will 
serve — that  of  the  Long  Conductor. 

§  713.  Look  for  a  long  conductor,  and  put  your  charged  bodv 
near  one  end  of  it ;  then  an  opposite  charge  is  induced  on  that  end, 
and  a  '  same  '  charge  drifts  off  to  the  far  end.  The  two  knobs. 
Fig.  300,  form  such  a  *  long  conductor.' 

The  Gold-leaf  Electroscope  has  a  metal  stem  bearing  at  the  top 
a  knob,  or  plate,  to  which  the  various  charged  bodies  to  be  tested 
are  presented.    The  stem  passes  down  through  an  insulating  plug  of 


574 


ELECTROSTATICS 


[§713 


wax  or  ebonite  into  a  draught -proof  box  of  metal  and  glass,  to  its 
flattened  lower  end  are  gummed  a  pair  of  strips  of  gold-leaf,  or, 
for  all  ordinary  use,  Dutch-metal,  or  better,  for  measuring  purposes, 
one  '  leaf  '  is  a  stiff  strip  of  metal  as  in  Fig.  306.  Charge  given  to 
the  top  of  the  stem  travels  down  and  is  shared  by  the  leaves,  which 
thereupon  open  out  by  mutual  repulsion. 

In  Fig.  307  the  cap-stem-and-leaves  of  the  gold-leaf  electroscope 
form  the  long  conductor — you  put  the  rod  near  the  top  plate 
because  all  the  lower  part  is  '  shielded '  by  the  metal  case — the 
leaves  have  acquired  '  same  '  charge.  For  this,  in  its  turn,  case 
and  earth  form  '  long  conductor,'  —  is  induced  up,  and  then  opposite 
charges  attract  and  pull  the  leaves  open,  Fig.  307  A.  Without 
the  metal  case,  electroscopes  are  very  fickle  in  action. 


^ 

» 

^ 

r=- 

<. 

^ 

1 

1 

1 

1 

1 

s 

-   + 
A 

\- 

B 

<-    - 
C 

\. 

Fig.  306. 


Fig.  307. 


If  you  scrape  the  rod  on  the  top  plate,  -j-  and  —  destroy  each 
other  there,  and  leave  all  +  on  the  stem-and-leaves ;  you  have 
'  charged,  by  contact,  with  the  same  sign.' 

But  in  Fig.  B,  where  you  are  touching  the  cap  of  the  electro- 
scope with  your  finger,  cap-stem-and-leaves  are  merely  the  near 
end  of  YOU,  the  Long  Conductor  ;  +  goes  far  away,  the  leaves  are 
collapsed. 

If  now  you  take  away,  first  your  finger,  and  then  the  rod  {not 
the  other  way  about),  the  —  spreads  all  over  cap-stem-and-leaves, 
and  the  leaves  open  out  with  the  opposite  sign  to  the  inducing  charge 
on  the  rod.  Fig.  C. 

You  have  '  charged,  by  induction,  with  opposite  sign,' 

The  puzzling  things  that  can  happen  when  the  case  of  the 
electroscope  is  insulated  are  a  mere  un kindness  to  students. 

§714.  The  Electrophorus  (electricity  carrier).  Fig.  308,  is  an 
important  instance  of  charging  by  induction.  It  is  the  simplest 
sort  of  '  electrical  machine '  by  means  of  which  considerable 
quantities  of  electricity  may  be  obtained  without  continual  waste 
of  labour  in  friction. 

On  the  table  lies  a  slab  of  ebonite,  glass,  etc.,  or  a  sheet  of 


§714] 


FRICTIOXAL   ELECTRICITY 


575 


scorched  brown  paper,  rubbed  or  brushed  to  electrify  it,  as  usual. 
Upon  this  is  laid  a  smaller  plate  of  thin  metal,  usually  a  disc  of 
tin  or  brass  3  to  6  in.  diam.,  to  which  is  attached  an  insulating 
handle.  Brown  paper,  and  the  lid  of  a  tin  which  has  l)een  stuck, 
while  hot,  on  to  half  a  stick  of  sealing-wax,  is  a  homely  com- 
bination that  works  as  well  as  anything. 

Any  sharp  corners  and  edges  on  the  metal  plate  should  be 
smoothed  off,  or  the  charge  would  readily  leak  from  them  into  the 
air,  §§  893,  895.  Hold  the  handle  by  its  upper  part  only,  or  the 
charge  may  leak  to  the  fingers.     The  handle  should  be  clean. 

Stand  the  plate  on  the  electrified  slab  (which,  as  the  lines  of  force 
show,  is  perturbing  the  whole  neighbourhood) :    it  comes  by  no 


Fig.  308. 


means  into  that  close  contact  with  it  which  is  necessary  to  actually 
pick  off  charge  from  the  electrified  surface,  §704;  this  probably 
occurs  at  only  a  few  small  patches,  therefore  it  is  shown  in  the  dia- 
gram as  distinctly  separated  :  it  gets  no  charge,  and  the  electric 
lines  continue  to  spread. 

Now  touch  the  plate — i.e.  earth  it — and  it  becomes  the  near  end 
of  that  Long  Conductor,  yourself ;  and  a  -f  charge  is  drawn  up 
along  you  into  it.  Plate  and  slab  lie  inert,  without  potential 
energy,  there  seems  to  be  no  electricity  about  an^'where. 

Removing  your  finger,  lift  the  plate  by  the  top  of  its  insulating 
handle  (iii),  and  you  find  it  more  and  more  willing  to  spark  off 
this  charge  as  it  rises  (iv).     Repeat  ad  lib. 

It  is  a  hefty  improvement  on  the  two-apple  experiment  of  Fig. 
300  :  it  corresponds  to  the  nearer  one.  In  one  of  its  many  varied 
forms — a  plate  with  insulating  handle,  a  tray  supported  on  tumblers, 
a  patch  of  tinfoil  on  a  glass  plate,  etc.,  etc. — it  is  a  most  convenient 
'  charged  body  '  for  experiments. 

Work  has  to  be  done  in  pulling  out  the  electric  lines  as  you  see 
in  (iii),  a  light  plate  feels  perceptibly  heavier  to  lift,  the  charged 


676 


ELECTROSTATICS 


[§714 


brown  paper  will  often  lift  with  it  and  have  to  be  torn  off.  As  the 
(slab  +  electrophorus),  or  equally  any  (rod  +  insulated  rubber), 
was  electrically  inert  before  separation,  it  is  evidently  this  Work 
done  in  pulling  apart  the  oppositely  charged  bodies  that  provides 
the  store  of  potential  energy  in  the  electric  field,  which  can  move 
light  stuff,  produce  the  heat,  light,  and  sound  of  electric  sparks,  etc. 
(cf.  §  666). 

§715.  Electrical  Machines.  The  early  machines  for  producing 
electricity  consisted  of  cylinders  or  large  circular  plates  of  glass 
which  were  rotated  against  leather-covered  pads  smeared  with 
tin  amalgam.  The  machine  had  to  be  thoroughly  warmed  and 
dried  before  use,  during  use  the  driver's  exertions  kept  it  and  him 
quite  warm  enough. 

These  '  Friction  Machines  '  have  been  superseded  by  '  Induction 
Machines ' — continuous  acting  improvements  on  the  electro- 
phorus— free    from    this    wasteful    heavy    friction.    Two    very 

different  patterns  of  these  will  be 
described,  the  Kelvin  sand-dropper  as 
an  illustration  of  principle,  and  the 
machine  invented  by  Mr.  Wimshurst  as 
a  successful  machine  in  practice,  stand- 
ing in  something  the  relation  to  the 
electrophorus  that  the  rotary  newspaper 
press  does  to  the  old  hand  platen. 

The  Kelvin  Sand-dropper,  Fig.  309, 
is  an  amusing  contrivance  you  can 
make  at  home.  You  want  two  big  tin 
funnels,  2  yd.  of  stiffish  wire,  two  cans 
such  as  syrup  tins,  and  two  cut-(flint-) 
glass  tumblers  ;  clean  and  dry  (and  don't 
scratch  them  with  sand,  or  this  book 
will  be  unpopular).  Some  seaside  friend 
will  be  glad  enough  to  send  you  half-a- 
stone  of  fine  sand  :  dune  sand  runs  best. 
Flatten  up  the  nozzles  of  the  funnels  to  leave  just  a  wire-hole.  Coil 
the  wire,  as  shown,  to  grip  inside  the  cans,  and  to  encircle  the  funnel- 
ends  without  touching.  Stand  the  cans  on  the  glass  insulators, 
and  support  the  funnels  firmly  above  them,  anyhow  you  can.  The 
cross  wires  must  not  touch  each  other,  earth,  or  anything,  by  a  clear 
half -inch.  Fill  up  with  sand  and  let  run  :  if  the  friction  of  the 
trickling  sand  does  not  suffice  to  start  it,  scrape  an  electrified  rod 
or  glass  on  one  of  the  cans. 

Suppose  Kg  and  its  wire  -f  \  the  end  of  (earthed)  funnel  J^  gets 
electrified  —  by  induction,  and  the  sand-grains  carry  off  bits  of 
this  —  charge  and  deposit  it  in  K^.  The  wire  from  K^  begins  to 
induce  -f  on  the  nozzle  of  Jg,  and  its  escaping  sand  increases  Kg's 
+  charge,  and  so  on,  always  intensifying  each  other,  until  the 
sand  streams  get  so  highly  electrified  that  they  spray  out,  miss  the 


Fig.  309. 


§715] 


FRICTIOyAL  ELECTRICrTY 


5T7 


(repelling)  cans,  from  which  you  can  now  get  tiny  »park«,  and 
splutter  all  over  the  table.     A  spark  has  just  been  taken  from  K,. 

It  is  the  energy  of  fall  of  the  sand,  of  course,  which  is  l)eing  con- 
verted into  electrical  energy. 

The  Wimshurst  machine.  In  this  two  glass  or  ebonite  dines, 
a  foot  or  more  diameter  and  J  in.  apart,  are  rotated  rapidly 
opposite  ways,  by  a  hand-wheel  and  open  and  crossed  driving  cords. 
In  Fig.  310  they  have  been  represented  as  concentric  drums,  in 
which  form  in  fact  they  are  occasionally  made.  Sixteen  or  mor© 
short  strips  of  tinfoil  are  gummed  on  the  outer  sides  of  the  plate* 
(inside  and  outside  of  drums) ;  these  help  spread  the  charge  over  the 


Fig.  310. 

plates.  At  opposite  ends  of  the  horizontal  diameter  are  double 
'combs'  attached  to  insulated  'prime  conductors.  There  are 
also  two  stiff  wires  fixed  across  the  machine  at  45°,  and  carrjmg 
tinsel  brushes  which  just  sweep  the  tinfoil  '  sectors  as  they  pM. 
These  cross  wires  are  the  essential  *  Long  Conductors,  the  axle 
connects  them  and  they  are  not  insulated.  ^  x    ,'^  ♦u^ 

Hold  a  '  starter  '  piece  of  electrified  ebonite  near  X.  but  on  the 
opposite  side  of  the  plates,  and  -f  charges  are  induced  by  it  up  aloi^ 
the  long  conductor  YX  :  they  cannot  get  through  the  glass  to  it 
(although  the  inductive  force  does,  even  better  than  t»»">"»»\«f ; 
§  733),  and  they  trickle  off  the  tinsel  brush  on  to  the  pUte.  and  get 
carried  along  to  positions  1,  2,  3.  «,u««^  it 

At  2  they  induce  -  charge  up  the  long  conductor  Jl.  whence  it 

u 


678  ELECTROSTATICS  [§715 

similarly  trickles  on  to  the  other  plate,  and  gets  carried  back  to  II, 
where  it  now  takes  up  the  task  of  inducing  +  up  the  cross  wire,  so 
you  can  take  away  your  starter  for  good  (if  dry  and  warm  the 
tickling  friction  of  the  brushes  suffices  for  starting). 

The  +  and  —  charges,  carried  on,  give  themselves  up  to  the 
enclosing  sharp-pointed  combs  on  the  prime  conductors.  The 
lower  halves  carry  opposite  charges  the  other  way. 

At  the  top  of  the  figure  is  a  view,  looking  down  on  the  edges  of 
plates,  reduced  to  its  very  simplest :  it  is  all  you  need  learn  to  draw. 

You  see  that  these  processes  go  on  in  a  mutually  intensifying 
fashion,  and  in  a  very  few  seconds  the  machine  is  prepared  to  give 
off  long  sparks  from  either  prime  conductor.  If  there  is  nothing 
near  enough  to  spark  to,  brush  and  glow  leakage,  §  895,  takes  place, 
and  the  whole  machine  fizzes,  shines  in  the  dark,  and  ozonizes 
the  air,  producing  a  strong  characteristic  smell. ^.^^ 


Fig.  311. 

When  active,  the  machine  is  much  harder  to  turn,  work  being 
done  in  pulling  apart  attracting  charges. 

The  Wimshurst  generates  an  exceedingly  small  '  continuous  ' 
or  '  direct '  current  of  '  high  voltage.'  To  increase  the  output  of 
current  it  is  occasionally  built  with  several  pairs  of  plates  :  a  limit 
is  set  to  the  voltage  by  the  breakdown  of  the  air  as  an  insulator. 
To  cope  with  this,  some  were  run  in  compressed  COg,  but  the  straight- 
forward way  is  to  make  all  distances  larger. 

An  old  Wimshurst  with  7 -ft.  plates  has  lived  in  honourable 
retirement  in  the  Science  Museum  ever  since  it  wrecked  two  fine 
batteries  of  ley  den  jars,  but  the  present-day  call  for  direct-current 
at  highest  voltage  has  been  met  by  an  American  modification  : 
instead  of  plates,  rubber  belts  are  used.  Fig.  311.  The  mutual 
induction  takes  place  in  the  middle,  where  they  move  close  together, 
and  the  deposited  charges  are  then  carried  away  and  entirely  given 
up  to  combs  inside  the  large  '  prime  conductor  '  spheres. 

This  makes  it  very  plain  how  the  terrific  kick  of  the  originally 
earth-potential  charge  is  given  to  it,  by  the  work  done  pushing  it 
*  up  the  potential  hill '  against  the  repulsion  of  the  accumulated 
charge  already  in  possession,  §  723.  A  machine  is  being  built 
with  7-ft.  spheres  in  the  effort  to  reach  towards  10  million  volts. 


FRICTIONAL  ELECTRICITY  579 


EXAM   QUESTIONS,   CHAFFER   XLIV 

j        1.  Describe  experiments  to  show  that  equal  and  opponitc  r)iarKe«  of  oJoc- 

tricity  are  always  produced.     What  deductions  do  you  draw  ? 
V        2.  Give  some  account  of  the  distribution   of  eloctricity  on  conductoni. 
Explain  the  action  of  the  lightning  conductor.     [See  §  900.] 

3.  How  and  under  what  conditions  can  on©  conductor  bo  mado  to  give 
up  all  its  charge  to  another?  How  find  which  of  two  conductoni  had  lh« 
greater  charge  ? 

4.  How  would  you  show  that  electricity  accumulatefl  to  a  groat^n*  aurfac<^ 
density  on  edges  and  points  ? 

Describe  two  practical  applications.     (  X  2) 

5.  Describe  any  form  of  sensitive  electroscope. 

How  would  you  use  it  to  show  the  absence  of  (a)  electric  force,  (b)  electric 
charge,  inside  a  closed  conductor?  Discuss  theee  facts  fn*m  the  theoretical 
point  of  view.     (  X  2) 

6.  Describe  a  gold  leaf  electroscope.  How  would  you  charge  it  negati\'oly 
'  by  induction  '  ?  How  would  you  use  it  to  find  the  signs  of  the  chargon  on 
two  spheres,  and  which  was  the  greater?     (  X  2) 

7.  How  would  you  use  an  electroscope  to  explore  the  diatributton  of  (o) 
charge,  (6)  potential  over  an  irregular  conductor? 

8.  Can  the  leaves  of  an  electroscope  be  made  to  diverge  if  they  are  kept 
at  zero  potential  ?  If  so,  describe  how,  and  explain  why.  [Earth  leaves, 
insulate  and  charge  case.] 

J  9.  Describe  the  electrophorus  and  explain  its  action.  How  woultl  you 
charge  an  electroscope  (a)  positively,  (6)  negatively,  by  means  of  on**,  an<l  how 
confirm  the  nature  of  the  charge  ?  What  is  the  source  of  itH  energy  ?  (  >  2) 
10.  Describe  some  form  of  electrostatic  induction  machine,  and  explain 
the  use  of  leyden  jars  in  connection  with  it.  How  and  why  in  the  ii|iark 
altered  by  disconnecting  the  jars  ?     [.See  §  895.] 

What  sort  of  current  is  obtainable  from  those  machines  and  for  what  can 
they  be  used  ?     (  X  5) 


CHAPTER   XLV 
ELECTRIC  FIELD,  POTENTIAL,  AND  CAPACITY 


§  72L  The  forces  acting  between  electrical  charges  at  a  distance 
can  be  investigated  in  a  way  resembling  that  of  §  682,  or  by  a 
torsion-balance,  as  was  done  by  Coulomb  in  1785,  but  the  best 
Proof  of  the  Inverse-Square  Law  remains  that  which  Cavendish, 
in  1772,  based  on  the  absence  of  electric  force  inside  a  hollow  closed 
charged  conductor. 

Suppose,  Fig.  312,  the  conductor  a  sphere  charged  uniformly 
with  e  units  per  sq.  cm.  of  its  surface.  Place  at  any  point  P  inside 
a  small  test  charge.  P  may  be  chosen  as  the  vertex  of  a  pair  of 
slender  cones  ;  the  axis  APB  of  these  meets  the  sphere  at  the  same 

inclination  at  both  ends,  and  hence  the 
areas  the  cones  cut  out  on  the  surface 
are  proportional  to  AP^  and  BP^,  and 
bear  charges  proportional  to  e  .  AP^  and 
e  .  BP2.  These  are  distant  AP  and  BP 
from  P,  and  together  produce  no  resultant 
force  along  APB  on  the  test  charge  at  P. 
This  condition  is  fulfilled  by  the  equa- 
tion, force  e  .  AP2/AP2  =  force  e  .  BP2/BP2, 
or  the  force  is  proportional  to  the  charge, 
and  inversely  as  the  square  of  its  distance. 
Fig.  312.  Since  the  whole   sphere  can  be  filled 

with  similar  pairs  of  cones  with  vertices 
at  P,  and  every  pair  must  fulfil  the  condition  independently,  this  is 
the  only  possible  solution. 

Defining  therefore,  electrostatically,  the  Unit  Charge  or  Quantity 
of  Electricity  as  that  which  repels  equal  charge  1  cm.  .away,  in  air, 
with  a  force  of  1  dyne,  there  is  between  charges  e  and  e' ,  d  cm.  apart 
in  air,  a 

repulsive  Force  =  --^  dynes. 

§  722.  The  strength  of  the  Electric  Field  F,  at  a  point,  or  the  '  Electric 
Intensity  '  at  the  point,  is  defined  as  equal  to  the  force  in  dynes  which 
would  act  on  a  unit  of  positive  charge  placed  at  the  point. 

The  force  on  charge  E  placed  in  field  F  is  therefore  EF  dynes. 

§  723.  If  E  is  pushed  1  cm.  forward  against  F,  EF  ergs  of  work 
must  have  been  expended  on  it,  and  to  push  forward  unit  charge 
distance  s  cm.  against  field  F,  F^  ergs  of  work  are  demanded. 

This  will  be  obtainable  again  by  letting  the  charge  move  back 

680 


§724]  ELECTRIC   FIELD   AND   POTENTIAL  581 

the  s  cm.  under  the  force  F.  It  has  been  stored  as  potential 
energy,  or,  as  we  say,  the  Electrical  Potential  of  the  charge  has  been 
increased,  by  an  amount  F^,  the  strength  a.  length  of  the  field. 

Haying  expended  100  ft.-lb.  of  work  on  a  pound  weight  by 
carrying  it  uphill,  we  have  increased  its  gravitational  potential 
energy  by  100  units,  we  have  carried  this  unit  uright  to  a  place 
of  100  units  higher  'gravitational  potential,'  simply  another 
way  of  saying  100  ft.  vertically  higher.  Measuring  the  work  done 
on  this  1  lb.  is  thus  a  method  of  measuring  difference  of  level. 
It  is  frequently  useful  to  think  of  '  charge  *  aa  electrical  weight 
and  '  difference  of  electrical  potential  '  as  difference  of  electrical 
level  through  which  it  is  lift  eel,  the  work  done  in  the  process  being 
the  product  of  the  two. 

The  work  you  did  lifting  the  elect rophorus  from  its  attracting 
base,  lifting  the  sand  into  the  funnels,  or  turning  the  machine  to 
push  its  charges  on  to  the  already  charged  prime  conductom, 
'  raised  the  charges  up  to  sparkling  potential.' 

§  724.  We  can  do  the  same  amount  of  work  on  a  unit  charge, 
and  therefore  rise  through  the  siime  difference  of  potential,  either 
by  working  against  an  intense  force  for  a  short  distance  or  a 
weaker  force  for  a  longer  distance,  just  as  we  can  reach  the  same 
height  by  scrambling  a  few  yards  up  the  face  of  the  hill  or  by 
walking  a  few  rods  on  the  sloping  back.  We  can  sfieak  therefore, 
with  everyday  meaning,  of  a  steeper  or  easier  *  potential  gradient/ 
and  we  can  draw  equlpotential  surfaces  analogous  to  the  '  contours 
of  equal  altitude  '  on  a  map. 

Contours  are  crowded  together  where  they  run  across  the  steep 
slope  ;  so  are  equipotential  surfaces  close  together  where  they 
cross  parts  of  the  electric  field  of  high  intensity,  their  closeneM 
proportional  to  the  intensity. 

In  the  alternative  more  picturesque  method  of  marking  hills, 
the  '  hill-shading  '  lines  are  packetl  closest  together  when*  they 
run  down  the  steepest  slopes ;  just  in  the  same  way  the  unit  line* 
of  force  are  closest  in  the  strongest  parts  of  the  field,  again  their 
number  per  sq.  cm.  proportional  to  the  intensity. 

Fig.  313  represents  the  equipotential  surfaces  and  the  lines  of 
force  between  a  -f  charged  egg  at  potential  8  and  a  —  charged  ball 
at  potential  —  4.  It  might  equally  represent  the  contour  lines  and 
hill  shading  of  a  flat-topped  hill  800  ft.  high  and  a  flat- hot tome<i 
pit  400  ft.  deep,  as  in  the  sectional  elevation  beneath,  which  gives 
a  side  view  of  the  whole  of  the  upper  figure.  The  contoum  or 
equipotentials  are  marked  with  their  ±  heights  above  the  tero 
level.  . 

Difference  of  potential  betwwn  two  places  is,  of  course,  evidence 
that  electric  force  would  be  acting  on  a  charge  place<l  U^twcrn 
them,  and  if  there  is  a  conducting  imth,  the  elwtrinty  will  Iw 
driven  along  it  from  the  place  of  higher  to  that  of  lower  potential. 
Hence  Potential  Difference,  P.D.,  is  commonly,  in  dealing  with 


582 


ELECTROSTATICS 


[§724 


electric  currents,  referred  to  as  electromotive  force,  E.M.F.    The 
unit  employed  there,  however — the  Volt — is  much  smaller  : 

1  Electrostatic  Unit  of  Potential  =  300  Volts. 

It  follows  that  if  electricity  is  at  rest  on  a  conductor,  the  whole 
conductor  is  at  one  potential,  at  the  same  electrical  level  through- 
out, the  surface  of  the  conductor  is  an  equipotential  surface. 

And  since  there  is  no  force  inside  a  charged  hollow  conductor 
due  to  any  charge  upon  or  outside  it,  i.e.  no  work  would  be  done 
in  moving  a  test  charge  about  anywhere  inside  it,  therefore  the 
inside  of  the  conductor  is  throughout  at  the  same  potential  as 
its  surface. 

Hence  the  flat  top  and  bottom  in  the  diagram  :   a  high  and  a  low 


Fig.  313. 


lake,  joined  by  many  a  hill-side  '  force  ' — stream  lines  of  water,  or 
of  electric  force. 

The  equipotential  surfaces  cut  the  lines  of  force  at  right  angles  ; 
for  if  not,  the  force  would  have  a  component  parallel  to  the  surface, 
which  would  cause  a  potential  difference  as  one  moved  along  the 
surface,  and  that  is  contradictory.  Similarly,  canals  dug  along 
contour  lines  would  be  full  of  stagnant  water,  and  would  cross  all 
the  hillside  streamlets  at  right  angles. 

Parallel  lines  of  force  indicate  a  field  of  uniform  strength,  and 
hence  are  cut  at  right  angles  by  equipotential  surfaces  spaced  at 
equal  distances. 

§  725.  Potential  due  to  a  charge  on  a  very  small  conductor.    Let 

the  conductor  have  charge  -\-  e.  At  r  from  it  this  produces  a 
field  strength  {i.e.  a  repulsive  force  on  a  unit  test  charge)  =  e/r^. 


§728]  ELECTRIC   FIELD   AND   POTENTIAL  593 

Push  the  test  charge  nearer  by  the  very  small  distance  d   the 
work  done  against  the  electrical  repulsion  =  d  x  e/r*   and  it  han 

arrived  at  distance  r  —  d  from  e. 

-ST  ^  c  ed 

^°^'  fTT^  -  r^  y.a  _  rd'  *"^  *'  ^  *®  ^^^  **"*"  <*"^  >^  <^*n  ^  »« 

small  as  ever  we  like  to  make  it)  rd  can  be  neglected  comparwi 
with  r2,  and  the  expression  becomes  our  ed  r*.  Hence  the  work 
done,  which  is  the  increase  in  potential,  has  been  expressed  as  the 
difference  of  two  similar  quantities,  each  being  (charge  -^  distance 
from  it).     Hence  : 

The  potential  at  a  point"  due  to  a  small  charged  conductor  in  the 

neighbourhood  {in  air)  =  charge  -^  distance  from  it,  e/r. 

Making  r  infinitely  great,  e/r  =  0,  and  the  actual  potential  is 
theoretically  the  work  done  in  bringing  unit  -f  charge  from  an 
infinite  distance  up  to  the  point.  Practically,  the  Potential  of  the 
Earth  is  arbitrarily  chosen  as  Zero,  and  potentials  are  reckoned 
above  or  below  it,  just  as  heights  and  depths  are  reckoned  from 
sea-level. 

§  726.  If  there  are  several  charged  bodies,  the  potential  at  the 
point  is  ±  ejri  ±  ejr^  ±  ejr^,  etc.,  -  l)eing  used  for  a  neffative 
charge.  This  is  understandable  enough  ;  a  house  30  ft.  high  is 
perched  on  the  side  of  a  small  hill  100  ft.  above  the  stream  at  its 
foot,  but  the  whole  district  is  on  the  long  slope  of  a  distant  ridge 
and  the  foot  of  the  little  hill  is  1000  ft.  above  sea-level.  V'erj* 
naturally  one  thinks  of  the  altitude  of  the  house-top  in  three  in- 
dependent steps,  30+100+  1000  or  1130  *  units  of  iwtentiaP 
(measured  by  carrying  1  lb.  up  to  it). 

§  727.  Potential  of  sphere,  radius  r,  due  to  charge  e  on  itself.    By 

symmetry  the  charge  will  spread  itself  uniformly  over  the  sphere, 
and  will  have  the  same  effect  at  external  points  as  if  concent  rate<l 
at  its  centre  (much  as  the  mass  of  the  earth  attracts  gravitationally 
as  if  situated  at  its  centre  of  mass).  For  if  it  were  eccentric,  the 
potential  of  the  nearer  side  would  be  higher,  and  electricity  would 
be  driven  round  to  equalize  it.  Every  point  on  the  sphere's  surface 
is  distant  r  from  its  centre,  and  hence  it^  potential,  in  air,  is  f  r. 

§  728.  Evidently  the  crowding  of  a  large  charge  on  to  an  isoUte<l 
conducting  sphere  would  raise  it  to  a  high  |X)tential,  but  it  by  no 
means  follows  that  every  heavily  charged  surface  is  at  a  ^igh 
potential,  for  there  may  be  large  negative  charges  on  neighbouring 
conductors,  lowering  the  potential  all  around  them,  just  like  lumps 
of  ice  cooling  their  whole  neighbourhood.  In  Fig.  308  there  is  a 
large  +  charge  on  the  plate,  yet,  on  trial,  the  whole  system  appears 
electrically  dead,  devoid  of  all  |)otential  energy,  at  zero  |M)t€*titial ; 
it  is  the  proximity  of  the  —  on  the  ebonite  slab  that  keeps  it  so. 


584  ELECTROSTATICS  [§  728 

There  can  be  large  quantities  of  electricity  at  low  potential,  and 
small  charges,  or  uncharged  conductors,  at  high  potential ;  just  as 
there  are  large  populations  in  the  plains,  and  few  or  no  inhabitants 
of  the  hill-tops.  Or  there  can  be  very  different  surface  densities 
of  electricity  at  different  places  on  one  conductor,  which  is  of  course 
at  the  same  potential  throughout.  If  a  wire,  held  by  a  sealing-wax 
handle,  be  brought  from  an  electroscope  and  touched  on  the  egg- 
shaped  conductor  of  Fig.  313,  the  leaves  will  open  to  the  same 
extent  wherever  it  touches,  for  the  conductor  (egg  +  wire  + 
electroscope)  is  throughout  at  one  potential.  But  if  a  proof  plane 
be  touched  on  the  pointed  end  and  then  carried  away  to  another 
electroscope  there  would  result  a  wider  opening  of  its  leaves 
than  if  touched  on  the  round  end.  The  closer  packing  of  lines 
shows  that  there  is  more  charge  per  square  centimetre — a  greater 
surface  density  of  electrification — on  the  little  end,  and  this  spreads 
to  the  proof  plane.  Now,  when  the  latter  gets  away  and  is  free 
from  the  potential-equalizing  influence  of  the  conducting  surface, 
it  will  have  a  higher  potential. 

Note The  lift  of  the  gold-leaf  in  an  electroscope  is  of  course 

a  measure  of  the  work  done  on  it,  and  now  stored  in  it  as  gravita- 
tional potential  energy,  i.e.  of  the  difference  of  Potential  between 
it  and  the  case. 

§  729.  Electrical,  or  electrostatic,  Capacity.  Only  very  small 
quantities  of  electricity  can  be  stored  on  isolated  conductors  of 
ordinary  size,  for  leakage  through  the  air  inevitably  begins  if  charged 
to  more  than  about  200  electrostatic  units  of  potential  (=  60,000 
volts). 

The  charge  or  quantity  of  electricity  that  raises  the  potential  of 
a  conductor  by  1  unit  is  the  measure  of  the  Electrical  Capacity  of 
the  conductor,  or  its  Capacitance. 

This  is  comparable  with  measuring  the  capacity  of  a  tank  by 
finding  how  much  water  would  fill  it  a  foot  deep.  Then  suppose 
that  all  tanks  begin  to  leak  under  the  pressure  of  200  ft.  height  of 
water. 

Charge  e  given  to  an  isolated  sphere  of  radius  r,  in  air,  raises  it 
to  potential  e/r,  §  727.  For  this  to  be  equal  to  1,  e  =  r,  and  now 
e  =  its  capacity. 

Hence  the  capacity  of  an  isolated  sphere  in  air  is  equal  to  its  radius 
in  centimetres.  Not  proportional  to  its  surface,  in  spite  of  the 
electricity  being  spread  there. 

Then  Total  Charge  =  Capacity  x  total  rise  in  potential. 

Thus  a  football  9  cm.  radius  could  at  most  hold  only  about 
9  X  200  =  1800  units  of  charge. 

The  capacity  of  an  isolated  disc  in  air  =  diameter  -^  tt. 

These  electrostatic  Capacity  Units  are  much  smaller  than  those 
with  which  you  are  acquainted  in  Radio  ;  it  takes  900,000  to  make 
one  microfarad,  and  3000  million  electrostatic  units  of  charge  =  1 
coulomb. 


§  731]  ELECTRIC   FIELD   AND   POTENTLAL  S85 

§730.  Take  advantage  of  §728,  keep  the  potential  of  the 
charge  down,  and  so  obviate  its  leaking  off,  by  providing  another 
charge  of  opposite  sign  close  to  it.  It  will  now  be  possible  to 
crowd  on  much  more  electricity,  and  arrangements  of  this  nature 
are  called  *  Condensers.' 

In  the  Concentric  Sphere  Condenser,  made  by  Faraday,  a  hollow 
sphere,  radius  b,  encloses  the  ball,  radius  a,  which  is  given  its  charge  e 
by  way  of  an  insulated  wire  passing  through  a  small  hole  in  the 
outer  shell.  By  §  727  this  causes  a  potential  e/a  all  over  the  ball's 
surface,  and  e/6  all  over  the  inside  of  the  shell.  Connecting  the 
shell  to  earth  lowers  its  potential  to  0,  and  that  of  the  inner  sphere 
to  e/a  —  e/6,  the  fixed  difference  between  them.  Putting  this 
potential  difference  equal  to  1,  c  becomes  equal  to  the  capacity 
of  the  inner  sphere,  called  the  capacity  C  of  the  whole  condenser. 

£_«       1     .•e  =  C=    "* 


a       b  ' '  b  —  a 

which  can  be  very  much  larger,  e.g.,  if  the  9  cm.  radius  football  were 
enclosed  in  a  10-cm.  earthed  shell,  its  capacity  would  be  increase<! 
tenfold. 

§  73L  A  Condenser  much  easier  to  construct  consists  of  a  pair 
of  large  fiat  Parallel  Plates  of  metal — sheets  of  tinfoil,  for  instance, 
gummed  on  the  inner  faces  of  two  pieces  of  plate  glass,  spacec!  apart 
by  the  thickness  of  distance-pieces  of  thin  glass  rod.  Giving  one 
plate  a  +  charge  by  an  electrophorus  or  machine,  and  keeping  the 
other  earthed  by  touching  it,  or  by  a  wire  to  the  nearest  gaspipe, 
practically  all  the  electric  lines  run  straight  across  from  one  plate 
to  the  other,  that  course  being  so  much  the  shortest .  Thus  one  plate 
catches  all  the  lines  from  the  other,  i.e.  the  charges  on  the  plates 
of  a  condenser  are  equal  and  opposite,  hence  only  one  is  considered 
in  stating  the  Capacity. 

To  calculate  this,  in  the  foregoing  formula  put  b  —  a  —  the 
small  distance  apart,  d.    Then  : 


I 


C  = 


ab  _  4nab  _  average  surface  area  of  spheree 
d  ~    4:Tzd  ~~  47rd 


from  which  expression  the  radii  have  disiippeared,  leaving  no 
obligation  to  keep  to  the  spherical  form,  only  d  is  to  be  small  com- 
pared with  the  other  dimensions  :  therefore  the  Capacity  of  an  air 
condenser  with  parallel  plates  each  of  S  sq.  cm.  area  and  d  cm.  apart  = 

S 
4nd 

The  electrophorus  and  its  slab  formed,  of  course,  just  such  a 
condenser ;  as  you  lifted  it  you  increased  d  and  rcthiced  its  capacity, 
therefore  the  'potential  rose  because  the  total  charge  remaine<i 
unchanged  and  equal  to  capacity  X  potential. 


586 


ELECTROSTATICS 


[§731 


You  are  still  puzzled  as  to  how  it  comes  about  that  a  Condenser 
can  hold  so  much  more  ?  Recollect  that  its  capacity  is  the  charge 
that  fills  it  to  unit  P.D.  ;  and  recollect  that  P.D.  =  Fs,  the  (strength 
X  length)  of  electrostatic  lines  of  field  ;  i.e.Fs  must  ==  1. 

If  s  is  large,  F  is  only  fractional ;  if  5  =  small  d,  F  must  be  large. 
But  F  the  field-strength  is  proportional  to  the  number  of  lines  of 
force  springing  through  unit  area  from  the  plate,  §  724,  and  each 
springs  from  a  unit  of  charge,  §  707.  So  that  when  you  bring  the 
plates  close  together,  you  have  to  crowd  on  a  large  charge  in  order 
to  maintain  unit  P.D.,  and  there  is  your  high-capacity  Condenser. 

§  732.  The  earliest  attempt  to  collect  electricity  from  a  machine 
(a  large  ball  of  sulphur  rotated  in  a  lathe  against  a  man's  hands) 
was  the  very  natural  one  of  holding  a  glass  of  water  so  that  a  chain 
hanging  from  the  '  prime  conductor,'  an  iron  bar  slung  overhead 
by  silken  cords,  dipped  into  it,  with  the  idea  that  the  electric 
fluid  might  run  down  the  chain  and  dissolve  in  the  water.  The 
attempt  succeeded,  for  on  going  to  lift  out  the  chain  with  his  other 
hand,  Cunseus  of  Leyden  got  a  shock  that  scared 
him  horribly,  '  not  for  the  crown  of  Holland  would 
he  suffer  it  again.'  But  visitors  of  course  flocked  in, 
and  electric  shocks  became  all  the  rage. 

See  now  the  resemblance  between  this  arrangement 
and  the  condensers  we  have  been  describing.  The 
hand  grasping  the  glass  is  an  earthed  conductor, 
which  closely  surrounds  the  charged  water  inside, 
being  insulated  therefrom  by  the  glass,  and  into  it  a 
large  opposing  charge  is  induced  up  from  the  earth. 
Touching  the  chain  of  course  connected  the  opposite 
charges,  and  they  flowed  together  through  the  arms 
and  chest.  It  was  soon  discovered  that  a  tinfoil 
coating  pasted  on  outside  and  inside  the  glass  did 
better  than  the  hand  and  the  water,  and  the  form 
of  electrical  condenser  called  the  Leyden  Jar  was 
evolved.  It  is  still  a  common  and  convenient 
pattern  for  high-potential  purposes.  Fig.  314.  An  open-mouthed 
jar  of  glass,  preferably  '  flint,'  has  tinfoil  pasted  on  inside  and  out, 
to  about  two -thirds  way  up  ;  the  glass  margin  is  cleaned  and 
varnished  with  shellac,  and  well  baked,  for  a  shellac  surface  retains 
its  insulating  power  better  than  a  glass  one.  From  a  thick  wooden 
disc  lying  in  the  bottom  of  the  jar  rises  a  brass  stem  and  knob  ; 
disc  and  lower  end  of  stem  are  wrapped  in  tinfoil  to  secure  good 
conducting  connection. 

Again,  '  Franklin's  Pane  '  is  a  sheet  of  glass  with  tinfoil  pasted 
on  both  sides,  leaving  a  wide  margin  all  round. 

§733.  In  all  these  practical  forms  of  Condenser  there  is  glass 
between  the  opposing  conductors  instead  of  air.  Does  this  make 
any  difference  ? 


Fig.  314. 


§  733]  ELECTRIC   FIELD   AND   POTENTIAL  687 

Faraday  filled  in  the  air  space  of  one  of  his  spherical  condensera 
with  shellac,  and  found  that  the  capacity  wa«  much  increa«e<l. 
It  took  3  times  as  many  sparks  from  the  elect rophorus  l)cfoit» 
refusing  more  charge  ;  when  it  was  made  to  share  its  charge  with 
a  similar  air  condenser  it  lost  only  a  quarter  instead  of  half.  &h  judge<l 
by  the  spreading  it  could  still  produce  in  an  electroncope.  And 
experiments  with  plate  condensers  show  that  glass  is  more  effective 
still,  the  capacity  with  glass  between  the  plates  is  6  or  7  times 
as  much  as  with  air. 

The  insulating  material  gets  the  name  of  the  Dielectric,  as  the 
inductive  action  takes  place  through  (rfiVi)  it,  and  the  ratio  of  the 
capacity  of  a  condenser  made  with  the  dielectric  to  that  of  an  equal- 
sized  one  with  air  only,  is  called  the  Specific  Inductive  Capacity  of 
the  Dielectric  {S.I.C.,  or  in  formulce,  k),  or  its  Dielectric  ConMant. 

The  Capacity  of  a  Parallel  Plate  Condenser,  area  of  plate  8, 

dielectric  of  s.i.c.  =  k  and  thickness  d,  is  therefore  SX*y4itd. 

Even  an  isolated  sphere  is  of  course  a  condenser,  for  the  lines 
from  it  end  on  walls,  etc.,  somewhere.  Hence  the  CajHicity  of  a 
Sphere  immersed  in  a  large  block  or  tank  of  dieleciric  is  k  times 
its  radius. 

Since  Charge  =  Capacity  x  Potential,  you  see  that  if  an  electri- 
fied ball,  to  which  is  connected  an  electroscope,  be  plunged  under 
oil,  the  electroscope  leaf  falls  to  a  smaller  deflection. 

And  the  same  would  be  observed  if  you  inserted  a  stout  slab  of 
wax  or  ebonite  between  the  two  plates  forming  a  charge<i  condenser 
{e.g.  two  upright  zinc  plates  stuck  on  paraffin-wax  feet)  connected 
to  the  electroscope. 

You  see  now  why  it  was  necessary  to  specify  *  in  air  *  in  defining 
Unit  Charge. 

The  capacity  of  a  pair  of  long  concentric  cylinders  such  as  a 
submarine  cable,  radii  a  and  6,  separated  by  thickness  6  —  a  of 
dielectric  k,  is  k  X  cm.  length  -^  4-6  (log  6  -  log  a). 

Your  100-ft.  '  aerial '  forms  an  air  condenser  with  the  ground 
20  ft.  beneath,  of  capacitance  roughly  160. 

Some  Specific  Inductive  Capacities  are,  approximately  : 

Paraffin  wax,  heavy  mineral  oil,  india-nibber  .  2-0  2-2 

Rosin,  vulcanized  rubber,  ebonite,  carbon  disulphitio  -J 

Shellac,  gutta-percha  (submarine  cables),  paper  .  * 

Sulphur,  mica  ....••  J^^* 

Glass ^ 

Alcohol Ji 

Water ~ 

In  addition  to  the  cain  of  capacity  by  using  glass  between  the 
plates,  there  is  the  advantage  that  much  greater  potential  differ- 
ence  may  be  applied  without  spark  di.st^hargo  en.HUing.  An  eighth- 
inch  air  gap  will  stand  onlv  about  40  units  of  P.D.,  even  if  it  can  be 
kept  free  of  threads  of  dust,  a  l/S-in.  glmw  plate  shouhl  caMily  with- 
stand  400  :  beyond  this  there  is  a  risk  of  the  glass  punctunng.  as  il 


688  ELECTROSTATICS  [§  733 

by  pressure  of  a  sharp  punch.  There  is  also  the  mechanical  advan- 
tage that  the  attraction  between  the  oppositely  charged  plates 
cannot  possibly  pull  them  into  contact,  §  735. 

§  734.  Energy  of  charged  condenser.  It  has  been  pointed  out  in 
§  723  that  the  work  done  in  carrying  a  charge  from  a  place  of  low 
to  one  of  high  potential  =  charge  X  difference  of  potential,  and 
that  this  work  is  stored  as  potential  energy  in  the  electric  field. 
The  energy  stored  by  a  conductor  which  has  been  raised  from  zero 
potential  to  V  by  giving  it  a  charge  of  E  units  is  not,  however,  the 
full  product  EV.  For  at  first  the  conductor  was  uncharged,  and 
the  first  small  fraction  of  the  charge  could  be  brought  up  to  it  on  a 
little  electrophorus  without  any  repulsion  having  to  be  overcome, 
i.e.  without  doing  any  work  ;  just  as  the  first  brick  of  a  wall  could  be 
pushed  along  the  ground  into  position  without  lifting  it,  and  possesses 
no  available  energy  because  it  cannot  fall.  The  next  1/r^th  fraction 
of  the  charge  has  to  be  brought  up  against  the  repulsion  of  the 
fraction  already  in  possession,  this  having  raised  the  potential 
of  the  conductor  to  I /nth  its  final  value.  The  third  l/nth  has  to 
be  lifted  to  a  place  of  2?iths  the  final  potential,  and  so  on,  just  as 
successive  bricks  have  to  be  lifted  higher  and  higher.  And  precisely 
as  the  total  gravitational  energy  stored  in  the  wall  (and  set  free  if 
it  falls)  is  found  by  considering  the  height  to  which  the  centre  of 
mass  has  been  raised,  and  is  half  the  product  of  its  mass  and  full 
height,  so  the  Electrical  Energy  of  a  body  which  has  been  raised 
from  potential  0  to  V  by  giving  it  charge  E  is  JEV. 

Thus  the  energy  of  the  football  of  §  729  is  i  X  1800  X  200  =: 
180,000  ergs  =  0-013  ft. -lb.,  just  enough  to  produce  a  slender 
thread  of  light,  a  little  heat,  a  tiny  crack,  and  a  brief  tingling  in 
the  knuckle  brought  up  to  receive  the  spark  discharging  the  ball. 

Ridiculously  small  as  this  sounds,  if  you  kicked  it  on  to  a  66,000- 
volt  wire  of  '  the  Grid,'  where  it  might  hang  and  be  charged  100  times 
per  second,  1-3  ft. -lb.  becomes  appreciable  ;  while,  if  it  were  a 
broadcasting  aerial  at  a  million  frequency  ? 

Thus  it  is  that  condensers  were  formerly  of  very  limited  utility  ; 
but  now,  for  telephony,  and  a  fortiori  for  radio-frequency  apparatus, 
are  manufactured  in  millions. 

§  735.  Force  between  plates  of  charged  condenser.  Electrometers. 
Take  the  Electrophorus,  apparently  inert  and  dead,  and  lift  the 
plate  a  little  distance  s,  drawing  out  a  field  of  strength  F  between 
plate  and  slab.  In  this  alternative  fashion  you  have  produced 
a  charged  condenser,  of  energy  JEV  =  JE  .  Fs  "(§  723)  =  |EF  .  s  = 
mechanical  force  X  distance  moved  against  it. 

Thus  J  charge  X  field  strength  is  the  Force  in  dynes  between  the 
plates. 

On  this  depend  Electroscopes — or  rather.  Electrometers,  because 
they  are  used  for  measuring — in  ever -increasing  variety,  from  large 


§  736]  ELECTRIC   FIELD   AND   POTENTIAL  km 

discs  hanging  too  far  apart  for  a  spark  to  jump,  clown  to  minute 
and  highly  sensitive  instruments  which  have  to  Iw  read  by  micro. 
scopes.  Fortunately,  modern  developments  of  our  old  Ck>ld-leaf 
friend  make  him  as  good  as  most,  so  let  us  stick  to  him. 

As  Electrometer,  he  often  has  a  single  narrow  aluminium  leaf 
and  an  attracting  plate,  Fig.  315  A  (as  to  the  usual  repulsion  see 
Fig.  303),  and  the  swing  of  the  leaf  is  watched  through  tele«^pe 
or  micrometer-microscope. 

Usually,  the  field  exists  simply  on  account  of  the  presence  of  the 
charge,  from  which  its  lines  arise ;  so  that  field  and  charge  are 
proportional  to  each  other,  and  Force  iEF 
is  proportional  to  the  square  of  either. 
The  movements  of  electrometers  used  in 
this  way  are  therefore  proportional  to  the 
square  of  the  charge,  or  to  the  square  of 
the  P.D. ;  they  move  very  little  indeed  at 
first,  a  common  gold-leaf  is  useless  below  ... 

1  electrostatic  unit  P.D.,  300  volts,  but  Fio.  315. 

afterwards  they  widen  out  rapidly. 

If,  however,  one  plate  be  kept  charged  by  an  ordinar>'  H.T. 
batter}',  so  that  F  has  a  high  value  from  the  start,  then  any  little 
alterations  in  E  (far  too  small  to  affect  F  appreciably)  cause* move- 
ments  proportional  to  themselves  (of.  §  819,  headphone).  Thi« 
is  done  also  in  the  '  tilted  '  form  of  the  instrument,  Fig.  315  B. 
which  can  measure  very  small  fractions  of  a  volt,  or  respond  to  the 
discharge  of  single  radioactive  particles.  In  its  latest  form,  an 
almost  invisible  conducting  fibre  hangs  in  vacuo. 

§  736.  In  the  condenser  we  manufactured  in  the  last  paragraph. 
by  lifting  the  electrophorus  plate,  the  energy  \K\  =  ^Kh  .s  i» 
proportional  to  the  little  distance  8  between  the  plates ;  but 
8  X  area  of  plate  =  volume  of  dielectric  (air)  now  traverscil  by  the 
electric  lines.  This  proportionality  rather  suggests  that  the  Energy 
is  really  contained  in  the  Dielectric,  which  is  strained  by  the 
electrical  stress,  and  should  therefore  contain  energ}',  §  148. 

That  this  strain  is  very  real  is  shown  by  the  puncturing  of  the 
glass  of  an  over-charged  Leyden  jar,  or  by  the  experiment  of  dis- 
secting a  charged  jar.  A  ginger-beer  glass  is  fitted  with  removable 
inner  and  outer  coatings  of  tin,  it  is  charged,  the  inside  tin  is  hoisted 
out  by  a  loop  of  silk,  the  outside  is  pulled  off  by  hand,  and  the  two 
are  laid  together  on  the  table.  Yet  when  the  inner  casing  is  dropped 
back,  and  the  outer  shell  put  on  again,  the  jar  will  give  the  usual 
strong  spark  to  the  discharging  tongs,  or  to  the  experimenter's 
knuckle,  whichever  he  prefers. 

Evidently  the  function  of  the  conductor  has  been  merely  to 
distribute  the  charge  over,  or  to  collect  it  quickly  from,  the 
insulating  surface  of  the  Dielectric.  The  tinfoils  on  a  Wimshumt 
act  mostly  in  the  same  way ;  with  good  brushes  the  machine 
can  work  without  any  sectors  at  all. 


590  ELECTROSTATICS  [§  737 

§  737.  Let  us  see  what  this  Dielectric  Strain  consists  in,  and  at 
the  same  time  solve  the  mystery  of  why  different  dielectrics  affect 
the  capacity  of  a  condenser  ;  and  go  on  and  tell  Gilbert  how  his 
electrics  and  non-electrics  differed,  and  why  damp  mattered,  and 
what  was  wrong  with  flame. 

An  atom  is  a  sort  of  solar  system  in  miniature  ;  it  consists  of  a 
small  positive  nucleus  round  about  which  are  flying  numerous 
negative  electrons,  of  total  charge  equal  to  that  of  the  nucleus, 
and  kept  from  straying  off  by  its  attraction.  In  a  solid  the  atoms 
cannot  wander  about,  and  in  a  dielectric  the  electrons  cannot  escape 
from  the  atom,  but  when  put  in  an  electric  field,  as  between  charged 
plates,  their  orbits  are  distorted,  and  bulge  over  towards  the  +  plate  ; 
so  that  all  that  face  of  the  dielectric  becomes  predominantly  negative, 
while  on  the  other  face  the  positive  nuclei  are  now  not  so  com- 
pletely shrouded  by  their  electrons,  and  that  becomes  positive. 

That  this  is  really  so  is  shown  by  certain  wax-resin  mixtures, 
which,  poured  melted  in  between  charged  plates,  allowed  to  solidify, 
and  then  split  loose,  form  slabs  of  dielectric  which  remain  per- 
manently electrified. 

Virtually,  therefore,  the  +  charge  on  a  tinfoil  is  partly  masked 
by  the  —  distribution  of  electrons  which  it  calls  up  towards  the 
dielectric  surface.  Even  though  it  spreads  over  the  surface,  it 
cannot  get  at  them,  it  cannot  break  them  loose  from  their  atoms, 
but  the  two  taken  together  present  to  the  opposite  plate  of  the 
condenser  a  less  charge  than  in  air.  Consequently  you  iiow  have 
to  pile  more  charge  on  the  plate  to  restore  the  status  quo  ante  ; 
and  the  more  readily  a  dielectric  strains,  in  response  to  the 
electrical  stress,  the  more  additional  charge  is  called  for  ;  and  that 
is  the  dielectric  of  high  '  constant '  or  S.I.C.  Meanwhile,  inside 
the  dielectric,  these  surface  charges  partially  neutralize  and  reduce 
the  electrical  force  (a  conductor,  where  it  would  be  zero,  might  be 
described  as  of  infinite  S.I.C). 

It  was  the  concept  of  this  quasi-elastic  '  electric  displacement ' 
which  led  Maxwell,  about  1874,  to  forecast  the  possibility  of  electric 
waves,  which  might  conceivably  become  of  use  in  telegraphy ; 
and  now  the  space  all  round  your  head — and  inside  it  too — is  per- 
petually being  stuffed  with  them. 

Per  contra,  in  a  metal,  Gilbert's  typical  non-electric,  some  electrons 
can  drift  from  atom  to  atom,  and  the  stream  of  them  is  the  electric 
current. 

In  Moisture,  and  also  in  Flame,  and  the  gases  which  have  only  lately 
left  it,  we  shall  see  later  that  there  are  abundant  atoms  with  either 
an  electron  too  many  or  an  electron  too  few,  and  these  —  and  -{- 
charged  '  ions  '  move  along,  and  sooner  or  later  neutralize  the  charges 
the  experimenter  has  been  producing.  A  candle  flame  tears  to 
pieces  in  a  strong  field. 

§  738.  Coupling  condensers  *  in  parallel.'  Any  number  of  jars, 
etc.,  are  coupled  in  parallel  by  joining,  by  wires  or  strips  of  tinfoil. 


§  739]  ELECTRIC   FIELD   AND   POTENTIAL  591 

all  the  right-hand  (or  inner)  plates  together  in  one  bunch,  and  all 
the  left-hand  (or  outer)  plates  together  in  another.  The  total 
capacity  of  this  '  leyden  jar  battery  '  is  just  the  sum  of  the  individual 
capacities  added  together,  and  the  P.D.  to  which  it  can  be  charged 
is  that  at  which  the  weakest  dielectric  breaks  down. 

In  the  familiar  Variable  Air  Condenser  of  a  Wireless  set  all  the 
moving  plates  are  in  parallel,  and  the  total  capacity  is  the  aggregate 
area  of  them  which  happens  to  be  in  between  the  fixeo^ plates 
divided  by  4it  times  the  average  air-gap.  Measure  both  eiden  of 
every  moving  plate,  but  no  fixed  plates  at  all. 

Compact  condensers  of  very  large  capacity,  for  P.D.*8  of  only 
a  few  hundred  volts,  are  made  of  alternate  tinfoils  and  larger  leave« 
of  thin  dielectric  ;  the  odd  foils  all  project  at  the  left-hand  end  and 
are  secured  together  metallically,  the  even  foils  are  similarly  groupe<l 
at  the  right. 

The  best  dielectric  is  ruby-red  mica,  which  splita  very  well  to 
1/1400  in.  thick,  will  then  withstand  2500  volts,  and  is  of  great 
permanence  in  every  way,  but  very  expensive. 

Much  cheaper  condensers  for  all  ordinary  work  are  made  of  0001- 
in.  cigarette  paper,  painted  with  precipitated  tin,  dried  and  plani.she<l 
on  hot  rolls  ;  rolled  tightly  two  together  with  intervening  plain 
papers,  baked  in  vacuo  and  impregnated  with  hard  paraffin  wax 
under  pressure  :  a  2 -microfarad  telephone  condenser  weighs  about 
J  lb.  complete. 

§739.  Coupling   «in   series'    or   Mn   cascade.*     The   left-hand 

plate  of  the  first  condenser  is  connected  to  the  machine.  Itn 
right-hand  plate  is  connected  to  the  left-hand  plate  of  the  secoml 
jar,  and  so  on,  as  in  Fig.  316  (i).     You  will  see  that  this  might 

m  in^  m  ili 

Fio.  316. 

almost  as  well  be  (ii),  and  now  the  intermediate  plates  are  doing 
nothing  and  might  be  left  out,  as  in  (iii).  So  that  assuming  the 
n  condensers  all  equal  in  size,  this  arrangement  nroduces  merely 
one  of  the  same  area  but  with  dielectric  n  times  as  thick.  The  joint 
capacity  Skj'^Ttind)  is  only  1/nth  of  one  of  them,  but  the  com- 
bination  is  n  times  stronger  to  resist  excessive  chargmg  preMurwi. 
In  mica  condensers  for  high  voltages  triple  sheets  of  thm  mica  are 

In  practice  leyden  jars  can  be  connected  up  as  in  (iv).  each 
well  insulated  on  a  glass  plate ;  or  as  in  (v),  the  common  way  of 
connecting  pairs  on  to  Wimshurst  machines  (don  t  put  your 
knuckle  to  a  machine  with  jars  attached). 


692  ELECTROSTATICS  [§  739 

If  unequal  capacities  are  cascaded,  suppose  C/C  =  3/2.  Then 
whatever  +  charge  is  induced  up  into  C,  leaves  an  equal  —  charge 
on  C,  and  it  =  CV  =  C'V,  /.  V/V  =  C'/C  =  2/3,  or  V/total 
(V  +  V)  available  =  2/5.  That  is,  the  larger  condenser  gets 
charged  only  to  P.D.  2  instead  of  the  whole  5,  so  the  available 
capacity  is  less  than  half.  An  algebraic  formula  can  be  concocted 
for  any  number,  but  is  not  worth  while. 


EXAM   QUESTIONS,    CHAPTER   XLV 

Wireless  and  long-distance  power  lines  have  brought  this  chapter  into 
prominence  again.  Potential  has  been  discussed  at  length,  as  an  under- 
standing of  it  is  very  necessary.  Some  of  the  calculations  below  may  not  be 
found  easy,  nor  worth  everybody's  while. 

1.  Define  *  electric  field  or  intensity,'  '  potential.' 

An  isolated  sphere  of  radius  5  cm.  is  given  a  positive  charge  of  20  electro- 
static units.  Plot  a  curve  to  show  the  variation  of  potential  along  a  line  10 
cm.  long  drawn  from  the  centre. 

2.  Two  pith  balls,  each  of  0-1  gm.  and  0-5  cm.  radius,  hang  from  the  same 
point  on  60-cm.  silk  threads.  They  stand  5  cm.  apart;  find  their  potential. 
(X2) 

3.  What  is  '  electrical  potential ',  ?  A  charge  of  10  units  is  distant  20  cm. 
from  one  of  —  30  units ;  what  is  the  potential  half-way  between  ?  What 
is  the  electric  force  between  the  charges  ? 

4.  If  a  sphere  of  radius  10  cm.  and  charged  with  30  units  is  joined  to  another 
of  radius  15  cm.,  originally  charged  with  5  units  of  like  electricity,  in  which 
direction  will  electricity  flow,  how  much,  and  what  are  the  potentials  before 
and  after  contact  ? 

5.  Define  potential  and  capacity.  A  1-cm.  and  a  20-cm.  radius  sphere 
are  given  equal  charges ;  compare  their  potentials.  Find  their  charges  when 
the  small  sphere  is  removed  after  touching  the  larger  (a)  inside,  (6)  outside. 

6.  Two  spheres  of  radii  3  and  6  cm.  are  charged  with  9  and  36  e.s.  units, 
and  joined  by  a  wire.  What  is  the  final  potential  and  how  much  charge  was 
transferred  ?     What  was  the  loss  of  energy  ? 

7.  Show  how  the  capacity  of  an  insulated  conductor  is  altered  by  bringing 
it  near  to  an  earthed  conductor.  How  would  you  demonstrate  this  ?  Mention 
applications.     (  X  2) 

8.  Two  large,  flat,  circular,  insulated  metal  plates  are  joined  to  the  poles 
of  a  battery  of  cells.  State  and  explain  what  effect  is  produced  on  the  charges, 
potential  difference,  and  force  of  attraction  of  the  plates  if  they  are  moved 
towards  one  another,  remaining  parallel. 

9.  Describe  the  construction  and  explain  the  action  of  a  modern  form  of 
electrical  condenser.  How  can  a  large  capacity  be  got  into  a  small  bulk, 
but  at  what  disability  ?  How  can  the  capacities  of  two  condensers  be 
compared  ?     (  X  2) 

10.  The  inner  coating  of  a  leyden  jar  is  connected  to  an  electroscope,  and 
charged.  How  would  the  opening  of  the  leaves  differ  according  as  the  jar 
stood  on  the  table  or  on  glass  ?  How  is  it  possible  to  touch  first  the  outside 
and  then  the  inside  of  a  jar  without  discharging  it  ? 


ELECTRIC  FIELD   AND    POTENTLVL  593 

11.  A  microcoulomb  given  to  a  condenaer  cauaee  a  kilovolt  betwMo  Um 
plates.  What  is  the  capacity  ?  Another  condenser  of  half  the  capacity  ia 
now  connected  in  parallel ;  what  happens  to  the  charge  ? 

12.  How  do  you  combine  two  condensore  to  have  greater  or  leM  capacity  ? 

Calculate  both,  for  1/2  and  1/3  microfarad. 

13.  How  would  you  ascertain  which  of  several  condoniiers  had  the  ■inaHwl 
capacity  ? 

Three  of  10,  5,  and  1  microfeu:ad8  being  available,  what  are  the  largest 
and  smallest  capacities  obtainable  by  using  any  or  all  ? 

14.  A  condenser  of  two  parallel  plates  10  cm.  radius  has  the  same  capacity 
as  a  sphere  of  the  same  radius ;   how  far  apart  are  the  plates  ? 

15.  A  leyden  jar  A  of  capacity  450  e.s.u.  is  insulated  and  its  outer  coating 
connected  to  the  inner  of  B  600  e.s.u.  If  a  charge  of  10  is  placed  on  tho 
inner  coating  of  .4,  what  are  the  differences  of  potential  in  each  condei 
and  what  is  the  energy  of  the  system  ? 

16.  Four  sheets  each  1  m.  square,  and  10  cm.  apart,  form  a  coi 
If  the  linear  dimensions  were  halved,  what  would  be  the  change  in 
for  constant  P.D  ? 

State  two  practical  uses  of  condensers. 

17.  Two  2-cm.  radius  balls,  10  cm.  apart,  have  equal  charges.  Sketch  the 
distribution  of  electric  field  between  them  if  the  charges  are  (a)  like,  (6)  on* 
like.     How  is  the  force  between  them  altered  in  oil  s.i.c.  2  ? 

18.  Two  gold  leaves  hanging  originally  in  contact  are  charged  and  repel 
each  other  ;  how  does  their  distance  apart  depend  on  (o)  charge,  (6)  medium  T 

19.  Define  Potential  :  why  can  a  larger  charge  be  put  on  a  plate  if  it  is 
nearer  the  earth  ?  What  would  be  the  effect  on  its  potential  oi  interposiiif 
a  thick  plate  of  (o)  ebonito,  (b)  metal,  or  (c)  of  raising  it  ?     (  X  2) 

20.  A  condenser  is  made  of  two  plates  each  50  sq.  cm.,  0*5  mm.  apart.  It 
is  immersed  in  oil,  and  then  has  capacity  140  cm.  What  is  the  dislattrki 
constant  of  the  oil  ? 

21.  Of  two  equal-sized  condensers,  one  has  its  plates  separated  by  air 
and  the  other  by  sulphur :  the  air  condenser  is  chan^  and  made  to  sbara. 
and  its  potential  falls  to  l/7th,  calculate  s.i.c.  of  sulphur. 

22.  Calculate  the  energy  of  two  40-cm.  diam.  plates  1  cm.  apart  in  air. 
charged  to  100  volts  P.D.  (1  volt  =  1/300  e.s.  unit). 

What  would  be  the  energy  if  now  filled  with  oil  of  s.i.c.  2  (o)  keeping  the 
P.D.  constant,  (6)  charge  constant  ? 

23.  Describe  the  essential  features  of  some  form  of  electrometer,  and  show 
how  to  use  it  to  measure  (o)  the  potential,  (6)  the  quantity  of  a  charge.  If 
it  leaks,  by  what  tests  would  you  decide  whether  this  is  due  to  poor  msulation 
or  to  the  presence  of  ionizing  radiation  ?     (  X  3) 

24.  Describe  a  modem  gold-leaf  electroscope  capable  of  comparing  the 
voltages  of  two  voltaic  cells. 


MAGNETISM  AND   ELECTRICITY 

CHAPTER   XLVI 

MAGNETIC  FIELDS  AND  ELECTRIC  CURRENTS 
ELECTRO-MAGNETIC  INDUCTION 


§  741 .  We  have  now  to  endeavour  to  find  some  connection  between 
Magnetism  and  Electricity. 

Experiments  made  in  any  of  the  ways  suggested  in  the  two  pre- 
ceding sections  of  the  book  would  disclose  none.  A  magnet  has 
no  more  effect  on  an  electrified  body  than  the  unmagnetized  steel 
would  have  ;  like  most  things,  it  is  a  conductor  of  electricity,  but 
nothing  more.  Steel  and  brass  balls  can  be  suspended  and  elec- 
trified, both  attract  a  pith  ball,  but  only  one  moves  towards  a 
magnet.  A  suspended  electrified  lath  makes  no  attempt  to  set 
N.  and  S.     So  far  there  is  no  connection. 

But  set  the  electricity  into  motion.  In  the  middle  of  a  2-ft. 
length  of  electric -light  wire  twist  a  little  helix  of  three  or  four 
turns,  lay  a  sewing -needle  in  the  coils,  and  bend  the  long  ends 
of  the  wire  to  touch  the  outer  coating,  and  come  near  the  knob, 
of  a  charged  leyden  jar.  A  spark  jumps,  the  electrical  charges 
travel  along  the  wire,  and  the  needle  will  be  found  able  to  pick 
up  iron  filings  or  to  set  N.  and  S.  ;  it  has  become  magnetized  by 
the  passage  of  a  '  current '  of  electricity  in  the  wire  encircling  it. 

In  experiments  made  by  Rowland  and  others,  a  charged  disc 
was  quite  prevented  from  exerting  any  electric  attraction  on  a 
delicate  magnetometer  needle,  by  the  interposition  of  an  earthed 
metal  plate.  But  when  the  disc  was  spun  rapidly,  the  moving 
charge  produced  a  magnetic  effect,  which  was  felt  through  the 
metal  plate,  for  the  needle  was  deflected. 

Lightning  has  frequently  been  observed  to  cause  magnetization 
or  demagnetization. 

Hence  electric  charges  in  motion  can  affect  a  magnet ;  in  other 
words,  an  Electric  Current  gives  rise  to  a  Magnetic  Field. 

§  742.  Several  devices  for  separating  electrical  -f  and  —  charges 
have  already  been  described  ;  the  flowing  together  again  of  these 
charges  constitutes  an  Electric  Current.  But  although  these  devices 
yield  high  electric  pressures  (difference  of  potential)  capable  of 
forcing  current  through  an  inch  or  two  of  air,  perhaps,  yet  the 
currents  they  supply  are  usually  too  intermittent,  and  always 
too  scanty  in  total  quantity,  to  be  of  much  practical  value.     The 

694 


§743]  FIELDS   AND   CURRENTS  596 

abundant  and  continuous  currents  from  Voltaic  Batteries  (Chap. 
LII),  in  which  electric  charges  are  being  separated  by  chemical 
action,  are  commonly  used  in  electro-magnetic  experiments. 
The  chemical  action  produces  only  a  very  small  electric  jiotential 
difference,  only  a  thousandth  or  less  of  that  required  to  produce 
a  very  small  spark  in  air,  consequently  a  current  path  of  good 
conducting  copper,  brass,  solder,  etc.,  must  be  provided  all  the 
way,  and  the  current  is  quite  unable  to  pass  out  of  this  into  the  air. 
And  on  wire  wound  in  close  coils  a  thin  coating  of  cotton,  nilk, 
enamel,  etc.,  forms  ample  insulation,  just  to  prevent  metallic  contact 
of  adjacent  turns,  through  which  current  might  *  short-circuit  * 
without  travelling  the  whole  length  of  the  coils. 

The  electric  current  obtainable  from  the  public  mains,  and 
produced  by  the  electro-magnetic  machinery  of  §  754,  is  of  100  to 
200  times  higher  pressure,  and  not  to  be  recommended  to  beginners 
for  laboratory  experiments,  but  it  is  not  until  the  '  extra  high- 
pressures  '  of  the  electrical  engineer,  200  to  500  times  those  of 
domestic  supply,  that  we  again  reach  the  long  sparks  and  the 
imperative  necessity  for  long  glass  etc.  insulators  that  we  found 
in  electro-static  experiments.  And  considering  that  all  the  un- 
pleasantness arising  from  a  leyden-jar  shock  is  causcnl  by  the 
passage  of  a  current  for  a  few  millionths  of  a  second,  you  can 
understand  the  extreme  precautions  taken  by  an  engineer  who  is 
supplying  current  at  these  pressures  constantly. 

§  743.  Having  at  command  a  current  of  50  amperes  or  more, 
which  is  quite  beyond  an  ordinary  physics  lab.,  though  a  trifle  to 
an  engineer — and  sending  it  through  lengths  of  three  or  four  yards 
of  cotton-covered  wire,  or  light  *  flex,'  stretched  close  side  by  side, 
one  can  show  : 

That  wires  carrying  currents  in  the  same  direction,  directly 
attract  each  other,  and  cling  together  ;  cf.  Fig.  319. 

That  wires  carrying  currents  in  opposite  directions,  repel 
each  other  straight  apart ;  cf.  Fig.  320. 

To  elucidate  these  actions,  we  pass  one  current  straight  down  a 
wire  through  a  horizontal  card,  on  which  we  scatter  iron  filings, 
and  we  get  Fig.  317  (with  about  20  amps,  natural  size),  showing 
magnetic  lines  encircUng  the  wire,  weaker  at  distance. 

Stretching  the  same  wire  horizontally  just  above  a  card  sprinklecl 
with  iron  filings,  these  arrange  themselves,  when  the  card  is  tapped. 
in  short  straight  lines  crossing  the  direction  of  the  current  at  right 
angles,  and  showing  that  Fig.  317  is  rei)eated  in  every  plane  per- 
pendicular  to  the  wire,  along  its  whole  length.  The  magnetic  Imcs 
are  distinct  rings  round  the  current— they  do  not  *  spiral  round  it. 
nor  drift  along  it  in  the  least.  i    u       -i 

With  the  two  wires  carrying  currents  both  down  through  the  carU, 
Fig.  319  appears ;  recollecting  that  these  are  lines  of  force,  we  aeo 
that  the  wires  are  magnetically  pulled  together. 


596 


MAGNETISM   AND   ELECTRICITY 


[§743 


With  one  current  down  and  one  up,  Fig.  320  appears  ;  no  lines 
link  the  two  :  simply  two  ring  systems  of  Fig.  317  are  crowding 
each  other,  and  forcing  the  wires  apart. 

As  these  are  evidently  magnetic  lines,  let  us  introduce  a  little 
magnet ;  Fig.  318  shows  a  current  coming  up  past  a  N.  pole  :  pretty 
plainly  the  lines  are  hooking  the  wire  off  to  the  right — and,  equally, 


r-'  -il  J'-^k^-f- 


Fig.  317. 


'■^KUJ^'j-l 


Fig.  318. 


Fig.  319. 


Fig.  320. 


the  N.  pole  off  to  the  left,  since  action  and  reaction  are  equal  and 
opposite. 

Presenting  a  stout  magnet  pole  to  the  50-ampere  current,  the 
wire  hops  across  it,  one  way  or  other,  according  to  direction  of 
current,  and  N.  or  S.  pole  :  it  has  no  direct  attraction  for  the  wire. 

§  744.  Presenting  a  movable  magnet  pole — and  now  you  need 
no  great  current,  but  can  get  all  you  want  from  a  single  voltaic  cell 
and  a  yard  of  wire  : — 


§  "^^S]  FIELDS   AND   CURRENTS  A97 

At  once  the  question  arises,  Which  way  Is  the  oumnt 
running?  Conventionally,  the  (positive)  current  runs  along  the  wire 
frmn  copper  or  carbon  to  zinc  of  the  cell.  It  is  admitt«l  nowacUvi* 
that,  actually,  it  is  a  stream  of  negative  electrons  flowing  through 
the  metal  the  other  way  ;  hut  metals  are  not  the  only  conductom 
of  electricity,  and  nohody  is  going  to  alter  the  convention. 

As  in  §  672,  a  small  compass  needle  is  more  sensitive  tkan  the 
filings,  and  also  tells  which  way  the  lines  are  running. 

Stretching  the  wire  E.  and  W.,  and  hringing  it  just  alxive  or  below 
the  compass,  will  not  tell  us  much,  for  we  have  just  seen  that  the 
field  due  to  current  is  perpendicular  to  it,  and  l)eing  thus  N.  and  S. 
is  merely  added  to,  or  subtracted  from,  the  earth's  controlling 
field,  without  altering  its  direction. 

But  holding  the  wire  more  or  less  N.and  S.,  parallel  to  the  needle. 
and  bringing  it  above  or  below,  the  needle  will  \ye  seen  to  deflect 
opposite  ways  in  the  two  cases,  and  ultimately  set  practically 
perpendicular  to  the  wire  when  very  close.  And  its  movement  will 
be  found  to  agree  with  the  Rule — Swimming  in  and  with  the  current. 
facing  the  magnet,  the  north  pole  moves  towards  your  left  hawl. 
As  the  N.  pole  sets  '  down  stream,'  this  Ampere's  RlUe  may  \yv  moff* 
generally  stated  thus— Swimming  in  and  with  the  current,  the  field 
in  front  of  you  runs  towards  your  left  hand. 

If  the  wire  is  stretched  on  a  level  with  the  comiwss,  the  nee<lle 
is  not  deflected  E.  or  W. — there  is  no  field  straight  towanb  oc 
away  from  the  wire — but  one  or  other  pole  ducks  down,  and  M 
you  would  have  to  swim  on  your  side  to  face  the  needle,  it  is 
evidently  obeying  the  Rule. 

Plainly  it  was  this  Rule  that  enabled  N.  and  up  to  be  marked  on 
Fig.  318  ;  or  it  might  have  been  S.  and  down.  Twist  and  turn  the 
figure  about,  and  in  the  laboratory  twist  and  turn  wire  and  compaas, 
for  Ampere's  Rule  you  must  know ;  and,  fortunately,  knowing  it, 
you  can  carry  right  through. 

§745.  Fig.  320  illustrates  also  the  action  of  a  coil  of  wire  of 
one  turn  (or  of  several  hundred  bunche<l  into  one)  round  which  the 
current  circulates.  Notice  that  in  the  middle  the  lines  are  all 
going  one  way  ;  and  just  in  the  centre  are  per|K»ndicular  to  the 
plane  of  the  coil,  uniformly  spaced,  and  shortly  parallel,  i.f.  the 
field  is  approximately  uniform  for  a  small  space  hereabouta. 

In  Fig.  321  the  current  is  going  down  the  two  wires  on  the 
right  and  coming  up  the  two  on  the  left ;  this  is  a  coil  of  two 
turns,  the  small  beginning  of  the  long  helical  coils  or  Solenoids 
(SwXev,  an  eel)  familiar  in  electrical  apparatus.  (You  see  how  it 
combines  Figs.  319  and  320.)  Notice  that  the  lines  nin  along  the 
axis  of  the  coil,  where  they  keep  fairly  uniform  and  pArallel. 
Consequently  a  pair  of  ring  coils  such  as  these,  or  a  long  coil,  is  of 
great  use  when  a  uniform  magnetic  field  is  re<|uired,  e.g.  for  measure- 
ment, or  for  magnetizing  steel  magnets  uniformly. 

The  running  of  lines  out  from  one  end  and  into  the  other  end. 


598 


MAGNETISM   AND   ELECTRICITY 


[§745 


shown  to  perfection  in  the  photograph  Fig.  322,  where  two 
magnets  have  been  placed  with  their  N.  poles  near  the  ends  of  the 
'  solenoid,'  indicates  that  the  coil  acts  like  a  magnet  (with  the 
distinction  that  now  the  return  of  the  stream  through  the  interior 


Fig.  321. 


$>^c^-:^--,.^>- 


Fig.  322. 


is  traceable).  A  few  dozen  turns  of  wire  wound  on  a  paper  tube, 
and  connected  to  a  voltaic  cell,  make  a  coil  the  opposite  ends  of 
which  attract  and  repel  a  compass  needle  just  like  rather  feeble 
magnet  poles.  It  does  not  matter  whether  the  coils  are  in  one 
or  more  long  layers  (solenoid)  or  bunched  into  a  ring. 

§  746.  If  the  inside  of  the  long  coil  is  filled  with  iron,  many  score 
times  more  lines  will  flow  through,  because  the  iron  is  so  very 
permeable,  and  we  obtain  a  strong  Electro-magnet. 

Thus  an  Electro-magnet  is  easily  made  by  winding  several  turns 
of  insulated  wire  round  a  wrought-iron  bolt  and  connecting  the 
ends  of  the  wire  to  a  battery.  The  turns  must  all  go  the  same 
way  round,  but  whether  they  run  up  or  down  the  iron,  or  in  how 
many  layers,  makes  no  difference.  If  only  a  weak  current  is 
available,  there  must  be  many  hundred  turns  :  the  total  flow  of 
current  round  each  cm.  length  of  iron  must  be  kept  large. 

In  winding  a  '  horseshoe  '  the  wire  must  cross  over  between 
the  legs  and  wind  on  them  opposite  ways.  Fig.  330,  to  produce 
the  opposite  poles  required.  Straightening  out  the  horseshoe, 
this  would  form  a  continuous  coil. 

The  N.  pole  of  the  iron,  from  which  lines  run  out,  is  towards  the 
swimmer's  left  as  he  faces  the  iron,  by  Ampere's  Rule.  The  current 
enters  the  magnet  in  Fig.  330  by  the  wire  marked  +,  and  goes 
behind  ;  and  see  Fig.  326. 

Soft-iron  electromagnets  are  much  stronger  than  permanent 
steel  ones  ;  they  let  go  when  the  current  is  cut  off,  and  are  extremely 


§747] 


FIELDS   AND   CURRENTS 


590 

useful  in  all  sorts  of  electrical  machinery.  Thev  a«8ume  protean 
torms,  according  to  their  purpose,  which  is  seldom  lifting  woighti* 
but  two  professional  strong  men  may  be  mentioned. 

The  first  is  the  surgeon's  electro-magnet  for  extracting  chiiw  of 
iron  from  the  eye,  or  from  wounds  :  it  is  a  single  stout  bar.  like  Fig. 
326,  but  with  conical  iron  extensions  of  various  lengths  which  can 
be  screwed  on  to  the  pole,  and  a  flexible  probe  of  iron -wire -rope. 
It  may  weight  J  cwt.  upwards,  according  to  what  he  is  prejMireil 
to  pay,  and  can  provide  current  for. 


Fig.  323. 


Fio.  324. 


Fio.  325. 


Fic;.  326. 


The  second  is  the  lifting  magnet  of  Fig.  325,  where  the  aluminium 
coil  lies  in  an  annular  recess  in  a  soft  steel  casting,  covennl  with  a 
thin  protective  bronze  plate  ;  the  central  stump  and  the  outer  ring 
form  the  poles.  These  are  made  up  to  5  ft.  diam.  and  4  ton«  weight 
and,  slung  from  cranes,  can  lift  plates,  girders,  scrap  iron,  etc., 
up  to  five  times  as  much. 

§  747.  Let  us  return  to  the  mutual  action  between  magnet  pole 
and  current-carrying  conductor,  and  consider  a  few  instanccH  : 

A  light  compass  needle  brought  near  the  art*- lamp  carbons  of 
Fig.  324  would  have  its  N.  pole  driven  to  the  left  ;  a  larger  magnet 
drives  off  the  current -carrying  flame  of  the  arc  itself  towanl.H  the 
right,  at  right  angles  to  the  direction  of  approach  of  the  pole,  and 
may  stretch  it  so  much  as  to  extinguish  it. 


600 


MAGNETISM   AND   ELECTRICITY 


[§747 


An  alternating-current  arc  spreads  out  into  a  golden  butterfly, 
but  if  you  carry  your  eye  quickly  down  past  it,  you  see  the  poor 
thing  has  but  one  wing,  which  it  flaps  right  and  left  as  the  current 
reverses. 

Fig.  326  shows  how  a  slack  current-carrying  wire  will  wind 
itself  round  the  leg  of  a  great  magnet,  or  will  unwind  and  wind  on 
the  opposite  way  when  the  current  is  reversed. 

In  Fig.  323  the  40-cm.  coil  of  §  774  is  carrying  10  amps,  in  its 
forty -two  turns  always  the  same  way  down  in  front  of  that  same 
magnet  pole  ;  it  revolves  fitfully,  putting  on  speed  every  time  it 
passes  the  pole,  just  like  the  horse  working  the  capstan  on  the 
beach  every  time  he  passes  the  boy  with  the  stick.  It  is  acting  as 
a  '  direct-current  electro-motor.' 

In  Fig.  327  a  heavy  magnetic  needle  is  pivoted  in  the  middle 
of  a  6-in.  coil  of  wire  hung  by  two  long  thin  wires  or  a  length  of 


Fig.  327. 


Fig.  328. 


'  flex,'  through  which  current  is  supplied.  The  coil  is  at  first  sus- 
pended in  the  magnetic  meridian,  so  that  magnet  and  coil  lie  to- 
gether much  as  in  Fig.  328.  The  current  circulating  perhaps  100 
times  round  the  coil  is  equivalent  to  a  100  times  greater  current 
passing  once  down  and  up  :  observe  how  the  lines  of  force  are 
wrenching  magnet  and  coil  round  opposite  ways.  In  the  experi- 
ment of  Fig.  327  the  magnet  swings  out  one  way,  but  comes  to 
rest  at  a  deflection  such  that  the  couple  exerted  on  it  by  the  earth 
is  equal  and  opposite  to  that  due  to  the  coil.  And  the  coil  swings 
round  the  other  way,  under  the  reaction  of  the  magnet,  until  the 
twist  on  the  suspending  wires  checks  it. 

§  748.  Now  look  at  Fig.  329.     A  is  Fig.  320  (turned  on  its  side 
merely)  with  its  down  and  up  currents  ;  B  is  Fig.  318. 

Only — the  magnetic  lines  due  to  one  current  only  are  shown,  and 


§749] 


FIELDS   AND   CURRENTS 


601 


Fio.  S29. 


this  one  current  may  be  regarded  as  entirely  replaced  by  iUi  maffnettc 
field.  And  in  B,  only  the  lines  from  the  magnet  \h}\c  are  shown, 
and  the  magnet  is  represented  by  its  field,  up  at  right  angles  to  which 
comes  the  current-carrying  conductor  U. 

In  A,  we  have  just  seen  that  the  action  is  a  motion  of  conductor 
U  directly  away  from  D,  Fig.  320.     In  B,  it  is  a  motion  of  U  to  the 
right  (Fig.  318).     We  see  that  both 
these  can  be  described  as  the  same 
action,  under  one  general  rule  : 

I.  A  conductor  carrying  a  current 
through  a  magnetic  field  is  forced 
to  move  so  as  to  cut  across  the 
magnetic  lines. 

The  conductor  will  always  en- 
deavour to  move  so  as  to  cut  most 
lines,  i.e.  at  right   angles  to  itself 

and  at  right  angles  to  the  magnetic  lines.  The  stone  rolls  the 
quickest  way  downhill. 

Looking  at  Figs.  323,  324  and  326  in  this  new  way,  it  will  eauily  lie 
seen  how  the  moving  wire  is  *  mowing  down  '  as*  many  magnetic 
lines  as  it  can. 

This  way  of  always  reducing  the  experimental  conditions  to  a 
current  flowing  across  a  magnetic  field  may  seem  a  onesided  way 
of  looking  at  the  problem,  but  it  is  the  way  along  which  the 
electrical  engineer  has  made  all  his  progress. 

II.  The  Direction  of  the  Motion  is  nhmys  obtainable  by  careful 
application  of  Ampere's  rule  ;  swimming  In  the  current  and  facing 
the  place  the  lines  come  from,  that  place  must  move  off  to  the  left, 
i.e.  the  conductor  is  pushed  to  the  right. 

Or  a  mnemonic  device  of  the  engineer  is  this  : — 
Hold  up  the  left  hand,  thumb  and  index-finger  outstr«tchc<l, 
middle  and  other  fingers  naturally  partly  bent ;  then  a  current 
flowing  out  along  the  middle  finger,  across  magnetic  lines  nmning 
out  parallel  to  the  index-finger,  is  acted  on  by  a  force  out  along  the 
thumb. 

§  749.  How  great  is  this  force  that  ads  on  the  conductor  carrying 

a  current  in  the  magnetic  field  ? 

Fig.  330  represents  an  apparatus  which,  though  incapable  of 
accurate  results,  serves  very  well  to  suggest  how  this  Question  is 
to  be  dealt  with.  ABCDEFG  is  a  frame  of  wire  pivoted  at  B  and 
F  in  mercury  cups  scooped  out  in  a  fixed  wooden  bar.  Through 
the  mercury  in  these  it  makes  good -conducting  connection  with  the 
remainder  of  a  circuit.  A  scale-pan  hangs  at  D  on  C'K.  »"<*  J.j»«^ 
whole  frame  is  exactly  balanced  by  counter- weights  at  AG.  The 
straight  wire  CE  moves  up  and  down,  parallel  to  itself,  and  at  right 


602 


MAGNETISM   AND   ELECTRICITY 


[§749 


angles  to  the  lines  of  the  magnetic  field  in  the  narrow  gap  between 

the  pole -pieces  of  a  magnet  NS. 

For    simplicity,    suppose    this    field    uniform    in    the    gap    and 

negligible  outside  it.     It  can  be  measured  by  magnetic  methods, 

and  we  can  therefore  tell  to  start 
with  how  many  lines  would  be  cut 
if  the  horizontal  wire  CE  moved  1 
cm.  vertically  ;  it  =  lines  per  sq. 
cm.  X  no.  of  sq.  cm.  the  wire 
sweeps  over  in  its  motion  =  field 
strength  X  length  of  wire  in  field  x 
1  cm. 

Let  this  total  number  =  n. 
Load  the  scale-pan  at  D  with  n 
dynes.  Send  a  current  along  CE 
so  as  to  lift  it,  and  adjust  the 
current  until  there  is  equilibrium 
again,  i.e.  the  upward  force  acting 
on  CE  is  equal  to  n  dynes.     This  is 

then  the  c.g.s.  unit  current  which  may  be  called  the  '  decampere.' 
The  Ampere  is  one-tenth  of  this  :   it  is  the  practical  unit.     On  a 

conductor  carrying  1  Ampere  10  cm.  across  unit  magnetic  field  there 

is  a  force,  at  right  angles  to  both,  of  1  dyne. 

To  get  a  great  force  a  large  current  must  cross  a  broad  and  strong 

magnetic  field. 

Ex.  1.     Calculate  the  total  force  acting  on  a  30 -cm.  length  of  wire  carrying 
20  amp.  at  right  angles  to  a  field  of  5000  gauss  (unit  lines  per  sq.  cm.). 

Force  =  20  x  5000  x  30/10  =  300,000  dynes;   over  300  gm.  wt. 


Fig.  330. 


§750.  The  direct-current  Electro-motor 
loop  of  wire  ABCDEF,  Fig.  331,  free  to 
cylindrical  space  between  the  pole-pieces 
magnetic  lines  are    running    across   as 
dotted.      A    current    is    sent    from    A 
round  to  F,  there  will  be  a  force  on 
BC  lifting  it  upward  and  on  DE  press- 
ing   it   downward    (for    as    you    swim 
from    B    to    C,    facing   N,    N    has    to 
appear  to  go  off  to  your  left),    so   the 
loop  will  turn  until  it  stands   vertical, 
when  the  vertical  forces  can  turn  it  no 
farther.  x 

Suppose,  however,  that  its  inertia 
carries  it  on,  and  also  that  as  it  passes 
this  vertical  dead-point  the  current  is 
reversed,  so  as  to  flow  from  C  to  B  and 
from  E  to  D  :  BC,  now  on  the  right,  is 
driven  down,  and  DE  is  driven  up  on  the  left 
tinues  to  rotate  in  the  direction  SCN. 


Suppose  a  rectangular 
rotate  on  axis  XY,  in  a 
NS  of  a  magnet,  where 


i.e.  the  loop  con- 


760] 


FIELDS   AND   CURRENTS 


603 

The  usual  way  of  making  the  machine  itself  effect  the  Revereal 
of  Current  is  shown  at  X.  The  wire  ends  are  attached  to  two 
half-cylinders  of  copper  enclosing  the  axle,  quite  separated  from 
each  other  by  insulating  material  (mica,  etc.).  Against  these 
press  two  fixed  '  brushes,'  formerly  strips  of  cop|)er,  as  shown 
nowadays  blocks  of  graphitic  carbon,  to  which  a  continuous  current 
is  supplied.  When  the  loop  is  vertical  the  insulating  gap  has  come 
under  the  brushes,  a  moment  later  the  copper  segment  (upwr)  just 
escaped  from  the  left-hand  brush,  slips  under  the  right,  and*  vice 
versa,  so  that  the  current  is  now  being  sent  into  the  loop  the  other 
way  round.  This  is  the  Split-ring  Commutator. 
^  The  actual  electro-motor  suited  to  work  with  continuous  (or 
direct  ')  current  is  this  machine  modified  in  detail :  Fig.  332. 


Fig.  332. 

(1)  The  cylindrical  space  is  nearly  filled  with  a  mass  of  soft  iron. 
This  enormously  increases  the  number  of  magnetic  lines,  and 
therefore  the  forces  acting.  Whether  this  iron  core  stands  still 
or  rotates  makes  little  magnetic  difference,  consequently,  for 
mechanical  strength,  the  wire  is  wound  in  grooves  on  the  iron 
(some  of  which  are  shown  empty  in  the  diagram),  and  this  whole 
massive  Armature  revolves.  The  iron  is  laminated,  as  indicated, 
see  §  824. 

(2)  There  are  many  similar  loops  of  wire  arranged  at  equal  angles 
to  fill  the  whole  periphery.  The  half-cylinders  of  the  Commutator 
are  slit  up  into  narrow  straight  strips,  separated  by  mica,  so  that 
each  loop  gets  its  pair  of  segments.  The  '  brushes  *  are  blocks 
of  graphite  (one  shown,  top  left)  which  minimize  sparking,  and  put 
a  polish  on  the  commutator  which  I  have  seen  at  Niagara  improve*! 
to  a  beautiful  patina  by  17  years'  running.  The  loops  are  also 
all  interconnected  (we  cannot  go  into  details,  one  firm  alone  knows 
1600  ways)  :  the  effect  is  to  get  a  stronger  and  steadier  rotation, 
the  principle  being  quite  unaltered. 

On  the  left  of  Fig.  332  is  the  magnet  carcase  NS,  which  has  less 
'  magnetic  leakage  '  than  Fig.  331  :  very  often  four  poles  are  used 
which  are  then  nsns,  and  two  pairs  of  brushes. 


604  MAGNETISM   AND   ELECTRICITY  [§  750 

For  Back  E.M.F.,  and  Starters,  see  §  756. 

Ex.  2.  A  single  turn  of  wire  in  the  form  of  a  rectangle  15  cm.  x  8  cm. 
can  turn  in  a  horizontal  field  of  strength  2500.  Indicate  the  forces  acting  on 
each  side  of  the  rectangle,  and  calculate  the  couple  acting  upon  it  when  its 
plane  makes  an  angle  of  45°  with  the  field  and  20  amp.  flows  in  it.     Fig.  331. 

There  is  no  action  on  the  short  sides,  which  do  not  cut  the  field  as  they 
move.  On  each  long  side  BC,  ED,  is  a  force  (15/10)  x  20  amp.  x  2500  = 
75,000  dynes  at  right  angles  to  field  (vertically  up,  or  down,  in  Fig.  331),  and 
therefore  exerting  turning  moment  75,000  X  4  sin  45°  cm.,  which  on  both 
together  amounts  to  850,000  dynes  X  cm. 

§751.  If  the  current  flowing  across  a  magnetic  field  causes 
a  body-moving  force  on  the  conductor,  what  will  happen  when  an 
empty  conductor  is  bodily  moved  across  a  magnetic  field  ?  Will 
there  arise  an  electricity-moving  (electromotive)  force  tending  to 
drive  a  current  along  the  conductor  ?  This  is  by  no  means  the  only 
thing  that  might  happen,  but  let  us  experiment  : — 

Connect  a  length  of  wire  to  a  reasonably  sensitive  Galvanometer 
(see  next  Chapter),  and  twist  the  slack  round  the  magnet  pole, 
Fig.  326.  The  galvanometer  is  deflected,  showing  a  current  flowing 
while  the  wire  is  moving. 

A  loop  of  the  wire  moved  up  and  down  in  the  polar  gap,  like  CE 
in  Fig.  330,  again  deflects  the  galvanometer  while  the  wire  is  moving. 

Coiled  into  two  or  three  turns,  i.e.  using  2  or  3  CE's  instead  of 
one — the  effect  correspondingly  increases,  and  using  a  coil  of  many 
turns  you  need  only  a  small  magnet.  Then  putting  a  resistance 
of  any  sort  into  the  circuit,  and  increasing  it,  the  current  diminishes. 
Hence  what  you  are  really  doing  is  inducing  an  Electromotive 
Force  in  the  moving  wire,  and  this  drives  what  current  it  can 
round  the  circuit. 

The  big  rotating  coil,  with  its  commutator,  that  acted  as  a 
'  motor,'  now,  when  turned  by  hand,  produces  a  current.  Indeed, 
you  may  need  no  big  magnet,  the  Earth's  field  being  enough. 

And,  if  you  connect  your  galvanometer  to  the  terminals  of  an 
actual  electromotor,  and  turn  it  very  gently,  you  will  probably 
get  a  good-sized  deflection  as  long  as  you  keep  on  turning. 

These  experiments  lead  to  the  conclusion  that 

III.  Forcibly  moving  a  conductor  across  a  magnetic  field  so  as  to 
cut  the  lines  excites  an  Electromotive  Force  in  it. 

.  As  before,   the   mechanical  moving  force,   the   magnetic   field, 
and  the  current,  are  at  right  angles. 

And  now  which  way  will  the  induced  current  flow  ?  Suppose  it 
went  the  same  way  as  before,  the  way  which  would  assist*  the  very 
motion  that  produced  the  current.  The  motion  would  go  on 
faster,  causing  a  greater  current,  which  would  help  more,  and  so 
on,  always  faster  and  stronger  without  any  help  from  without. 
This  would  be  the  Perpetual  Motion,  ever  vainly  sought  for 
through  the  centuries.     Therefore 


§751]  FIELDS   AND   CURRENTS  605 

ly.  The  current  is  always  in  such  a  direction  as  to  oppose  the 
motion  inducing  it.  Its  direction  is  the  reveree  of  that  found  in 
§  '48. 

This  is  Lenz's  Law  ;  it  is  another  fundamental  Htatcment  of 
electromagnetic  induction,  it  is  the  appropriate  form  of  the 
prmciple  of  the  Conservation  of  Energy.  It  can  he  put  in  a  more 
general  way  still :  *  Whatever  you  do  in  eiectromagnetlcs,  the  system 
opposes  you  every  way  it  can.' 

This  law  you  will  study  in  the  hiboratory  by  pushing  a  magnet 
pole  up  to  the  face  of  a  coil  in  which  you  can  see  the  direction  of 
the  wires,  and  finding  that  the  current  induced  in  the  coil  is  such 
as  to  develop  an  opposing  pole  on  that  face  of  the  coil,  usinir 
Ampere's  rule.     See  further  Chapter  LI. 

With  more  power  available,  it  can  be  striliingly  shown  with 
the  aid  of  a  stout  4-in.  copper  ring.  This  is  slung*^  by  stringH  no 
that  it  can  swing  on  to  the  pole  of 
the  big  electromagnet,  laid  horizontal. 
Fig.  333,  left. 

With  no  current  on  the  magnet, 
the  ring  swings  over  the  pole  and 
dangles  about  freely  for  a  long  time. 
But  when  the  magnet  is  energized,  the 
ring  swinging  towards  it  suddenly 
checks,  and  crawls  and  slinks  about,  Fio.  333. 

and  may  have  to  be  pushed  over  the 
pole  by  hand,  to  which  it  feels  as  if  held  by  treacle  or  invisible  glue. 

As  it  fell  towards  the  pole,  it  had  to  cut  through  all  the  great 
sheaf  of  lines  spreading  from  it ;  that  induced  an  E.M.F.  in  it,  which 
circulated  a  big  current  (for  the  copper  is  an  extremely  good  con- 
ductor), and  in  such  a  direction  as  to  send  lines  out  of  the  face  of 
the  coil  ill  opposition  to  the  oncoming  pole. 

Now  switch  off  the  magnet,  and  the  ring  leaps  along  it  :  you 
are  shrinking  that  sheaf  of  lines  down  to  nothing,  and  the  ring 
makes  an  effort  not  to  lose  them. 

Let  it  hang  at  the  pole  again,  and  switch  on,  and  it  jumps  away 
from  your  attempt  to  push  magnetic  lines  through  it. 

Of  course  it  creeps  back,  for  it  is  only  during  motion  that  currents 
are  induced,  but  if  you  could  keep  changing  the  pole,  you  might 
stave  it  off  altogether. 

This  is  exactly  what  can  be  done  with  a  laminated -iron  magnet 
fed  by  alternating  current,  which  changes  its  polarity  100  or  more 
times  a  second.  Fig.  333,  right.  The  ring  floats  in  tlie  air,  for  the 
alternating  currents  induced  in  it,  by  the  lines  thrashing  in  and  out, 
oppose  its  attempt  to  fall  over  the  pole. 

That  the  ring  soon  gets  too  hot  to  hold  is  plain  evidence  of  currents 
of  many  hundreds  of  amperes.  So,  too,  a  couple  of  copper  rings 
from  a  water-bath  will  float,  and  cling  together:  the  mutual 
attraction  of  parallel  currents  the  same  way,  §  743. 


606  MAGNETISM   AND   ELECTRICITY  [§  752 

§  752.  How  great  is  this  electromotive  force  that  drives  a  current 
along  a  conductor  moving  across  a  magnetic  field  ? 

Turn  to  Fig.  330  again,  move  the  wire  CE  at  the  steady  speed  of 
1  cm.  per  sec.,  so  that  it  '  mows  down  '  magnetic  lines  at  the  rate  of 
n  per  sec.  The  electromotive  force  caused  in  CE  =  n  units  of 
E.M.F. 

The  Electromotive  Force  in  a  conductor  is  equal  to  the  number 
of  unit  magnetic  lines  it  cuts  per  second.  This  is  the  modern  form 
of  Faraday's  Law  of  Electromagnetic  Induction. 

If  a  conductor  is  moved  so  as  to  cut  one  unit  magnetic  line  per 
second,  Unit  Electromotive  Force  arises  in  it. 

The  Volt  =  100  million  times  tliis  unit  (and  even  then  proves  to 
be  only  1 /300th  the  electrostatic  unit  of  potential  difference). 

To  get  a  high  E.M.F.  a  great  length  of  wire  must  be  moved  rapidly 
across  a  strong  magnetic  field. 

What  current  the  electromotive  force  succeeds  in  setting  going 
depends  on  how  good-conducting  is  the  circuit  of  which  the  moving 
conductor  forms  part. 

Ex.  3.  The  wire  in  Ex.  1  is  moved  at  right  angles  to  itself  and  to  the  field 
at  a  speed  of  15  cm.  per  sec.     What  E.M.F.  is  induced  in  it  ? 

5000  X  30  X  15  =  2,250,000  lines  cut  per  sec. 
=  0-0225  volt 

§  753.  Now  let  us  turn  to  further  instances  of  the  production 
of  electric  current  by  moving  a  conductor  across  a  magnetic  field. 
Consider  first  the  Earth  Inductor,  shown,  in  section  by  the  North- 
South  plane,  in  Fig.  334. 

Taking  a  rectangular  loop  of  a  few  score  turns  of  wire,  with  its 
ends  connected  to  a  sensitive  galvanometer  G,  hold  one  horizontal 

side  magnetic  E.  and  W.  and 
'  I  1 1 1  '  lU^UAl-fl '  /  /  /'  ^  steadily  rotate  the  loop,  on  this  as 
^ ' ' ' i/^i  ' I  I^^Hs^l l\  ^\  ^-^^^  ^'  ^^  ^^  earth's  magnetic 
^/l^l'il  ' I ijWN  V^  y  field.  This  side  does  not  move, 
,  ,.  //  'i  'i/0'lf/{//^^'Q-^  two  sides  move  in  planes  parallel 
H'/  ''/    6/4m^n  /  /   }W  yJ~^  to  the  lines  and  cut  none;   atten- 

,  i\' I    '  I  i'^ij/;f'i==Lffi^''^  ^  tion  can  therefore  be  confined   to 

^1 1%  I '  I  ' ' 1 1 1 //I  the     fourth    side    only.      As   this 

ii.! I  'hul U IjJ'  moves  near  A  it  is   cutting  lines 

i^v^//p//777/,^///  fast,   and  the   electromotive  force 

Fig.  334.  induced  in  it  drives  a  current  which 

deflects  the  galvanometer  needle 
strongly  to  the  right.  Approaching  B,  it  is  cutting  across  lines 
much  slower,  and  the  galvanometer  needle  creeps  back  towards 
zero.  At  B  there  is  momentarily  no  cutting,  towards  C  it  begins 
to  cut  lines  the  other  way,  and  the  galvanometer  swings  to  the 
left,  reaching  maximum  at  C,  zero  at  D,  and  so  on. 

Thus  an  Alternating  Current  is  being  produced  :  a  fine  one  can  be 
got  from  the  suspended  coil  of  Fig.  327,  by  giving  it  a  good  spin. 


§  754]  FIELDS   AND   CURRENTS  607 

There  is  no  obligation  to  use  one  side  as  axis.  For  supposo  the 
axis  at  X,  the  fourth  side  moves  only  half  as  fast,  hut  the  first  side, 
in  which  the  wire  runs  back,  is  now  also  cutting  lines  the  other  way ; 
i  —  {—  i)  =  I  ',  i-e.  rotating  the  coil  on  a  central  or  any  other 
parallel  axis  has  the  same  effect. 

And  further,  the  shape  of  the  coil  does  not  matter,  so  long  as 
its  area  remains  the  same.  At  OB  the  number  of  lines  paming 
through  =  field  strength  x  area  of  coil ;  arrived  at  OD,  all  these 
pass  through  the  reverse  way.  The  total  change  =  total  lines 
cut  =  twice  field  strength  x  area  of  coil,  i.e.  if  the  area  is  the  same 
the  induced  E.M.F.  is  the  same,  whatever  parts  of  the  wire  happen 
to  do  the  actual  cutting. 

So  any  coil,  held  in  the  hand,  will  serve  as  Earth  Inductor. 
The  electromotive  force  =  rate  at  which  lines  are  being  cui. 
So  long  as  no  additional  obstruction  is  placed  in  a  wire  circuit, 
the  current  moved  in  it  is  proportional  to  the  electromotive  force 
[Ohm's  Law,  §  772]. 

Hence  Current  is  proportional  to  rate  of  cutting  lines,  e.g.  in  this 
apparatus,  to  speed  of  rotation. 

Multiplying  both  sides  by  the  Time  spent  in  the  process 
Current  X  time  of  flow  oc  rate  of  cutting  lines  x  time  spent. 
The  left-hand  side  is  the  total  Quantity  of  Electricity  induced 
to  move  past  any  particular  point  in  the  circuit. 

.'.  Quantity  cc  total  number  of  lines  cut. 
This  is  a  general  and  important  result.  The  rush  of  electricity 
is  often  too  rapid  for  the  moving  parts  of  a  galvanometer  to  keep 
pace  with,  but  a  heavy  slow-moving  '  ballistic  '  galvanometer  will 
give  a  scale-swing  proportional  to  the  total  Quantity  that  passe<l 
in  the  rush,  just  as  a  heavy  pendulum  swings  out  proportionally 
to  the  whole  momentum  of  a  bullet  shot  into  it. 

In  the  Earth  Inductor,  for  instance,  rotation  from  B  to  D  gives 
a  swing  proportional  to  whole  area  of  coil  X  earth's  total  field 
(§  696).  Turning  over  from  H  to  H'  gives  a  less  throw,  it  misse* 
lines  at  the  start  and  cuts  some  backwards  at  the  end.  But 
resolving  the  field  into  the  Horizontal  and  Vertical  Componenta, 
this  turning  a  coil  over  flat  on  the  table  gives  a  throw  proportiona 
to  V,  and  turning  over  from  N  to  Z  (or  better,  about  a  vertica 
axis)  from  facing  north  to  facing  south,  gives  a  throw  projxjrt lonal 
to  H.     Hence  the  apparatus  can  be  used  to  find  the  Dip,  etc.,  §  095 

Moving  the  coil  parallel  to  itself  produces  no  current,  for  the 
following  half,  in  which  the  wire  is  coming  biick,  cuts  as  many 
lines  as  the  leading  half,  and  neutralizes  the  induced  E.M.F. 

§  754.  The  Dynamo.  As  in  §  750,  replace  the  Earth's  field  by 
the  20,000  times  stronger  one  of  a  magnet.  Take  Fig.  331,  and 
instead  of  supplying  current,  turn  the  loop  round  by  hand. 

The  two  sides  of  the  loop  assist  each  other  in  producing  an 
Alternating  Current.    This  can  be  led  out  as  it  is,  through  two 


608 


MAGNETISM   AND   ELECTRICITY 


[§754 


'  slip  rings,'  solid  insulated  rings  each  under  its  own  brush  ;  or  it 
can  be  led  to  the  split-ring  Commutator,  from  which  the  brushes 
gather  the  rushes  of  it,  always  passing  out  at  the  same  brush,  as 
Direct  Current. 

Then,  just  as  before,  by  multiplying  loops  of  wire,  and  using  soft 
iron,  one  gets  a  more  uniform  current  from  a  much  more  compact 
machine,  the  electromagnetic  '  Generator,'  or  Dynamo. 

It  is  precisely  the  same  machine  as  before,  Fig.  332,  with  a  new 
name  and  function. 

As  Motor  it  is  supplied  with  current  and  does  work. 

As  Dynamo  it  is  supplied  with  mechanical  energy  and  produces 
current. 

The  brushes  may  need  shifting  round  a  bit,  that  is  all. 

One  question  occurs  to  you  :  where  does  it  get  its  magnetic 
field  to  start  with  ?  Its  cast-iron  field-magnets  retain  a  little, 
enough  to  start  a  small  current,  which  feeds  them,  and  the  machine 
'  builds  up  '  before  many  seconds. 

Sometimes  one  badly  wants  full  current  quicker  than  this,  and 


Fig.  335. 


then  one  has  to  forgo  the  great  strength  of  the  electro-magnet,  and 
use  a  permanent  tungsten-  or  cobalt-steel  one,  and  call  the  machine 
a  Magneto. 

The  duty  of  the  familiar  petrol -engine  '  Mag  '  is  to  give  strong 
snatches  of  current,  no  matter  which  way — the  briefer  the  better. 
This  purpose  the  primitive  '  shuttle '  armature  of  the  earUest 
d3rQamos  fulfils  admirably  :  you  see  in  Fig.  335  how,  in  no  more 
than  an  eighth  of  a  turn,  the  lines  that  were  flowing  through  from 
the  steel  '  horseshoe  '  magnet,  through  the  concave  soft-iron  cheeks 
or  pole-pieces,  and  through  the  shuttle  or  I-shaped  iron  armature, 
one  way,  are  torn  out  of  it  and  its  encircling  coils  (dotted  in  cross 
section)  and  then  driven  in  the  other  way  about,  which  of  course 
just  doubles  the  effect.     See  further  §  827. 

The  Gramophone  Pick-up  is  a  miniature  magneto.  The  needle 
waggles  an  iron  tongue  near  to  one  or  other  pole,  so  that  lines  are 
sent  through  it  one  way  or  the  other,  therefore  inducing  little 
currents  in  a  diminutive  fixed  coil  surrounding  the  tongue ;  and 
these  are  amplified  in  the  valve -system. 

§  755.  So  long  as  there  is  a  mutual  cutting  of  magnetic  Unes  and 
conducting  circuits,  it  does  not  matter  in  the  least  whether  the 


§756] 


FIELDS   AND   CURRENTS 


600 


circuits  move  and  the  lines  stand  still,  or  the  Hues  move  and  the 
circuits  stand  still.     Let  us  take  instances  of  moving  linen  : 

In  the  push-bike  lighting  magneto,  Fig.  336,  a  four- legged  steel 
magnet  '  rotor  '  (shown  lifted  high  out  of  its  place)  is  spun  over  four 
coils  in  series,  wound  opposite  ways  (see  plan)  on  a 
soft-iron  cross  *  stator  '  :  you  can  make  out  for  your- 
self easily  enough  how  they  all  help  one  another,  and 
the  current  reverses  four  times  per  revolution.  You  see 
how  the  absence  of  sliding  contacts  for  collecting  cur- 
rent simplifies  this  smallest  of  lighting  generators  in 
England. 

So  it  does  the  largest,  started  up  to-day,  the 
75,000-h.p.  turbo-generators  of  the  County  of  London 
Co.  at  Barking. 

§  756.  Starting  a  direct-current  motor  ;  Its  Back 
E.M.F.  No  more  resistance  than  can  be  heli)ed  is  left 
in  any  electromotor,  for  it  would  mean  *  nmning  with 
a  brake  on.'  Consequently,  the  first  rush  of  current 
into  the  motor  is  terrific  ;  no  machinery  above  an 
eighth  h.p.  can  stand  it,  and  Starting  Resistances 
have  to  be  inserted  and  cut  out  of  circuit  in  succession  as  it  speeds 
up. 

When  running,  the  armature  conductors  are,  of  course,  cutting 
the  field,  and  therefore  they  generate  an  E.M.F.  and,  of  course,  this 
opposes ;  it  is  a  Back  E.M.F.,  partly  damming  back  the  current 
you  drive  in  hy  farce  majeure. 

At  low  speed  the  back  E.M.F.  can  be  only  small,  and  that  is  why 
starting  resistances  are  necessary,  but  it  rises  with  the  spee<l,  until 
just  enough  current  gets  in  over  it  to  keep  up  that  speed  under  loatl. 
If  the  load  is  taken  off,  the  speed  runs 
up,  until  the  back  E.M.F.  nearly  equals  the 
applied  E.M.F.,  only  a  little  current  getting 
in  ;  so  that  electromotors  are  not  wasteful 
of  current.  Different  ways  of  supplying 
the  field-magnets,  calle<l  series,  shunt,  and 
compound,  windings,  are  used  to  adapt 
them  best  to  different  purposes;  aeries 
gives  heavy  starting  effort  ;  shunt,  steedier 
speed  ;   compound,  a  compromise. 

A  common  face-plate  pattern  of  Starter 
for  a  shunt-wound  d.-c.  motor  is  «ketche<l 
in  Fig.  337.  When  the  braas  handle  H 
it  connects,  and  keeps  connected, 


^Tf^ 


Fig.  337. 


line  L 


is  moved  to  the  right  .  •     i  v 

with   the   brass  sector  S,  thereby  passmg  current  by  terminal  !• 
to  the  field  magnet  of  the  motor,  below.     Also,  by  way  of  the 
first  stud,  it  admits  current  from  L  through  the  whole  length  of 
the  starting  resistance,  and  terminal  A,  to  the  armature. 
As  this  speeds  up,  its  back  E.M.F.  enables  resistance  to  be  dis- 

X 


610  MAGNETISM   AND   ELECTRICITY  [§  750 

pensed  with,  and  gradually  moving  the  handle  right  over,  an  iron 
plate  on  it  presently  sticks  to  the  electromagnet  NV.  This  is 
only  strong  enough  to  hold  on  if  supplied  by  full  voltage,  conse- 
quently, if  the  mains  current  fails,  this  no-voltage-release  lets  go,  and 
the  switch  handle  is  pulled  clean  off  by  the  spring. 

If  the  current  through  the  armature  increases  unduly,  the  over- 
load release  magnet  O  picks  up  its  iron  bar  and  presses,  a  little 
copper  V  spring  against  two  pins,  short-circuiting  NV,  whicK,  again 
lets  go. 


EXAM   QUESTIONS,   CHAPTER   XLVI 

'  First  catch  your  hare,'  says  Mrs.  Beeton.  So  we  start  by  considering  how 
the  electric  current  in  use  in  the  world  is  generated — only  millionths  are 
produced  any  other  way.  A  great  and  ever-increasing  proportion  of  the 
power  on  which  modern  civilization  so  largely  depends  is  electrical,  and  the 
days  when  the  subject  of  this  chapter  could  be  relegated  to  the  end  of  the  book, 
as  hard  to  understand,  and  the  solitary  instruction  be  given  '  to  push  a  magnet 
into  a  coil  of  wire,'  are  a  century  past. 

Do  every  experiment  you  can,  and  see  as  many  more  done  as  possible, 
until  you  realize  the  essential  simplicity  of  it  all.  Learn  I,  II,  III,  IV,  know 
what  they  mean  and  how  to  illustrate  them.     Learn  Ampere  and  Volt. 

A  hare  hasn't  much  of  a  tail,  and  here  it  is  only  the  latter  half  of  the  last  §. 

Note. — Strength  of  field  due  to  a  long  straight  wire  ==  2  (amperes  /lO)  -^ 
cm.  distance  from  wire. 

1.  An  electric  current  is  flowing  along  a  wire.  Making  use  of  a  compass, 
how  would  you  determine  the  direction  of  the  current  if  the  wire  is,  (a)  hori- 
zontal, (b)  vertical,  (c)  coiled  up  in  the  form  of  a  solenoid  ?  Draw  the  lines 
of  force. 

2.  What  happens  if  a  heavy -current -carrying  wire  is  dipped  in  iron  filings  ? 

3.  How  would  you  investigate  the  magnetic  field  at  different  distances 
from  a  long  vertical  wire  carrying  a  current  ?  Show  in  a  figure  the  position 
of  the  apparatus  used. 

4.  A  long  vertical  wire  carries  a  descending  current,  and  a  small  compass 
needle  is  placed  successively  N.,  S.,  E.,  and  W.,  at  equal  distances  from  it. 
How  will  the  position  of  rest  of  the  needle  and  its  time  of  oscillation  vary  at 
these  different  points  ? 

5.  Parallel  wires  carrying  currents  opposite  ways  are  observed  to  repel 
each  other  :  connect  this  action  clearly  with  Anapere's  '  swimming  '  rule, 
sketch  in  lines  of  force;  and  calculate  the  action  between  two  wires  2  cm. 
apart  each  carrying  1  amp.     (  X  2) 

6.  Describe  experiments  to  show  how  magnetic  field  is  associated  with 
current.  Under  what  conditions  would  you  rely  upon  measurements  of 
magnetic  field  as  measuring  current  ? 

7.  Describe  the  construction  of  an  electromagnet.  What  factors  affect 
the  strength  of  field  between  the  poles  ?  In  what  respect  is  it  better  than  a 
permanent  magnet  ?     (  X  3) 


FIELDS  AND  CURRENTS  OH 

8.  Give  some  account  of  electromagnetic  induction,  briefly  de«cril>ing 
illustrative  experiments.     State  laws,  and  indicate  how  they  apply  to  your 

experiments.     (  X  3) 

9.  What  are  direction  and  magnitude  of  force  acting  per  centimetre  on  a 
current-carrying  conductor  in  a  magnetic  field  ?     Illustrato  by  some  current- 

measuring  instrument. 

10.  A  coil  carrying  a  current  is  mounted  so  that  it  ran  turn  freely  about 
a  horizontal  axis.  How  will  it  set.  itself  in  the  Earth's  field  when  the  axis 
is  (1)  E.  and  W.,  (2)  N.  and  S.,  (3)  in  any  other  horizontal  direction  ? 

11.  Show  that  a  circuit  carrying  a  current  will  experience  a  turning  couple 
when  placed  in  a  magnetic  field,  and  explain  the  factors  which  dotormino 
its  magnitude.  How  is  it  applied  in  the  construction  of  a  simple  motor? 
(X  2) 

12.  Describe  the  principles  involved  in  the  construction  of  an  electric 
motor. 

What  is  meant  by  its  back  E.M.F.  ? 

If  the  impressed  volt€ige  is  100,  and  the  current  in  the  armature,  of  re- 
sistance 0-5  ohm,  is  10  ainp.,  what  is  the  back  E.M.F.  ? 

How  are  resistances  used  in  starting  and  controlling  a  motor?     (  X  3) 

13.  There  can  be  shown  to  be  a  potential  diffbronce  between  the  ends  of 
the  axles  of  a  train  in  motion,  increasing  with  the  speetl  of  the  train.  Explain 
how  this  arises. 

14.  State  the  laws  of  induced  currents  and  describe  experiments  to  illustrate 
them.  A  N.  pole  is  brought  down  to  the  middle  of  a  coil  lying  on  the  table. 
Which  way  is  the  induced  current,  and  how  would  you  pn)ve  this  ? 

15.  What  effect  is  shown  by  a  galvanometer  connected  to  a  hohzontAl 
coil  in  these  two  cases  :  (a)  vertical  magnet  placed  half-way  through  coil  is 
dropped,  (6)  horizontal  magnet  at  middle  of  coil  is  dropped. 

16.  A  cardboard  tube  about  a  foot  long  is  wound  over  with  wire  and  con- 
nected to  a  sensitive  galvanometer.  Describe  all  that  you  can  obser\*e  as  a 
magnet  is  slowly  pushed  right  through  the  tube. 

1 7.  Sketch  the  distribution  of  currents  in  a  large  copper  plate  drawn  between 
the  poles  of  a  horseshoe  magnet.  Why  cannot  a  coin  be  spun  in  a  strong 
field? 

18.  State  Lenz's  Law  of  induced  currents  and  say  exactly  how  you  woukl 
prove  it  experimentally. 

19.  A  copper  disc  is  situated  underneath  a  pivoted  magnet.  Describe  ami 
explain  what  happens  when  the  disc  is  rotated. 

20.  A  copper  hoop  is  spun  on  a  diameter  (o)  parallel  to  the  lines  of  force 
of  a  magnetic  field  (6)  perpendicular  to  them.  Why  does  it  come  to  rest 
quicker  one  way  ? 

21.  Describe  and  explain  the  nature  of  the  currents  induced  in  a  metallic 
hoop  rotating  in  the  earth's  magnetic  field  about  an  axis  in  its  own  plane. 

How  does  the  current  change  during  the  revolution,  and  how  can  direct 
current  be  obtained  from  a  coil  ?     (  X  5) 

22.  What  information  about  the  magnetic  field  of  the  earth  have  you  been 
able  to  obtain  by  rotating  a  coil  of  wire  ?     Describe  your  experiments  briefly. 

23.  Describe,  giving  suitable  diagrams  and  such  tletail  as  you  can,  the 
mechanical  production  of  Direct  Cun-ent.     (  X  2) 

24.  State  the  principles  of  electromagnetic  induction,  and  show  how  they 
are  applied  in  the  commercial  production  and  distribution  of  electricity. 


CHAPTER   XLVII 


THE   MEASUREMENT   OF  ELECTRIC  CURRENT 


§761.  Electric  currents  are  most  commonly  measured  by  their 
magnetic  effect,  using  one  or  other  adaptation  of  the  actions  de- 
scribed in  the  last  Chapter. 

The  simplest  instruments  are  Moving  Iron  Instruments  ;  of  these 
there  are  many  varieties. 

In  a  rough  pocket  pattern  a  half-inch  of  wire  nail  is  fastened 
at  right  angles  to  the  middle  of  a  straight  strip  of  watch-spring 
stretched  across  the  case.  The  iron  dips  into  a  little  coil  of  wire, 
and  when  the  current  is  sent  through  this,  it  becomes  magnetized 
and  is  drawn  farther  in,  the  motion  being  transmitted  by  a  link  to 
the  short  arm  of  a  lever,  of  which  the  long  arm  is  the  pointer  on  the 
dial. 

In  another  pattern.  Fig.  338,  two  strips  of  soft  iron  lie  side  by  side, 
parallel  to  the  axis  of  an  encircling  coil.     Current  through  this 
magnetizes  both  ahke,  and  they  repel  each  other  N.N.  and  S.S.  : 
one  is  fixed  and  the  other  swings  out  pendu- 
lum-wise,   like    the    weight    on    a     letter- 
balance,  or  the  leaf  of  a  gold-leaf  electro- 
scope,   until    magnetic    and    gravitational 
forces  balance.     This  is  a  '  gravity  control ' 
instrument,  and  must  stand  level,  but   it 
can  be  made  portable    by    balancing    the 
moving  iron    and    pointer   exactly   on   its 
pivots,  and  bringing  the  movement  under 
the  '  spring  control '  of  a  flat  spiral  spring. 
Fig.  338.  just  like  the  '  hair  spring  '  of  a  watch,  only^of 

phosphor-bronze  to  avoid  magnetic  trouble. 
Pivot-friction  has,  of  course,  to  be  kept  as  small  as  possible,  or 
the  instrument  would  stick  ;    and  then  to  prevent  it  going  on 
wagging,  some  sort  of  air- vane  (not  shown)  has  to  be  provided  to 
'  air- damp  '  such  useless  oscillation. 

In  actual  practice,  shaped  plates  of  thin  sheet  iron,  curved  to 
a  circular  arc,  and  one  moving  inside  the  other  like  your  half -closed 
hands,  replace  the  straight  strips  of  Fig.  338,  and  go  far  to  remedy 
the  excessive  inequalities  of  scale -divisions  from  which  that  would 
suffer. 

Every  moving-iron  instrument  is  a  law  unto  itself,  and  must 
be  tested  by  the  maker  against  a  standard  at  half-a-dozen  different 
currents,  and  have  a  scale  drawn  out  and  fitted  accordingly. 

612 


762] 


MEASUREMENT   OF  CURRENT 


613 


That  these  scales  are  inevitably  far  from  uniform  in  length 
of  division  is  the  chief  drawback  to  moving-iron  inHtnimentn. 
Another  is,  that,  though  quite  accurate  on  alternating  current, 
some  read  high  on  falling  direct  current,  &n  the  iron  temporarily 
retains  too  much  magnetism,  but  modern  iron  ha«  reduced  thii, 
and  moving-iron  instruments  of  perfectly  satisfactory  '  switchboard 
accuracy  '  are  obtainable  cheaply  for  every  purpose. 

§  762.  Grenerically,  electro-magnetic  current -measuring  instru- 
ments are  called  Galvanometers.  If  their  scales  are  graduated  to 
read  direct  in  amperes  they  are  Ammeters  (and  milli-amp6re  meters, 
etc.),  while  they  can  be  converted  also  into  Voltmeiers  ;  these  special 
adaptations  are  dealt  with  in  §  794. 

The  simplest  of  *  Moving-Coil  Galvanometers '  is  the  '  String 
Galvanometer,'  sketched  in  Fig.  339,  which  you  will  see  is  practically 
Fig.  330  laid  on  its  side.  The  electromagnet  shown  is  a  square 
iron  ring,  energized  by  2  or  3  h.p. 
in  the  thick  coil  at  the  back,  and 
producing  an  intense  field  across 
the  long  narrow  polar  gap  in 
front.  Down  the  length  of  this 
gap  stretches  a  glass  thread, 
1/300  mm.  thick  (1/30  as  thick 
as  this  paper),  silvered  over  to 
make  it  conductive. 

Current  passing  along  this 
'  string  '  moves  it  at  right  angles 
to  itself  and  to  the  field,  '  in  and 
out  '  of  the  gap,  and  this  move- 
ment is  watched  by  the  micro- 
scope, X  600,  which,  with  its  illuminating  condenser,  occupies  a 
hole  bored  through  the  pole-pieces.  i-  ,  .,     u      ^       • 

The  thread  is  drawn,  between  the  two  dots,  shghtly  bowed  out 
towards  the  front,  current  is  evidently  going  up  it ;  m  the  m verting 
microscope  field  it  has  moved  over  to  the  right-hand  position. 

Used  as  Electro-Cardiograph,  for  heart  currents,  the  tenMon  of 
the  string  is  made  about  100  dynes,  and  gives  a  period  of  1/lOUtJi 
second,  §  437,  and  1  micro-ampere  deflects  it  40  eyepiece- micrometer 

"^^Thi^Ts  not  astonishingly  sensitive,  but  the  spffjjj. ^***"^^^„^® 
instrument  is  that  the  string  moves  over  in  a  hundredth  of  a  ^ond 
and  stays  there,  without  vibration-it  is    dead-beat,    and  loUons 
faithfully  current  fluctuations  of  this  ^req»«^n<^.y.  . 

This  galvanometer  adapted  for  Sound-ranging,  §414.  has  haU^ 
dozen  copper  wires  1/8  mm.  diameter,  which  ^«'»^^X\  .•«^*^'?; 
phones  in  observation  posts;  their  image,  magnified  about  8  is 
recoX  on  a  continuous-moving  film  marked  m  hundredths  of 
rsecond,  and  this  is  developed  an<f  fixed  and  ready  for  measurement 
in  19  seconds. 


Fio.  339. 


614 


MAGNETISM   AND   ELECTRICITY 


[§763 


§  763.  Just  as  the  simple  but  costly  string -galvanometer  is  a 
repetition  of  Fig.  330,  so  the  ordinary  Moving-coil  Galvanometer 
is  just  Fig.  331,  the  Electro-motor,  with  only  this  essential  difference, 
that  whereas  the  motor  coil  is  encouraged  to  run  on  and  on,  the 
galvanometer  coil  is  held  in  leash  by  a  spring,  and  the  yielding  of 
this,  shown  by  the  attached  pointer,  measures  the  force  with  which 
the  current  tugs,  and  that  is  proportional  to  the  current  itself. 

There  are  minor  differences  between  motor  and  galvanometer, 
one,  naturally,  is  size  ;  the  coils  in  Fig.  340  are  somewhere  about 
half  actual  size.  The  coil  contains  from  dozens  to  hundreds  of 
turns  of  thin  copper  wire,  in  commercial  instruments  it  moves  on 
pivots  in  jewelled  bearings,  the  upper  pivot  cock,  suggested  by 
dotted  lines,  has  been  removed  in  the  figure.     Two  ways  of  leading 

current  in  are  shown  on  the  right, 
through  a  limp  silver  strip  too  thin 
and  flexible  to  interfere  measurably 
with  the  control,  which  is  that  of  a 
non-magnetic  phosphor-bronze  flat 
spiral  spring,  like  that  of  the  balance 
of  a  watch  or  small  clock,  or  else 
through  the  spring  itself. 

Just  as  in  the  actual  motor,  a 
mass  of  soft  iron  fills  in  most  of  the 
space,  and  keeps  the  magnetic  flux 
from  the  magnet,  through  its  soft-iron 
concave  pole-pieces,  radial,  uniform 
and  strong  across  the  narrow  annular 
air-gap  in  which  the  coil  moves. 
But  as  there  is  no  need  for  continuous  rotation,  and  as  its  weight 
and  inertia  would  be  a  nuisance,  the  iron  core  is  screwed  to  the 
framework,  and  the  coil  slips  round  it  without  touching. 

Instead  of  an  electromagnet,  in  all  direct-current  instruments 
a  Permanent  Magnet  of  tungsten-  or  cobalt -steel  is  used,  magnetized, 
not  up  to  its  maximum,  but  to  the  utmost  constancy.  Upon  this  the 
instrument  directly  depends,  for  from  §  749,  you  see  that  the  force 
on  each  straight  side  of  the  coil,  where  it  passes  through  the  polar 
gap,  is  the  product  of  its  length,  the  no.  of  wires  in  it  X  the  current 
in  each,  and  the  field- strength.  The  total  turning  moment  about 
the  pivot-axis  is  this,  X  2  for  the  two  sides  x  distance  of  either  side 
from  the  axis. 

Or  all  put  together.  Area  of  coil  X  turns  X  current  X  field. 
These  dainty  moving  parts  can  carry  no  more  than  a  tenth  of 
an  ampere,  the  bulk  of  larger  currents — no  matter  how  large — is 
by-passed  in  a  Shunt,  §  781. 

§  764.  In  more  delicate  laboratory  moving-coil  galvanometers 
(which  you  should  examine)  the  coil  is  often  squeezed  into  a  narrow 
hank,  filling  a  linear  polar  gap,  central  soft  iro^  being  done  away 
with,  and  control  is  by  the  stiffness  of  the  thin  suspending  strip 


Fig.  340. 


§  765]  MEASUREMENT   OF   CURRENT  615 

of  phosphor-bronze.  The  current  also  comes  down  this  and  out 
by  a  limp  silver  heUx  at  the  bottom,  Fig.  340,  upper  inset. 

Lamp  and  scale.  Sensitive  deflection  instruments  cannot  bo 
loaded  with  a  pointer,  but  carry  instead  a  Httle  mirror  (at  top  of 
coil  in  figure)  to  reflect  light  from  a  brightly  illuminated  slit,  or 
a  '  moon,'  back  on  to  a  paper  mm.  scale.  Slit  and  scale  being  1  m. 
away,  the  mirror  is  made  concave  of  1  m.  radius  of  curvature,  so  that 
all  light  from  the  slit  meets  it  radially,  and  returns  to  form  a  focussed 
image.  Figs.  176,  211.  Since  the  reflected  ray  moves  through 
twice  the  angle  turned  by  the  mirror,  this  is  equivalent  to  a  pointer 
2  m.  long. 

With  a  resistance  of  100  ohms,  and  period  of  swing  4  sec.,  gal- 
vanometers of  this  type  deflect  as  much  as  400  mm.  for  1  micro- 
ampere. 

Moving-coil  instruments  altogether  excel  moving-iron  instniments 
in  accuracy,  because  they  do  not  depend  on  the  magnetic  idiosyn- 
crasies of  iron  (and  this  applies  also  to  loud-speakers).  Their  scaleit 
are  long  and  almost  equally  divided  throughout,  and  their  blade- 
on-edge  pointers  deserve  to  be  read  with  a  magnify ing-glass, 
keeping  them  covering  their  reflections  in  an  anti-parallax  mirror 
beneath. 

They  have  two  great  advantages  over  the  moving-magnet 
galvanometers  to  be  described  later,  first,  that  since  their  controlling 
field  is  the  enormously  strong  one — perhaps  5000  gauss,  of  a  per- 
manent magnet — they  are  utterly  indifferent  to  stray  magnetic 
fields  ;  and  second,  that  as  their  coils  move  in  this  strong  field, 
currents  are  induced  in  them,  which,  by  Lenz, '  damp  '  their  swinging 
(cf.  the  copper  ring  of  §751),  and  they  can  be  made  very  nearly 
'  dead-beat,'  which  is  a  great  comfort  to  the  user,  and  lengthens 
his  life,  unless  it  be  overdone  so  that  they  crawl. 

§  765.  Various  current  meters.  Moving-coil  instruments  for 
Alternating  Current  must  have  their  field-magnets  of  laminated 
iron,  and  magnetized  by  the  current  itself,  so  that  the  force  oc 
(current )2  and  is  always  in  the  same  direction.  The  most  accurate 
ones  dispense  with  all  iron,  on  account  of  its  magnetic  peculiarities, 
Fig.  288,  and  with  all  lumps  of  metal,  for  fear  of  eddy  currenU, 
and  simply  hang  one  coil  in  the  field  of  a  larger  one. 

Very  irregular  jerky  currents,  such  as  those  supplied  by  an 
induction  coil  to  an  X-ray  tube,  are  '  smoothed  out  '  by  letting 
them  charge  a  condenser  packed  in  the  back  of  the  case,  this  dis- 
charging more  steadily  through  the  milli-ammeter  in  parallel  with  it. 

If  the  currents  are  irregular,  and  alternating,  such  as  those  through 
a  telephone,  they  can  be  sent  through  a  very  small  length  of  extreniely 
thin  resistance  wire,  which  they  heat  up  proportionally  to  their 
total  energy  value,  §  815.  In  contact,  or  almost  m  contact,  with 
this  '  heater  '  is  a  very  small  and  delicate  thermo- junction,  §  799. 
and  this  sends  a  current  into  the  micro-ammeter  proportional  to 
the  heating. 


616 


MAGNETISM   AND   ELECTRICITY 


[§765 


The  Hot-wire  Ammeter,  which  is  quite  non-magnetic,  either 
depends  on  the  expansion  of  wire  heated  by  the  current,  §  816,  or 
on  its  efifect  on  a  thermo-junction,  §  799. 

In  another  special  alternating-current  ammeter  two  coils  inter- 
secting at  right  angles  produce  a  '  rotating  field,'  which  drags 
round  an  enclosed  aluminium  disc  against  spring-control :  it  is 
the  2 -phase  motor- starter  of  §  831  simplified. 

Self  recorders.  Any  instrument  can  have  its  pointer  tipped 
with  a  pen,  and  be  told  to  write  on  a  clockwork-driven  chart,  but 
the  friction  of  the  pen  is  fatal  to  delicate  action. 

In  the  '  siphon  recorders  '  which  first  wagged  out  the  messages 
of  Atlantic  cables,  William  Thomson,  Lord  Kelvin,  electrified  the 
capillary  pen,  so  that  it  sprayed  the  ink  across  to  the  paper.  But 
cable-stations  are  damp  ;   and  that  was  superseded  by  making  the 

paper  oscillate  rapidly,  so  that  the 
record  is  a  line  of  close  dots.      A 
slow   chart -writer   of   the   present 
day  is  outlined  in  Fig.  341  :   the 
motor-clockwork   drags  the   chart 
over   a   prism-edge,    above   which 
stretches  a  typewriter-ribbon,  slow- 
ly fed  along.    Above  this  the  knife- 
edged  pointer   of   the   instrument 
swings  free  the  whole  width  of  the 
chart,  and  hanging   over  all  is  a 
heavy  '  boom.'     Every  half -minute  the  clock  drops  this  boom  on 
to  the  pointer,  chopping  it  down  on  the  ribbon,  and  so  dotting  the 
chart. 

Since  the  length  of  the  chart  represents  Time,  and  the  height  of 
the  curve  above  a  zero  base-line  is  Current,  the  product,  the  area 
under  the  curve,  represents  total  Quantity  of  electricity  passed, 
amperes  x  seconds  =  Coulombs  (or  Ampere-Hours  form  a  more 
sizeable  commercial  unit). 


Fig.  341. 


§  766.  This,  however,  involves  the  labour  of  replacing  and 
measuring-up  charts,  and  often,  especially  for  domestic  supply, 
Electric  Quantity  Meters  are  more  serviceable. 

These  are  just  electro-motors,  the  size  of  peg-tops,  of  types  suitable 
for  direct  (§  750)  or  alternating  (§  831)  current,  as  the  case  may  be, 
and  upon  them  the  Company  lavishes  gold  commutators,  iridium 
pivots,  sapphire  caps  and  other  anti-friction  contrivances,  lest  they 
stick,  and  the  customer  get  small  currents  for  nothing. 

But  that  facilitates  their  running  away,  once  started,  and  a  brake 
must  come  into  play.  This  is  a  thin  aluminium  '  fly-wheel ' 
disc,  which  rotates  between  the  jaws  of  little  permanent  magnets, 
their  lines  of  force  piercing  the  disc.  Eddy  currents  are  therefore 
induced,  proportional  to  the  speed,  and  are  absorbed  in  the  resistance 
of  the  metal.  This  keeps  the  speed  proportional  to  the  current  in 
the  meter  (it  is  the  fluid  friction  of  §  335),  and  accordingly  the  number 


§767] 


MEASUREMENT   OF  CURRENT 


«17 


of  revolutions  recorded  during  the  Quarter,  on  the  counting  diaU, 
is  proportional  to  the  total  Quantity  of  electricity  supplied  you, 
and  is  the  basis  of  your  bill. 

In  some  A.C.  meters  the  aluminium  brake-disc  itself  is  the  motor 
armature,  being  driven  by  a  two-phase  starter  device  as  in  §831, 
but,  like  many  degenerate  parasitic  organisms,  so  reduced  as  to 
be  almost  beyond  recognition. 

Small  Quantities  of  electricity,  such  as  single  charges  of  con- 
densers, are  proportional  to  the  kicks  they  produce  when  suddenly 
discharged  through  any  very  slightly  damped  galvanometer,  called 
in  this  use  a  Ballistic  Galvanometer.  They  can  \ye  shown  to  be 
equal  in  Coulombs  to  (Time  of  complete  oscillation  of  galvanometer 
/27r)  times  the  current  in  Amperes  which  causes  a  steiuiy  deflection 
equal  to  the  kick. 

§  767.  But  with  all  these  instruments  we  have  not  yet  arrived 
at  one  which  can  tell  us  the  absolute  value  of  a  current  in  ami)6rei» 
without  having  first  of  all  been  set  by  comparison  with  some 
manufacturer's  standard  :  all  of  these  in  case  of  accident  would  have 
to  go  back  to  the  maker  for  repair  and  recalibration.  We  must 
look  further. 

The  standard  current -measurer  of  this  country  is  the  Ampdre 
Balance,  in  the  coils  of  which  an  ampere,  flowing  through  a  field 
of  its  own  making,  balances 
a  weight  of  7  gm.  The 
second  question  its  eminent 
calculator  ever  asked  me 
was  whether  I  knew  of  a 
really  reliable  book  of  ten- 
figure  logarithms  :  let  us 
pass  on  to  the  older  stan- 
dard, the  type  of  all  mov- 
ing-magnet instruments,  the 

Tangent  Galvanometer 

Looking  at  Fig.  328,  you 
can  see  how  a  magnet  placed 
in  the  plane  of  an  encircling 
ring-coil  of  wire  will  be 
twisted  out  of  it  when  a 
current  flows.  .    x?  1 1  j 

Looking  also  at  Fig.  320,  you  see  that  the  magnetic  field  due  to 
a  ring-coil  is,  jtist  in  the  middle,  uniform  in  strength  and  at  right 
angles  to  the  plane  of  the  ring. 

Accordingly,  if  we  place  round  a  small  compass  needle  a  targe 
vertical  ring-coU  of  wires,  with  its  plane  magnetic  north  and  south, 
we  shall  have  the  magnetic  forces  at  right  angles  considered  in 
§  687,  and  this  arrangement  forms  a  Tangent  Galvanometer. 

In  Fig.  342  let  m  be  the  strength  of  the  neetUe's  pole,  practically 
at  the  centre  of  the  ring  of  radius  r  cm.,  composed  of  n  turns  of  wire 


Fio.  342. 


618  MAGNETISM   AND   ELECTRICITY  [§  767 

carrying  a  current  A  amperes,  which,  be  it  remembered,  is  only 
A/10  c.g.s.  units  of  current,  §  749. 

All  over  the  sphere  of  radius  t,  surrounding  the  pole  m,  the  mag- 
netic field  is  radial,  and  of  strength  m/r^,  §  683,  i.e.  this  number 
of  unit  magnetic  lines  sticks  out  of  each  sq.  cm.  of  it,  like  the  spikes 
on  a  horse  chestnut. 

If  the  encircling  belt  of  n  turns  of  wire,  each  2v:r  cm.  long,  were 
to  move  1  cm.  at  right  angles  to  itself,  as  if  slipping  off  the  imagined 
sphere,  there  would  therefore  be  2iirn  x  w/r^  cuttings  of  unit  lines 
and  wire. 

Therefore,  by  §  749,  the  coil  is  acted  on  by  a  force  in  this  direction 
(say,  West)  =  2Tzrn  x  mlr'^  x  A/10  =  2Tznm  A/lOr  dynes,  and,  of 
course,  the  equal  reaction  on  the  pole  drives  it  East. 

The  horizontal  component  of  the  earth's  field,  H,  pulls  the  pole 
North  with  force  Hm  dynes  ;  consequently  there  results,  as  in 
Fig.  294,  a  deflection  D  away  from  the  North  such  that 

^       2Tznm  A/lOr       2TznA. 

tanT>  = ^j— ^ =  ttt-tj 

Km  lOr  H 

or  A  amperes  =  ^ —  H  tan  D 

For  a  numerical  example  see  §  774,  II.  Notice  that  the  strength 
of  the  magnet  has  disappeared,  so  that  variations  in  this  have  no 
effect  on  accuracy. 

§  768.  In  the  laboratory  you  will  learn  to  recognize  the  Tangent 
Galvanometer  by  its  large  coil,  and  short  pivoted  magnet  provided 
with  a  long  light  pointer  stuck  on  at  right  angles,  i.e.  magnetic 
E.  and  W.,  so  that  when  you  have  set  the  coil  Magnetic  N.  and  S., 
'  in  the  magnetic  meridian,'  parallel  to  the  magnetic  '  needle,'  the 
scale  zeros  come  under  the  ends  of  the  pointer,  which  is  always 
well  clear  of  the  over-shadowing  coil.  The  numbers  marked  between 
the  terminals  are  the  n  turns  of  the  coils. 

You  learn  to  tap  the  coil  on  top,  so  as  to  minimize  pivot-friction, 
before  taking  a  reading  :  if  there  is  much  sticking,  the  pivot  wants 
sharpening  up,  to  a  60°  cone,  with  a  stone.  You  learn  also  to  keep 
your  eye  vertically  above  the  pointer,  helped  by  a  mirror  beneath 
it,  and  to  read  both  ends,  and  then  to  reverse  the  current  and  repeat, 
getting /oi^r  readings  every  time,  and  taking  the  mean. 

If  Tangent  Galvanometers  were  made  as  accurately  as  ship's 
compasses,  this  four-reading  technique  would  be  unnecessary, 
but  they  have  long  since  been  superseded  in  all  commercial  applica- 
tions by  the  vastly  more  convenient  instruments  we  have  been 
describing  in  §§  761  to  765.  They  survive  only  as  the  Students' 
Standard  Galvanometer  for  the  absolute  measure  of  current  in 
Amperes — I  have  told  you  the  alternative,  it  took  years  to  build 
and  cost  thousands — and,  within  limits,  the  rougher  they  are  made 
the  better ;    so  that  you  can  realize  that  all  you  have  to  do  is  to 


§  769]  MEASUREMENT   OF  CURRENT  619 

stick  a  little  compass  at  the  middle,  by  eye,  of  a  child'H  wuodfii 
hoop,  with  a  few  turns  of  wire  wound  on  it,  and  to  set  that  vertical, 
and  north  and  south,  and  then  to  take  these  four  readings,  and  make 
little  mechanical  adjustments  until  you  have  {lersuaued  them  to 
be  reasonably  equal,  and  there  is  your  Absolute  Standard  Measurer 
of  Current,  for  the  accuracy  of  which  you  are  beholden  to  nobody 
but  yourself,  all  its  faulty  centring  and  skew  right  angles  rendered 
harmless  by  your  patient  method  of  procedure. 

You  see  that,  without  knowing  H,  the  relative  values  of  currenta 
are  proportional  to  the  tangents  of  their  mean  deflections.  Chooee 
the  coil  of  n  turns  which  gives  you  sizeable  deflections  with  the 
currents  available,  but  do  not  exceed  60°,  for  tangents  then  increase 
too  fast  for  accurate  reading. 

You  will  probably  be  asked  to  plot  tan  D  against  the  reciprocal 
of  the  Resistance  of  the  Circuit,  which  is  its  Conductance,  §772. 
or  its  current-carrying  ability.  Do  not  let  anybody  persuade  you 
to  plot  cot  D,  for  that  would  be  the  reciprocal  of  a  current,  which 
is  sheer  nonsense. 

To  make  your  measurements  absolute  you  must  know  H,  and  you 
can  measure  it  with  the  self-same  compass-box  and  a  metre  scale, 
propped  on  books  or  blocks,  and  a  small  bar-magnet,  and  a  watch, 
by  the  method  of  §  692.  If  you  hdve  actually  taken  the  trouble 
to  try  this  method,  you  have  found  that  its  apimrently  formidable 
mathematical  difficulties  are  a  mere  bogey. 

With  some  tangent  galvanometers  the  coils  are  concealed,  and 
neither  n  nor  r  can  be  discovered,  but  there  is  always  a  Gal- 
vanometer Constant,  G,  measurable  once  for  all  by  comparing  a 
deflection  with  that  of  an  exposed  coil  galvanometer  carrying  the 
same  current— at  a  non-interfering  distance— and  in  general  one 
may  write 

A  =  5  tan  D. 

And  when  always  used  at  one  place,  H/G  may  conveniently  bo 
calculated  out  as  the  Reduction  Factor,  K,  by  which  one  converts 

the  tangent  of  the  observed  deflection  straight  mto  amperes 

A  =  K  tan  D 

The  Reduction  Factor  of  anv  instrument  whatever  is  the  factor 
by  which  you  multiply  its  indication  to  get  what  you  want,  eg. 
the  price  per  pound  multiplying  the  weight  on  the  scale  gives  the 
cost  of  the  sirloin.  In  this  present  case  it  is,  of  course,  complicated 
by  having  to  look  up  tangents  first. 

S769.  The  tangent  galvanometer  is  by  no  means  sensitive: 
how  can  it  be  when  current  and  pole  are  kept  so  far  apart  r 
Never  attempt  to  use  it  for  '  bridge  '  work,  or  any  null  method^ 
for  there  you  want  something  distinctly  more  sensitive ,    lU.uuu 

times,  say. 


620  MAGNETISM   AND   ELECTRICITY  [§  769 

To  make  it  more  responsive  it  will  plainly  be  necessary  to  wind 
on  more  turns  of  wire,  and  closer  to  the  magnet,  thus  increasing  n 
and  diminishing  r  in  the  formula,  and  allowing  A  to  be  smaller. 

Unfortunately,  the  needle -poles  have  now  got  into  parts  of  the 
field,  near  the  wires,  Fig.  320,  where  the  lines  are  far  from  equi- 
spaced  and  parallel ;  the  Tangent  Law  fails,  and  there  is  none  to 
replace  it ;  the  scale  has  to  be  calibrated  step  by  step. 

However,  in  the  majority  of  these  nondescript  galvanometers, 
one  assumes  that  the  current  is  simply  proportional  to  the  deflection 
for  some  few  divisions  either  side  of  zero,  especially  with  lamp  and 
scale,  and  usually  that  is  near  enough,  and  all  that  is  wanted. 

Having  thus  taken  the  first  two  steps  in  the  evolution  of  a 
Sensitive  moving-magnet  Galvanometer,  and  being  unable  to  vary 
10,  and  2tc,  let  us  attack  the  next  possibility,  H  :  Weaken  the 
controlling  field  of  the  Earth. 

This  can  be  done  by  laying  a  control  magnet  near  the  galvano- 
meter, so  that  the  needle  is  near  a  Neutral  Point,  Fig.  291,  of  the 
resultant  field.  You  know  when  you  have  done  so,  because  the 
needle  swings  much  more  slowly,  §  691. 

Unfortunately,  this  lays  the  uncontrolled  needle  more  than  ever 
open  to  the  influence  of  accidental  stray  fields — a  pocket-knife, 
or  a  passing  car — and  the  zero  begins  to  wander.  '  Give  me  a 
galvanometer  with  a  mind  of  its  own,'  says  Lord  Rutherford 
'  I  recollect  deciding  between  this  and  that  theory  because  the 
galvanometer  deflected  a  millimetre  and  a  half  instead  of  a  milli- 
metre.' 

An  Astatic  Pair  of  needles  is  better.  They  are  fastened  together, 
pointing  opposite  ways,  as  in  Fig.  343  A,  and  hanging  by  a  cocoon 
fibre  of  silk.  If  they  are  equally  strong 
magnets,  the  earth's  actions  in  them  are 
equal  and  opposite,  and  they  have  no 
standing  place  at  all — a-static — while  a  coil, 
wound  round  one  of  them  only,  has  its  full 
effect. 

In  that  ancient   pattern,    however,    the 

lower    needle    often    gets   badly  weakened 

by   the   accidental   heavy    discharges    that 

Fig.  343.  students  have  been  known  to  send  through 

galvanometers,    and    then   one    would    be 

better   off  without  the  upper  needle. 

In  the  modern  Broca  astatic  pair,  two  needles  are  fastened 
side  by  side  and  suspended  vertically,  Fig.  343  B.  They  are  mag- 
netized to  have  N.  and  S.  '  consequent  poles  '  in  the  middle,  having 
therefore  opposite  poles  of  half -strength  at  the  ends.  The  coil,  in 
two  flat  halves,  surrounds  the  middle  poles  only.  You  can  see 
easily  how  much  better  this  arrangement  is  in  every  way  :  control 
is  provided  by  a  magnet  fixed  nearer  one  end  of  the  pair. 

Finally,  tan  D  can  be  made  the  most  of  by  using  Lamp  and, 
Scale  as  in  §  764. 


§  769]  MEASUREMENT   OF   CURRENT  621 

Moving-magnet  Galvanometers  all  being  really  compasses,  you 
must  stand  them  the  way  they  wish.  They  are  usually  about  as 
sensitive  as  moving-coil  galvanometers  (fortunately,  you  will  never 
need  the  ultra-sensitive  ones),  they  are  cheaper  and  more  easily 
mended,  but,  excepting  only  the  best  astatics,  are  entirely  at  the 
mercy  of  the  outside  magnetic  field.  They  are  also  much  less  con- 
trollable in  their  swinging,  the  only  really  practicable  damping 
being  by  air  vanes  attached  to  the  needle,  so  that  they  waste  a 
lot  of  time  settling  down  to  rest. 

All  are  useless  with  alternating  current. 

It  is  curious  that  the  great  inventor  of  the  moving-coil  siphon- 
recorder  never  applied  it  to  laboratory  or  engineering  purposes, 
but  developed  many  moving- magnet  patterns.  True,  there  was 
no  great  laboratory  demand,  his  own  pioneer  one  of  Scotland 
grew  out  of  a  converted  cellar  ;  but  for  an  electric-lighting  engineer 
in  his  '  central  station  '  (often  driven  by  a  portable  engine  of  farm 
type)  to  have  to  retire  at  least  a  score  yards  from  his  leaky  dynamo, 
there  adjust  a  sliding  compass-box  on  a  modified  t.g.,  and  then 
look  up  the  meaning  of  its  two  scale-readings  on  a  cross-reference 
ready-reckoner — whereas  his  grandson  gives  one  glance  at  the  close- 
by  switchboard — will  show  you  well  enough  why  the  tangent - 
galvanometer,  inevitable  in  the  junior  laboratory,  nowadays  never 
leaves  it. 


EXAM   QUESTIONS,   CHAPTER   XLVII 

This  chapter  goes  on  to  apply  the  principles  of  the  preceding  chapter  to 
the  measurement  of  the  currents  therein  generated.  The  first  half-dozen 
§§  are  descriptive  of  instruments  you  can  find  in  the  laboratory,  etc.  :  don  * 
attempt  to  learn  them  unless  the  instruments  are  in  front  of  you.  But  at 
§§  767,  768  you  absolutely  must  sit  up  and  take  notice  :  it  is  your  absohite 
current  measurer. 

1  Describe  the  construction  and  action  of  a  moving-coil  galvanomoter. 
How  is  the  motion  '  damped,'  and  what  other  advantages  has  it  over  moving- 
magnet  or  moving-iron  instruments  ?     (  X  2) 

2.  Describe  any  pattern  of  sensitive  mirror  galvanometer,  explaining  how 
it  satisfies  essential  conditions.     (  X  4) 

3.  Define  the  terms  MiUi-volt  and  Micro-ampere 

Explain  the  construction  and  mode  of  action  of  an  mstrument  which  could 
be  used  for  measuring  one  of  these  quantities. 

4  Sketch  the  lines  of  magnetic  force  due  to  «»  «»e^^"^*;"7^"^„^^"^ 
round  a  circular  coil,  neglecting  the  earth's  field.  How  would  the  lines  near 
thrcentre  be  affect^  by  a  field  parallel  to  the  nlane  of  the  coil  ?  Explain 
the  method  of  determining  the  cuii^nt  by  the  defection  of  a  magnetic  needle 
suspended  at  the  centre. 


622  MAGNETISM   AND   ELECTRICITY 

5.  Explain  the  statement  that  the  horizontal  intensity  of  the  Earth's 
magnetic  field  in  a  certain  place  is  0-18  gauss. 

How  would  you  compare  it  with  the  strength  of  the  magnetic  field  at  the 
centre  of  a  circular  coil  carrying  an  electric  current  ? 

6.  A  current  flowing  in  a  circular  coil  produces  a  field  at  its  centre  of  4  units. 
The  same  current  in  another  coil  of  half  the  radius  produces  a  field  of  5  units. 
Compare  the  lengths  of  wire  in  the  two  coils. 

7.  What  relation  exists  between  a  current  and  its  field  ?  Which  produces 
the  greater  field  at  its  centre,  a  ring  of  5  turns  of  wire  20  cm.  diam.,  carrying 
5  amps.,  or  one  of  10  turns,  10  cm.  diam.,  carrying  1  amp.  ? 

8.  A  tangent  galvanometer  has  two  coplanar  coils,  one  of  3  turns  of  radius 
10  cm.,  and  the  other  of  2  turns  of  radius  15  cm.  Calculate  the  deflections 
produced  by  a  current  of  1  ampere  flowing  through  both  coils,  when  their 
effect  is  to  deflect  (a)  in  the  same,  and  (6)  in  opposite  directions.     H  =  0-18. 

9.  The  coil  of  a  galvanometer  is  placed  at  right  angles  to  the  magnetic 
meridian,  and  a  steady  current  is  passed  through  it.  The  magnet  when  set 
oscillating  makes  15  complete  vibrations  per  minute,  and  9  in  the  same  time 
when  the  current  is  reversed.  Compare  the  magnetic  force  due  to  the  current, 
with  the  earth's  field.     (  X  4) 

10.  Explain  the  principles  underlying  the  tangent  galvanometer,  and 
deduce  an  expression  connecting  the  current  with  the  deflection. 

Show  how  the  instrument  may  be  made  more  sensitive,  and  give  a  diagram 
of  a  galvanometer  embodying  these  suggestions.     (  X  2) 

11.  Define  the  electromagnetic  c.g.s.  units  of  current  and  potential,  and  the 
ampere  and  volt. 

Give  the  theory  and  use  of  the  tangent  galvanometer.  Why  is  the  needle 
short  ?     (  X  2) 

12.  A  current  flowing  through  a  tangent  galvanometer  of  10  turns,  radius 
8  cm.,  deflects  45°,  when  H  =  0-18.  What  alterations  would  you  make 
to  get  this  same  deflection  for  a  milliamp.  ?     (  X  2) 

13.  What  is  the  *  reduction  factor '  of  a  galvanometer?  1-2  amp.  is  sent 
through  a  tangent  galvanometer  of  10  ohms,  shunted  by  a  wire  of  5  ohms. 
The  deflection  is  45°.  If  the  radius  is  11  cm.  and  there  are  7  turns,  calculate 
H.     (  X  2) 

14.  A  2-0-volt  battery  gives  a  deflection  of  45°  on  a  tangent  galvanometer 
through  200  ohms,  and  this  is  reduced  to  30°  by  an  extra  300  ohms.  Calculate 
the  reduction  factor. 


CHAPTER   XLVIII 
RESISTANCE 


§  771.  As  stated  in  §  742,  a  complete  Circuit  of  good  conducting 
or  '  low-resistance  '  material,  preferably  Metal,  properly  insulatea 
with  '  non  '-conductor  of  exceedingly  *  high  resistance,'  to  prevent 
accidental  contacts  or  '  short  circuits,'  is  almost  indispensable  in 
Current  Electricity  :  when  metallic  connection  is  broken — *  the 
circuit  opened,'  by  intention,  or  by  a  '  bad  contact,'  which  will 
account  for  nineteen-twentieths  of  your  troubles — the  current 
cannot  pass. 

Consequently,  substantial  binding  screws  and  keys  and  switches 
are  employed  to  ensure  good  contacts  through  which  the  electronu 
can  stream  unchecked  :  with  a  variety  of  these  you  must  familiarize 
yourself  in  the  laboratory.  Contact  surfaces  must  be  neither 
lacquered,  i.e.  varnished  with  insulating  shellac,  as  is  most  visible 
brassware,  nor  corroded  with  non-conducting  oxide,  nor  dirty  with 
grease,  etc.,  but  must  be  kept  clean  and  smooth  in  the  usual  ways. 

Wire  ends  must  be  scraped,  if  at  all  dirty.  Not  a  trace  of  insulating 
cotton  or  silk  or  enamel  must  enter  the  binding-screw,  or  you  may 
easily  get  no  contact  at  all.  Hook  the  wire  round  the  stem  of  the 
binding-screw,  but  do  not  bring  its  tail  round  over  it,  or  when  you 
screw  down  hard,  tail  and  wire  are  apt  to  cut  each  other  ot!  :  hook 
two  wires  on  opposite  ways,  they  seldom  cut.  Hooking  wire-ends 
together  is  quite  useless,  even  twisted  they  are  suspect,  the  twisting 
must  be  long  and  tight  and  unshakeable.  Do  not  coil  temporary' 
wiring  into  pretty-pretty  '  spirals,'  but  get  rid  of  slack  by  folding 
it  '  sheepshank '  style.  Soldered  joints  must  be  scrupulously 
cleansed  from  corrosive  flux. 

§  772.  Now,  however  good  the  conductor  may  be,  the  passage 
of  a  considerable  electric  current  will  presently  make  it  warm. 

This  is  not  a  conversion  of  '  Electricity  '  into  *  Heat,'  for 
wherever  a  current -measuring  instrument  is  put  into  the  circuit 
it  will  show  the  same  current.     No  electricity  is  lost. 

It  is  a  production  of  heat  by  a  dissipation  of  electrical  Energ\-  : 
the  electromotive  force  driving  the  current  gets  lessened  in  the 
conductor :  unless  E.M.F.  is  kept  up,  by  power  spent  in  the 
dynamo,  or  chemical  activity  in  the  battery,  the  current  stops, 
much  as  a  body  moving  in  a  viscous  fluid  stops  as  soon  as  the 
driving  force  ceases. 

Thus  a  sort  of  friction  dogs  the  motion  of  electricity.  In  metals 
(solid  or  liquid)  it  is  comparable  to  fluid  friction,  for  the  slightest 
electromotive  force  can  always  cause  a  feeble  current;    nor  will 

623 


624  MAGNETISM   AND   ELECTRICITY  [§  772 

any  resistance  that  the  metal  can  offer,  however  great,  bring  about 
a  total  stoppage  of  current. 

In  other  liquids,  and  in  gases,  it  is  more  comparable  to  solid 
friction,  for  below  a  certain  starting  pressure  no  current  will  move. 

In  Metals,  indeed,  the  most  careful  experiments  have  proved  that 
The  current  flowing  in  a  conductor  is  exactly  proportional  to  the 
driving  potential  difference  (or  electromotive  force)  between  the  ends 
of  the  conductor,  provided  that  the  conductor  is  kept  at  a  constant 
temperature.    This  is  called  Ohm's  Law  of  Electrical  Conduction. 

With  gases,  every  alteration  of  volume  alters  the  temperature 
(§202),  yet  the  simple  Boyle's  Law  which  postulates  unchanging 
temperature  is  found  useful ;  in  just  the  same  way  here,  although 
the  slightest  current  raises  the  temperature  of  the  conductor  and 
upsets  the  proportionality,  Ohm's  Law  is  vastly  convenient.  So 
much  so  that  alloys  were  insistently  sought  for  in  which  moderate 
heating  should  not  upset  the  proportionality,  and,  now  found,  they 
provide  most  useful  electrical  standards.  So  much  so,  that  although 
it  applies  in  the  first  instance  only  to  Metals,  Liquid  Solutions 
have  been  brought  in  under  cover  of  a  fixed  starting  handicap, 
the  polarization  E.M.F.,  and  Insulators  are  treated  as  obeying  it, 
although  all  sorts  of  reservations  have  to  be  made  about  them 
individually.     But  don't  apply  it  to  gases. 

The  better  the  conductor  the  larger  the  current,  and  we  can 
get  rid  of  '  proportional  to  '  by  defining  a  conducting  power,  or 
*  Conductance,'  constant  for  a  given  conductor  at  a  given 
temperature 

Current  =  electromotive  force  X  conductance , 

though  it  is  more  usual  in  physical  laboratories  to  talk  about  the 
resisting  power  or  *  Resistance  '  offered  to  the  passage  of  the  current. 
This  is  evidently  the  Reciprocal  of  Conductance,  for  halving  the 
resisting  power  means  doubling  the  conducting  power,  etc.,  so  that 

Current  —  electromotive  force  -f-  resistance 
C-|      CR  =  E      orR  =  5 

§  773.  We  have  already  defined  units  of  current  and  electro- 
motive force,  we  must  therefore  define  the  Unit  of  Resistance 
as  follows  : — 

When  unit  electromotive  force  applied  to  the  ends  of  a  conductor 
causes  unit  current  to  flow  through,  the  conductor  possesses  Unit 
Resistance. 

For  practical  purposes,  as  stated  in  §  752,  the  Volt  is  the  Unit  of 
Electromotive  Force  and  is  100  million  (10^)  times  the  fundamental 
unit  there  defined.  Roughly  speaking,  it  is  a  little  less  than  the 
potential  difference  between  the  metals  of  the  original  Volta's  cell, 
where  copper  and  zinc  dip  in  salt  water. 


§774] 


RESISTANCE 


026 


The  Ampere  is  the  practical  Unit  of  Current  and  was  most  un- 
fortunately chosen  as  one-tenth  the  fundamental  unit  of  §749 
(which  is  hence  the  decampere). 

The  unit  of  resistance  is  the  Ohm.  If  1  volt  applied  to  the  ends 
of  a  wire  causes  1  ampere  to  flow  in  it,  the  wire  has  a  resistance  of 
1  ohm. 

Amperes  =  -p^ —       Ampires  x  Ohms  =  Volts      Ohms  =  -3 5 — 

■^  Ohms  ^  Amperes 

The  Ohm  has  one  decided  advantage  over  the  Ampere  and  Volt. 
It  is  the  property  of  a  portable  piece  of  metal,  and,  once  made  up, 
there  is  no  need  to  keep  turning  cranks  and  things  to  get  it.  Con- 
sequently the  electrician's  methods  have  mostly  been  devised  to  lead 
to  measurements  in  terms  of  resistances,  just  as  the  chemist  works 
down  to  his  box  of  weights. 

A  Megohm  is  a  million  Ohms  :  A  Microhm  is  a  millionth  of  an 
Ohm. 

§  774.  An  absolute  determi- 
nation of  Ohm,  Volt  and 
Ampdre  ;  H,  Dip,  etc.  In- 
stead of  making  any  attempt 
to  describe  the  elaborate  Am- 
pere Balance  and  Ohm  Ma- 
chine of  the  National  Physical 
Laboratory,  here  is  a  lecture 
experiment  made  10/5/32  with 
ordinary  apparatus  that  came 
to  hand,  which  does  the  whole 
business. 

In  Fig.  344,  an  '  earth  in- 
ductor '  coil  of  42  turns  of 
wire  20  cm.  radius  .*.  of  total 
area  43  X  tt  x  20^  =  53,000 
sq.  cm. 

I.  was  rotated  about  a 
vertical  axis  at  5/4  rev. /sec. 
(listening  to  watch  ticks), 
each  revolution  cutting  all  the 
lines  of  the  earth's  field  H, 
twice,  and  therefore  inducing 
in  circuit  an  E.M.F.  = 
53,000  xHx  2  X5/4-M08 
=  000132  H  volts. 

This  was  led  off  through  the 
split-ring  commutator  to  a 
moving-coil  galvanometer,   as 


Fio.  344. 


movmg-cou  gaivanuuict..i,  «o  a  current  of  value  00013-  HR 
amperes,  causing  scale  deflection  d,  R  being  the  resistance  of  (coil 
and  galvanometer  curcuit)  in  Ohms,  nau^  to  be  determined. 


626  MAGNETISM   AND    ELECTRICITY  [§  774 

II.  The  coil  was  stopped  N.  and  S.,  and  a  compass  put  at  its 
centre,  converting  it  into  a  Tangent  Galvanometer. 

In  series  with  the  m.-c.  galvanometer  is  put  a  large  resistance, 
known,  by  ordinary  comparison  methods,  §§  788,  797,  to  be  100,000 
times  the  small  Shunt,  §  781. 

A  current  from  a  battery  is  regulated  by  rheostat  until  the  m.-c. 
galvanometer,  now  getting  1 /100,000th  of  it,  again  reads  deflection 
d.     The  tangent  galvanometer  read  56J°  and  therefore  the  current 

7z tan  D  =  — ?5 TTi —  X  1-51  =  1-145  H  amperes. 

This  is  100,000  times  the  original  current 

.000132  H  ^  ^^^^^  ^  ^  j^g  jj 

.*.  R  =  115-5  ohms. 

III.  The  dotted  metre  scale  was  fixed  E.  and  W.,  and  the  M  and 
H  determination  detailed  in  §  692  was  performed  there  and  then, 
and  gave  H  =  0-192  gauss  (structural  iron  about). 

The  Voltage  0-00132  H  was  therefore  0-000254  volt. 
The  Current  M45  H  ^  100,000  was  2-20  microamperes. 

The  m.-c.  galvanometer  reading  d  was  165  mm.,  consequently 
165/2-2  =  75  mm.  on  its  scale  meant  1  micro-ampere. 

IV.  To  round  off  the  experiment,  reverting  to  the  original  connec- 
tions, the  coil  was  rotated  again,  and  its  axis  gradually  slanted 
from  the  vertical,  its  lower  end  towards  magnetic  north,  until  an 
angle  was  found  where  the  galvanometer  showed  no  current  at  all. 
This  was  66°  ;  it  is  the  Dip,  cf.  §§  695,  753. 

For  only  when  the  axis  of  rotation  coincides  with  the  direction 
of  the  Total  Field  is  one  half  of  the  coil  always  exactly  re-cutting 
backwards  every  line  the  other  half  cuts. 

Then,  §  696,  Earth's  vertical  component  field 

V  =:  H  tan  66°  =  0-432 
Total  field  T  =  V/sin  66°  =  0-472 

So  you  see  these  mysteries  are  soon  run  to  earth,  and  as  all 
other  practical  electrical  units  are  bred  of  them,  we  have  captured 
the  lot. 

§  775.  By  comparatively  easy  processes,  §  784,  etc.,  any  desired 
multiple  or  fraction  of  this  115-5  ohms  that  we  have  found  can  be 
made  up.  Internationally  the  Ohm  has  been  defined  as  the  resis- 
tance of  a  uniform  thread  of  pure  mercury  at  0°  C,  106-3  cm.  long 
and  weighing  14-4521  gm.  (which  gives  a  cross-section  of  1  sq.  mm.). 

The  original  arbitrary  unit  was  a  1-m.  thread  1  sq.  mm.  section  ; 
and  it  was  the  attempt  to  retain  something  like  this  that  led  to  the 
decimation  of  the  unit  of  current. 


§776] 


RESISTANCE 


627 


Fio.  345. 


The  idea  of  using  a  liquid  metal  is  to  be  free  of  the  internal 
strains  sure  to  be  present  in  a  solid  wire.  It  has,  however  been 
found  that  although  this  tube  is  kept  in  ice,  the  heat  produced  bv 
the  necessary  intermittent  testing  current,  from  a  single  dry  cell, 
in  10  min.,  warms  it  so  much  that  no  further  really  accurate  work 
can  be  undertaken  with  it  that  day. 

National  Standards  Laboratories  prefer  to  in.stal  the  two 
machines  referred  to  in  §  774,  and  to  prepare,  and  keep  under  observa- 
tion, and  occasionally  exchange,  Standard  Resistance  Coils  made  of 
Manganin  {a,  reddish  e^Uoy  of  87%  copper,  9-5  manganese,  and  3-5 
nickel)  and  constructed  as  in  Fig.  345.  A  short  piece  of  wide  brass 
tubing  is  wrapped  with  silk,  shellac 
varnished,  and  baked.  A  loop  of 
double-silk-covered  manganin  wire, 
a  little  above  the  required  resis- 
tance, has  its  ends  hard-soldered 
to  the  stout  copper  terminal  bars, 
and  is  then  wound  double  on  the 
cylinder,  thickly  varnished  with 
shellac,  and  heated  to  140°  C.  for 
8  hr.  by  sending  a  considerable  cur- 
rent through  it. 

This  drives  off  all  moisture,  so 
that  there  is  no  risk  of  current  leaking  from  one  turn  to  its  neigh- 
bour, and  also  anneals  the  wire,  which  was  '  hard  '  from  the 
wire-drawing  and  winding.  The  annealing  lowers  the  resistance 
2  or  3%,  and  thereafter  the  coil  remains  practically  unchanged 
for  years  :   without  it  the  resistance  crawls  down. 

The  coil  is  connected,  through  mercury  cups  in  which  its  thick 
copper  terminals  rest,  with  an  apparatus  of  the  principle  to  be 
described  in  §  784,  and  is  adjusted  to  equality  with  a  primary 
standard  by  shortening  one  end  of  the  wire,  or  by  shunting  part  of 
it  with  a  thin  branch  wire  (§781). 

The  coil  is  sealed  in  moisture-free  paraffin  oil  in  an  outer  case, 
and  is  kept  in  a  bath  at  constant  temperature. 

The  '  go  and  return  '  winding  prevents  the  establishment  of 
the  magnetic  field  existing  in  a  single- wound  coil.  There  is  no 
cutting  of  magnetic  lines  as  the  current  alters,  the  coil  is  '  non- 
inductive,'  and  the  current  does  not  run  on  afterwards,  an  action 
which  would  destroy  the  usefulness  of  the  resistance  for  many 
measuring  purposes. 

§  776.  Ordinary  resistance  coils  are  wound  in  the  same  way 
on  bobbins  2  or  3  in.  long,  and  are  best  of  Manganin,  though 
Eureka  or  Coiistantan  alloy,  copper  with  40%  nickel,  white,  strong 
and  tough  (very  likely  known  to  you  already  as  the  Electrum  of 
which  the  best  drawing  instruments  are  made)  is  equally  good  for 
most  purposes,  a  kindlier  metal  to  manage,  and  almost  non- cor- 
roding. 


628  MAGNETISM   AND   ELECTRICITY  [§  776 

It  has,  however,  the  objection  of  giving  a  thermo-electric  force, 
§  798,  against  copper,  of  40  microvolts  per  °C.,  whereas  manganin 
gives  only  2. 

Low  coils  are  of  short  thick  wire,  high  of  long  fine  wire.  The 
resistance  of  both  alloys  is  high,  so  that  but  little  wire  is  wanted, 
and  its  change  with  Temperature  is  so  small  as  to  be  quite  negligible 
in  all  ordinary  work. 

The  coils  are  attached  to  the  under  side  of  an  insulating  panel, 
and  the  ends  of  the  wire  are  soldered  to  brass  rods  rising  through  it 
to  brass  blocks  on  its  upper  surface  (Fig.  346,  §786).  When  the 
taper  plug  is  inserted  between  the  blocks,  the  current  goes  through 
the  plug,  the  conductance  of  which  is  vastly  greater  than  that 
of  the  coil ;  when  the  plug  is  taken  out,  the  current  must  go  all 
through  the  coil :  unplugging  a  coil  brings  it  into  use. 

Boxes  of  coils  of  1,  2,  2,  5,  10,  20,  20,  50,  etc.,  ohms  are  commonly 
made  up.  In  another  arrangement  a  turning  lever  moves  over 
eleven  studs,  between  each  pair  of  which  is  attached  a  1-ohm  coil. 
The  current  is  led  up  to  the  first  stud,  and  must  travel  through 
the  coils  in  succession,  until  it  reaches  the  stud  on  which  the  lever 
happens  to  rest,  then  it  passes  through  the  lever,  and  goes  to  the 
first  stud  of  a  batch  of  ten  10-ohm  coils,  and  so  on.  Either  of  these 
arrangements  permits  any  resistance  to  be  made  up  from  I  ohm 
to  total  of  box  :  the  turning-head  arrangement,  if  the  contacts  are 
kept  dean  and  tight,  is  by  far  the  handier. 

Resistance  boxes  are  intended  for  use  as  permanent  measuring 
instruments,  with  small  intermittent  currents,  and  not  as  electric 
stoves  ;  so  please  be  careful  how  you  connect  batteries  to  them, 
and  do  not  risk  a  burn-out.  The  maximum  continuous  power 
dissipation  in  any  ordinary  box  coil  should  not  exceed  0-1 
watt ;  beyond  this,  coils  must  be  looser  wound  and  ventilated. 

Variable  Resistances  or  Rheostats  are  often  very  useful  for 
regulating  current.  A  common  pattern  consists  of  bare  Eureka 
wire,  coiled  tightly,  with  narrow  air  space  between  the  turns, 
on  a  bar  of  slate.  The  current  enters  at  one  end  and  travels 
through  the  wire  until  it  reaches  a  movable  slider  which  carries  it 
off  along  a  brass  rod. 

In  the  Carbon  Rheostat  the  resistance  of  a  pile  of  plates  of  carbon 
is  reduced  by  squeezing  them  together  more  tightly  by  a  screw. 

All  regulating  resistances  act  by  dissipating  the  energy  as 
heat,  and  must  therefore  be  well  ventilated  and  heat-proof. 
Wire  wound  on  a  slate,  electric  stoves,  lamps,  wire-and-asbestos- 
wound  mats,  wire  netting,  expanded  metal — all  serve. 

So  does  copper  sulphate  solution  between  copper  plates,  or  a 
potful  of  water  with  a  handful  of  washing-soda  stirred  in,  and  a 
couple  of  iron  plates  or  rods. 

The  Resistance  of  a  one-centimetre  Cube,  from  fa^e  to  opposite  face^ 
is  the  Specific  Resistance  or  Resistivity  of  a  material.  Its  reciprocal 
is  Conductivity. 


§  777]  RESISTANCE  629 

§  777.  Some  Resistivities,  at  the  ordinary  temperature,  are  : 


Material. 


Reeiativity. 


per  r  C. 
warmer. 


Conductivity 
in  mho*. 


I.  Metals,  in  miUionths  of  an  ohm,  or  mierohms. 


Silver,  pure  electrolytic  annealed 

Copper,     do.     1-55,  wire 

AlTjminium,  wire 

Lead 

Platinum,  pure  . 

Iron,  mild  steel  wire,  about 

Mercury,  liquid  . 

Brass 

Manganin  . 

Constantan  or  Eureka 

Resistance  steels,  motor  controls 

Nichrome,  4Ni,Cr,  for  stoves 

Graphitic  carbon 


Hydrochloric  acid,  20% 

Sulphuric  acid,  25%,  battery 

Caustic  soda,  15% 

Common  salt,  15% 

Sal  ammoniac,  15%     . 

Copper  and  zinc  sulphates,  15% 

Distilled  water   . 


III.  Insulating  Matebials,  in 


Cotton,  Paper,  well  air-dried 
Asbestos  do.,  marble,  slate 
Wood,  dry 

Bakelite     .... 
Glass         soft     . 
„  window 

flint    . 
pyrex 
Porcelain 

Mica  .  .  .  • 

Paraffin  oils,  water- freed 
Silk,   and  cellulose  enamel   (wire 

insulation)       .  .  •       , 

Varnished   cambric    (machine   in 

insulation) 
Oiled  paper  (for  cable  insulation) 
Vulcanized  bitumen  (      „       .»      ) 
,,  rubber  (  „       .t       ) 

Gutta  percha  (submarine  cable)    . 
Rosin,  shellac,  sulphur,  ebonite, 
paraffin  wax,  fused  silica 


1-5 

0-4          1 

660.000 

1-62 

04          1 

620.000 

2-7 

0-4          1 

370.000 

20 

0-4 

50.000 

8-5 

0-37 

120.000 

10-5 

0-7 

95.000 

94 

0-72 

10.600 

6-7-5 

015 

abt.  150.000 

36-45 

0001 

..       25.000 

45-50 

0002 

..       20.000 

40-80 

Oil 

..       13.000 

97 

004 

..       10.000 

4500-9000 

-005 

220-110 

)N8  IN  Wat 

EK,  in  ohms. 

1-3 

-1-5 

0-77 

jwest  known 

1-5 

-1-5 

0-67 

2-5 

-1-9 

0-4 

6 

-21 

0-16 

4 

-1-7 

0-26 

24 

-23 

004 

01-25 

quick  fall 

10  to  004 

megohms 

micromhos 

millions  of  megohms,  or  10"  ohtns. 

Breakdown 

kilovoltn 

1  mm.  thick. 

5 

6 

0-5 
10  ^30 

_10  to  10 


to  01 

to  01 

1-50 

01-10 

0-5 

10 

50 

over  100 

1-1,000 

100-2,000 

100  up 

200 

400 

to  2.000 

400 

2,000 

400 

several 

thousand 


—6 

0  to  —5 


-  10  to  -  20 
•» 

-'20 
-5  to  -20 


over  15 

to  10 

tofiO 

7 


20 

10 
10 

1-60 


630  MAGNETISM   AND   ELECTRICITY  [§778 

§  778.  Theory  of  Metallic  Conduction.  In  §  737,  under  Dielectrics, 
it  was  pointed  out  that  an  atom  consists  of  a  massive  positively- 
charged  nucleus  round  which  minute  negative  electrons  are  in 
orbital  motion,  and  the  displacement  of  some  of  these  electrons, 
without  their  removal  from  the  atom,  was  found  sufficient  to 
explain  the  inductive  action  of  a  dielectric,  a  Non-conductor. 

In  Conductors  it  appears  that  a  number  of  electrons  escape  from 
the  atoms,  and  wander  freely  in  the  mass,  though  they  cannot  leave 
it  at  any  ordinary  temperature.  To  an  electron,  the  mass  is  by  no 
means  dense  ;  a  church,  containing  a  mouse,  and  some  very  small 
flies,  is  a  fair  scale  model  of  an  atom. 

In  liquids  you  have  watched  the  Brownian  movement  of  particles 
of  some  size,  and  have  inferred  from  it  the  incessant  activity  of 
the  molecular  dance.  In  Solids  there  is  no  reason  to  suppose  that 
this  dance  is  frozen  into  utter  immobility  :  in  fact,  the  slow  spon- 
taneous recrystalUzation  changes  that  go  on  in  metals,  in  setting 
concrete,  etc.,  forbid  one  to  fix  even  whole  atoms  too  rigidly — any- 
way, we  mustn't  grudge  the  mouse  his  little  activities — and  every 
individual  massive  particle  retains  its  energy  ^mv^  of  thermal 
motion  (which  is  its  temperature),  now,  as  a  vibration  about  a 
mean  position. 

Therefore,  as  we  insisted  in  §  367,  etc.,  that  all-moving  particles 
must  share  alike,  so  now  we  endow  each  wandering  electron  with 
the  average  energy  of  the  molecule  at  that  temperature.  Inciden- 
tally, let  us  find  their  speed  of  wandering,  from  the  relation  of 
§  202,  that  Pressure  in  dynes /cm.^  =  J  density  x  v^. 

For  hydrogen,  at  0°  C.  and  76  cm.  pressure,  this  gives  : 

76  X  13-6  X  981  =  J  X  0-0000898  gm./c.c.  x  v^ 
so  that  H2  flies  about  at  ?;  =  184,000  cm. /sec.  =  1-84  km. /sec. 

The  mass  of  the  electron  is  H/1840,  §  883,  therefore  to  keep 
equal  energy,  the  natural  electronic  velocity  V  will  be  1-84  x 
^(2  X  1840)  =  112  km./sec. 

Their  '  mean  free  path  '  L,  between  collisions  with  one  another 
or  with  atoms,  is,  however,  exceedingly  small ;  it  evidently  occupies 
a  time  L/V  seconds. 

Now,  if  an  electric  field  of  strength  F  be  set  up,  it  will  push  on 
each  electronic  charge  e  with  Fe  dynes,  which  will  give,  to  its  mass 
m,  acceleration  Fe/m,  which  in  time  L/V  enables  it  to  acquire  a 
final  speed  along  the  conductor  Fe/m  x  L/V,  averaging  half  as 
much,  §  17,  FeL/2mV. 

If  there  be  N  free  electrons  per  c.c,  the  result  is  a  charge  carried 
per  second,  i.e.  a  current,  Ne  X  FeL/2mV,  per  sq.  cm. 

Good  conductors  probably  differ  from  poor  ones  by  possessing 
a  larger  N. 

This  Current  is  proportional  to  F,  the  Electromotive  Force ; 
and  that  is  Ohm's  Law. 


\ 


§  778]  RESISTANCE  631 

Again,  F/current  =  Resistivity  =  2mVfSe^L 
and  since  V  increases  with  temperature,  so  must  the  Resistivity. 
The  Table  shows  that  for  commercially  pure  metals  it  increases 
about  1/250  per  °C.,  and  for  pure  Platinum  just  the  1/273  of  the 
gas  thermometer,  and  indeed  the  Resistivity  of  most  pure  metals  is 
approximately  proportional  to  the  Absolute  Temperatures. 

On  this  is  based  the  Platinum  Resistance  Thermometer,  which  is 
essentially  a  small  coil  of  pure  platinum  wire,  often  occupying  the 
'  bulb  '  end  of  a  hard-glass  or  porcelain  tube  about  the  size  of  a 
thermometer,  and  with  leads  brought  out  to  a  Bridge,  or  a  Potentio- 
meter arranged  for  measuring  resistance.  The  tube  contains  also 
dummy  leads,  which  are  put  into  the  opposite  arm  of  the  bridge, 
and  exactly  compensate  any  effect  of  temperature  except  that  on 
the  coil  itself. 

The  above  rule  being  only  approximate,  each  thermometer  has 
to  be  standardized,  as  usual,  in  melting  ice,  and  in  steam  at  76  cm., 
and  also  in  boiling  sulphur  vapour  at  76  cm.,  444-55°  C,  and  if  for 
use  at  low  temperatures,  in  oxygen  boiling  at  76  cm.  —  183°  C. ; 
it  it  then  capable  of  an  accuracy  of  at  least  0-1°  C.  between  about 
—  250°  C.  and  1063°  C,  the  freezing  point  of  gold.  It  is  incomixirMy 
handier  than  a  gas  thermometer,  and  much  more  accurate  than  any 
other.     Sensitive  patterns  respond  to  1/3000°  over  small  ranges. 

The  acceleration  of  the  electrons  by  the  field  F,  slight  as  it  is — 
and  the  actual  speed  of  travel  of  electrons  along  a  wire  carrying  any 
ordinary  current  is  probably  quite  small — gradually  increases  the 
general  speed  of  movement. 

The  free  electrons  and  the  atoms  of  the  metal  must  share  the 
increase  in  \mv^  between  them  ;  to  the  electrons  this  sharing-out 
represents  a  hindrance,  the  Resistance  of  the  metal ;  to  the  atoms 
it  is  an  addition  to  their  energy  of  thermal  oscillation,  the  tem- 
perature rises.  This  is  the  Heating  of  a  Conductor  by  a  Current, 
and  you  can  see  at  once  that  it  is  proportional  to  the  square  of  the 
current  strength,  §  815,  because  a  doubled  speed  of  electronic  drift 
means  twice  as  many  collisions,  twice  as  hard,  with  the  atoms. 

If  you  will  compare  the  last  column  of  this  Table  with  that  of 
Thermal  Conductivities  in  §241,  you  will  see  that  the  Ratio  of 
electrical  to  thermal  conductivity  is  (100,000  times)  Ag  6,  Cu  6-5, 
Al  7,  Fe  6-7,  Pb  6,  Hg  5-5,  Brass  6-2,  Manganin  and  Eureka 
about  4. 

This  suggests  that  the  comparatively  high  Conductivity  for 
Heat  {or  cold)  of  Metals  is  due  to  their  possession  of  these  very  mobile 
free  electrons,  which  diffuse  readily  through  it,  carrying  their 
AmV2,  and  going  share  and  share  alike  with  the  atoms  wherever 

they  are. 

There  is  now  no  question  of  a  natural  speed  dependent  on  tem- 
perature, and  an  impressed  one  quite  independent  of  it,  and  the 
worked-out  expression  shows  that  if  resistivity  oc  T,  thermal 
conductivity  should  be  independent  of  temperature  :    and  so  it  is. 


632  MAGNETISM   AND   ELECTRICITY  [§  778 

Alloys  are  often  much  more  resistant  than  their  constituent 
metals,  but  vary  a  great  deal  less  with  temperature. 

Consideration  of  Section  II  of  the  Table  is  deferred  to  Chapter  LII. 

The  resistivities  of  Insulating  materials  in  Section  III  have  nothing 
of  the  definiteness  of  those  of  metals,  but  indicate  that  the  con- 
duction is  electrolytic,  as  in  II.  This  is  especially  evident  for  paper  : 
in  Telephone  Cables  each  pair  of  wires  is  insulated  by  a  continuous 
strip  of  paper  folded  in  S,  many  hundreds  are  threaded  into  a  leaden 
tube,  which  is  then  drawn  down  tightly  over  them.  Their  manu- 
facture in  our  damp  climate  was  long  a  difficulty ;  they  are  kept 
sealed,  but  portable  emergency  air-compressors,  with  drying-tubes, 
sometimes  find  employment  drying  them  out. 

But  even  so,  look  at  the  millionths  in  I,  and  the  millions  of  millions 
in  III,  and  figure  out  for  yourself,  that  no  more  current  would  leak 
from  face  to  face  of  a  dry  4x7  cm.  cigarette  paper  0-001  in.  thick 
than  would  be  driven  by  the  same  voltage  5  times  round  the  world 
on  a  No.  22  copper  wire,  the  thickness  of  a  thin  pin. 

Tightly  lapped  brown  paper,  soaked  in  oil,  is  largely  used  as 
insulation  in  electric  power  cables. 

§  779.  Relation  of  Resistance  to  size  of  conductor.  Putting  2  or 
3  1-cm.  cubes  for  current  to  flow  through,  side  by  side,  gives  it 
2  or  3  times  the  opportunity.  So  that  a  conductor  of  cross-section 
A  sq.  cm.  and  length  1  cm.  would  have  a  conductance  A  times  that 
of  a  single  1-cm.  cube, 

its  Conductance  =  conductivity  x  A 
or  reciprocally.  Resistance  =  resistivity  ~  A 

Putting,  however,  2,  3,  etc.,  1-cm.  cubes  '  end  on,'  so  that  the 
current  must  flow  through  them  in  succession,  evidently  doubles, 
trebles,  etc.,  the  resistance  in  its  path,  i.e.  the  Resistance  of  a 
column  of  L  1-cm.  cubes  =  L  x  resistivity. 

Taking  these  two  together, 

For  a  conductor  A  sq.  cm.  cross-sectional  area  and  L  cm.  long 

the  Resistance  =  ^  X  resistivity  ; 

and  inverting  everything,  Conductance  =  conductivity  x  A/L. 

Ex.  1.  Calculate  the  resistance  of  1  yard  (91-5  cm.)  of  No.  22  copper  wire 
diameter  0-71  mm.). 

Ql.FJ 

«  =  ;r^  (0-5  X  0-071)'  X  0-00000165  =  0-0385  ohm. 

Ex.  2.  Calculate  the  resistance  of  an  '  accumulator  '  in  which  two  plates 
15  cm.  square  are  separated  by  0-8  cm.  of  sulphuric  acid. 

^  =  T^^^^Tk  X  1-5  =  00053  ohm. 


§  781]  RESISTANCE  *  633 

§  780.  Conductors  in   series  '   and   *  in   parallel.'    Carrying   the 

argument  further,  the  Resistance  of  a  number  of  conductors  through 
which  the  current  must  pass  in  succession  or  '  in  series  '  is  the  sum 
of  their  resistances 

I^  =  ^l  +  ^2  +  ^3+       •       .       . 

This  does  not  necessarily  apply  to  classes  II  and  III  above. 

The  Conductance  of  a  number  of  conductors,  by  any  of  which 
current  can  flow  from  P  to  Q,  is  the  sum  of  the  individual  con- 
ductances, just  as  the  traffic-carrying  capacity  of  all  the  roadM, 
rails,  canals,  etc.,  from  one  place  to  another,  is  the  sum  of  their 
individual  carrying  capacities. 

C  =  Ci  +  Cg  +  C3  +      .     .     . 

or  writing  conductance  as  the  reciprocal  of  resistance, 

R=;^  +  i^  +  r3+  •  •  • 

Thus  the  Resistance  of  any  number  of  conductors  '  in  multiple  arc  * 
or  '  in  parallel,''  Fig.  347,  has  to  be  found  by  first 
taking  the  reciprocal  of  their  individual  resis- 
tances, adding  these  reciprocals  together,  and 
then  taking  the  reciprocal  of  this. 

The  Currents  in   the   various   branches  are 
proportional   to    their    Conductances   {i.e.   in-  ^'o-  ^7. 

versely    as    their    resistances),    and    are    the 
fractions  cJC,  Cg/C,  etc.  (or  R/rj,  R/fg,  etc.),  of  the  total  current. 

Ex.  3.  Three  wires  of  resistances  2,  4,  and  6  ohms  are  joined  in  parallel 
and  together  carry  110  amp.  Find  their  joint  resistance  and  the  current 
in  each  wire. 

R       2^  4^  6       12 
.-.   R  =  12/11  ohm 

Current  in  2  ohms  =  2/12  ^  ^^^  *""P*  ^  ^  *"^P' 


4/12 


110  amp.  =  30     „ 

„      "     6      „      =i|y  X  110  amp.  =  20     .. 

Ex.  4.  A  coil  intended  for  a  5-ohm  standard  is  found  when  teeteil  to  have 
a  resistance  5033  ohms;  what  fine  wire  must  be  put  in  parallel  with  it  (as 
a '  shunt ')  to  reduce  the  joint  resistance  to  5  ohms  ? 

1/5  =  1/5033  +  l/«  or  0-2  -  0-19868  =  1/x. 
.-.  X  =  1/000132  =  760  ohms. 

§781.  Shunts.  The  arrangement  of  conductors  in  parallel 
is  often  made  use  of  to  get  a  definite  small  fraction  of  a  current, 
so  that  a  galvanometer  or  ammeter  suitable  for  measuring  small 
currents  may  also  be  available  for  large  currents. 


634 


MAGNETISM   AND   ELECTRICITY 


[§781 


The  galvanometer  has  its  terminals  AB  connected  together  by 
a  wire  of  resistance  less  than  that  of  its  own  coils  ;  most  of  the 
current  arriving  at  A  flows  past  the  galvanometer  through  the  shunt 
to  B,  and  only  a  fraction  traverses  the  coils  and  actuates  the  instru- 
ment. Thus,  if  the  shunt  has  1 /4th  the  galvanometer  resistance,  its 
conductance  is  4  times  that  of  the  galvanometer,  their  joint  con- 
ductance is  5  times,  therefore  4/5ths  of  the  current  goes  through  the 
shunt  and  only  l/5th  through  the  galvanometer.  If  the  shunt  = 
1/99  galvanometer,  only  0-01  passes  to  galvanometer,  etc. 

Ex.  6.  What  shunts  are  necessary  to  reduce  the  sensitiveness  of  a  500- 
ohm  galvanometer  to  1/3,  1/5  and  1/lOth? 

The  first  has  resistance  1/(3 — 1)  of  galvanometer,  for  then  its  conductance 
=  2/1  galvanometer's  and  it  takes  2  parts  of  current  while  galvanometer 
takes  1. 

The  second  has  resistance  1/(5 — 1)  of  galvanometer  and  the  third  1/(10 — 1). 
Shunts  250,  125,  and  55-5  ohms. 

Temporarily  shunting  a  galvanometer  with  a  few  inches  of  thin 
wire  is  a  precaution  that  saves  time  when  far  from  balance  in 
bridge  experiments,  etc. 


Methods  of  Comparing  Resistance 

The  comparison  of  resistances  with  one  another  is  an  important 
electrical  operation.     Some  methods  follow  : — 

§  782.  Replacement  method.  A  voltaic  battery  and  the  un- 
known resistance  are  wired  in  series  with  any  sort  of  galvanometer 
which  will  then  give  a  conveniently  large  deflection.  The  unknown 
resistance  is  removed  from  the  circuit,  a  resistance  box  put  into 
its  place,  and  resistances  unplugged  until  exactly  the  same  deflection 
is  obtained. 

The  unknown  =  total  of  known  coils  unplugged. 

§  783.  *  Halving '  method.  A  circuit  is  made  up  as  in  Fig.  348 
of  an  accumulator,  the  unknown  resistance,  a  resistance  box,  and 

a  tangent  galvanometer  or  amme- 
ter. Let  E  be  the  electromotive 
force  of  the  battery,  b  its  resistance, 
w  resistance  of  wires,  g  resistance 
of  galvanometer,  X  the  unknown. 
First  get  a  reading  of  current  C 
with  all  plugs  in  box,  then  by 
Ohm's  law. 


Fig.  348. 


■E  =  C{b  +  w-^g  +  X). 


Then  unplug  coils  in  the  box  to 
a  total  R  until  the  galvanometer  reading  shows  the  current  is 
halved. 

Then  E  =  1C(6  +  w;  +  gr  +  X  +  R). 


§  784]  RESISTANCE  635 

Now  6  of  an  accumulator,  w  of  thick  wires,  and  g  of  an  ammeter 
for  currents  up  to  1  amp.,  are  negligibly  small  (if  not,  they  must 
be  known  beforehand),  therefore,  very  nearly,  X  =  H. 

There  is  no  need  to  just  halve  the  current,  it  may  Ikj  reduced 
to  1/nth  its  value,  where  n  may  be  anything.  That  means  the 
resistance  is  now  n  times  what  it  was,  i.e.  you  have  put  in  an  R 
which  is  {n  —  1)  times  the  X  which  stays  there  all  the  time,  /.  X  is 
now  R/(7i  —  1),  of  which  the  foregoing  is  the  special  ca«e  for  n  =  2. 

See  also  Battery  Resistance,  §  875. 

§784.  The  Wheatstone  Bridge  is  an  arrangement,  dating  from 
1833,  which  enables  resistances  to  be  very  accurately  compared. 
Then  if  one  of  them  is  known  in  ohms,  the  actual  value  of  the  other 
is  this  multiplied  by  their  ratio. 

In  Fig.  349  a  battery  circuit  divides  at  A  and  rejoins  at  B.  A  is 
at  a  higher  potential  than  B,  C  is  at  an  intermediate  potential; 
evidently  there  must  be  some  point  1)  discoverable  in  the  other 
branch  which  is  at  the  same  intermediate  potential  as  C.  D  is 
tried  for,  and  found  when  a  sensi- 
tive galvanometer  in  the  bridging 
wire  CD  shows  no  deviation,  for  if 
there  were  any  potential  difference 
between  C  and  D  it  would  surely 
drive  a  current  through  CD. 

A  River  is  flowing  down,  both 
sides  of  a  biggish  island,  from  the 
common  level  A  to  the  lower  com- 
mon level  B.  C  is  part  way  down 
one  branch,  and  you  take  a  spade 
and  dig  a  trench  across  to  the 
other  stream,  water  following  as 
you  dig.  Arrived  within  sight  of 
it,  you  dig  along,  either  upstream  or  downstream,  until  you  judge 
the  stream  level  is  the  same  as  that  of  the  water  in  your  trench  ; 
then  you  dig  out  into  it.  A  floating  dead  leaf  is  your  galvanometer 
needle,  to  tell  vou  whether  your  trench  water  r»ms  either  way,  or 
remains  stagnant ;  if  it  does  nothing,  C  and  D  are  at  the  wune  level, 
and  in  whatever  ratio  of  levels  C  divides  the  fall  of  level  A  to  B, 
D  divides  it  in  the  same  ratio. 

Fall  of  level  is  a  river's  way  of  overcoming  the  resistance  of  rough 
bed  and  obstructing  weeds  :  fall  of  potential  provides  the  electron- 
moving  force,  and  Ohm's  Law  assures  iis  that  when  it  produce 
the  same  flow,  it  is  overcoming  resistance  proportional  to  it«elf, 
P  __  "P/R 

When  '  balanced,'  no  current  leaks  across  (T),  but  Cj  flows  all 
along  ACB,  and  c,  all  along  ADB.  Therefore  the  ratio  of 
Resistances  into  which  C  divides  ACB  is  the  same  as  that  into  which 
D  divides  ADB 

.*,  -  =  i^   or    =  - 

X      q  r      q 


636 


MAGNETISM   AND   ELECTRICITY 


[§784 


Corresponding  to  the  easy  algebraic  interchange  is  the  electrical 
fact  that  battery  and  galvanometer  can  be  interchanged  in  the  Bridge 
without  affecting  the  measurements. 

§  785.  In  the  *  Metre '  Bridge  arrangement,  Fig.  350,  of  the 
Wheatstone  conductors,  ADB  is  a  straight  strong  wire  of  resist- 
ance metal  stretched  along  a  scale.  The  wire  is  quite  uniform, 
the  resistance  of  every  cm.  of  it  is  the  same,  hence 

Resistance  p  :  resistance  q 
=  length  of  wire  AD  :  remaining  length  DB. 

The  corner  points  A,  C,  B  are  represented  by  thick  straps  of 
copper    of   no    appreciable   resistance,    and   provided   with    stout 


Fig.  350. 


binding  screws.  A  known  resistance  R  is  connected  into  the  gap 
AC  by  short  stout  clean  wires,  the  unknown  X  is  similarly  connected 
into  the  gap  CB.  Then  a  sliding  contact-maker  is  moved  along 
the  wire  until  a  point  is  found  where  the  delicate  galvanometer 
in  the  long  wire  CD  is  not  deflected  at  all  from  its  rest  position. 
Then 

resistance  R  _  length  p  of  stretched  wire  AD 
resistance  X       length  q  of  stretched  wire  DB 

In  the  above,  use  a  single  Leclanche  cell,  with  a  key. 

Don't  warm  the  contact  D  with  your  finger,  or  a  thermo-E.M.F. 
arises. 

Liquid  Resistances,  e.g.  batteries,  are  measured  by  using 
alternating  current,  supplied  from  a  little  toy  medical  induction 
coil,  and  getting  silence  in  a  telephone  ;   do  not  use  high  coils  in  R. 

§  786.  The  accuracy  of  the  metre  bridge  depends  on  the  perfect 
uniformity  of  the  stretched  wire.  This  is  difficult  to  maintain 
when  the  instrument  has  to  withstand  workshop  and  outdoor  use, 
and  the  Post  Office  introduced  the  Box,  Fig.  351,  of  resistance 
coils  so  arranged  as  to  dispense  with  the  stretched  wire. 

In  its  stead  is  a  '  bar  '  ADB  of  6  resistance  coils  [A]  1000,  100,  10, 
[D]  10,  100,  1000,  [B].  One  of  these  either  side  of  D  being  un- 
plugged, we  can  get  ratios  pjq  =  100/1,  10/1,  equal,  1/10,  1/100. 


(86] 


RESISTANCE 


637 

Attached  at  A,  by  a  link  under  two  binding  HcrewH.  or  by  an 
*  intmity  plug  without  coil,  is  a  complete  set  of  coils  from  1  to  50U0 
ohms,  enabling  R  to  be  made  anything  from  1  to  10.000  ohms 
X  to  be  measured  is  attached  to  the  binding  screws  CB  The 
battery  is  connected  to  A  and  B  through  a  tapping  key.  aiul  the 
delicate  galvanometer  to  C  and  D  through  another,* the  wiret* 
shown  dotted  being  contained  inside  the  box. 

In  use,  after  seemg  that  all  the  plugs  fit  snugly,  lo  ohms  is 
unplugged  each  side  of  D,  so  that  ;j  =  q.  One  ohm  is  unpluggwl 
m  AC,  the  battery  key  held  down,  and  a  quick  tap  on  galvanometer 
key  shows  a  deflection  towards  the  left,  say.  5000  is  unplugge<I 
in  AC,  and  a  quick  tap  sends  galvanometer  to  right.  Making  r 
10  only,  still  to  right ;  r  =  5  to  left,  6  to  left,  7  to  left,  8  to  right. 
p  :  q  =  R  :  X.     .*.  X  is  between  7  and  8  ohms. 

In  AD  unplug  100  and  plug  in  10,  ;> :  ^  ==  10  :  1.  Try  R  --  75 : 
to  left,  76  to  left,  77  to  right.     .*.  X  is  between  7-«  and  7-7. 


Fig.  351. 


Fio.  346. 


In  AD  unplug  1000  and  plug  in  the  100  ;  make  R  -  700  :  to  left. 
765  to  right,  761  feeble  left,  762,  763  doubtful,  7U  feeble  right. 
Therefore  7-625  is  the  nearest  value. 

You  see  that  the  P.O.  Box  can  not  only  measure  single  ohms 
by  lOOths,  but  also  resistances  up  to  100  X  11,000  =  M  megohm  : 
with  the  same  average  accuracy  of  1  in  500. 

The  binding  screws  are  usually  marked  G,  G  for  galvanometer 
attachment ;  B,  B  or  C,  Z  for  battery  (copper,  zinc) ;  X.  X  or 
L,  E  for  unknown  (line,  earth).  As  mentioned  above,  l)atter>'  and 
galvanometer  can  be  interchanged,  and  sometimes  this  is  actually 
desirable. 

Sometimes  R  is  made  up  into  three  bars  instead  of  two.  when  the 
keys  change  sides  and  make  a  rather  neater  lay-out  of  connections. 

Turning-lever  patterns  of  Box  are  made  in  which  p  and  q  are 
levers  for  studs  of  10,  100,  1000,  and  R  a  4-face  set  of  10  each, 
I's,  lO's,  lOO's,  lOOO's.     See  §776. 

A  single  Leclanchd  cell  should  be  used,  and  a  sensitive  galvano- 
meter, best  kept  shunted  until  near  balance. 

It  is  often  better  to  get  the  last  decimal  place  by  «rale  defttctions 
rather  than  by  the  1000  :  10  ratio,  which  is  apt  to  prove  insensitive. 

Go  over  every  plug  before  you  start,  and  turn  to  the  right  under 
gentle  pressure  until  it  seizes ;    ebonite  is  very  exiMinsible,  and  on 


638  MAGNETISM   AND   ELECTRICITY  [§  786 

a  hot  day  every  plug  may  be  making  bad  contact ;  if  it  will  not  seize 
it  is  '  bottoming,'  try  to  find  it  another  home.  The  wedging  action 
of  the  plugs  is  apt  to  make  all  the  blocks  a  trifle  loose  in  course  of 
time  ;  whenever  you  take  a  plug  out  (twisting  to  the  right),  just 
test  both  its  neighbours.  Various  special  plugs  or  caps  have  been 
invented  to  avoid  this  trouble  ;  but  never  let  one  remain  slack. 

In  an  Inductive  Resistance,  such  as  a  large  coil  of  wire,  or  an 
electromagnet  coil,  the  current  rises  only  gradually  on  making 
circuit,  §  829,  whereas  that  in  the  non-inductive  R,  rising  instantly, 
gets  first  kick  at  the  galvanometer,  and  obscures  the  steady  balance 
position.  The  two  keys  in  the  box  are  intended  to  obviate  this  ; 
press  the  battery  key  first,  wait  a  fraction  of  a  second  for  the  current 
to  grow  to  its  full  value,  and  then  press  galvanometer  key  :  release 
it  before  breaking  current. 

§  787.  Ammeter  and  Voltmeter  method.  A  good  practical  way 
of  measuring  resistance  is  by  the  use  of  Ammeter  and  Voltmeter, 
Fig.  360.  The  Ammeter  is  connected  in  series  with  the  resistance 
PQ,  and  the  current  A  flowing  through  both  is  observed.  The 
Voltmeter  is  connected  as  a  shunt  across  the  ends  of  the  resistance, 
and  the  Volts  V  between  the  ends  of  R  observed.  (A  jumps  up 
a  trifle,  but  this  is  the  extra  current  actuating  the  Voltmeter  and 
must  be  disregarded.) 

Then  R  ohms  =  V  volts  ~  A  amperes. 

In  this  way  it  is  easy  to  measure  a  resistance  such  as  that  of  a 
glow  lamp  connected  to  the  mains  and  actually  working.  And  by 
using  a  spiral  of  iron  wire  and  heating  it  in  a  flame  the  rise  of  R 
with  temperature  is  strikingly  shown  by  the  fall  in  current. 

The  accuracy  of  the  method  is  that  of  the  instruments  employed, 
and  of  their  employer,  who  should  look  out  for  possible  zero  errors. 

The  Ohm-meter  is  an  instrument,  much  in  technical  use,  which 
carries  out  this  method  in  one  reading.  As  sketched  in  Fig.  362 
(in  contrast  with  the  more  important  Watt-meter,  with  which  it 
is  liable  to  be  confused)  it  has  two  coils  fixed  together  cross-wise, 
and  moving  in  the  field  of  a  permanent  magnet.  The  two  coils 
are  connected  in  series  and  in  shunt  respectively,  the  first  would 
give  a  deflection  proportional  to  the  amperes,  and  the  second  to 
the  volts,  but  their  zero  readings  are  at  right  angles,  and  the  tangent 
law  applies,  the  resultant  position  being  such  that 

tan  D  =  volts /amps.  =  R. 
Instead  of  degrees,  the  scale  is  graduated  to  read  Ohms  direct. 

Body  resistance.  Your  own  body  really  acts  as  a  liquid  resistance, 
and  alternating  current,  as  in  §  785,  ought  to  be  used,  but  most 
people  flnd  it  painful.  A  near- enough  measurement  may  be  made 
by  using  a  H.-T.  battery  of  perhaps  40  cells,  in  series  with  yourself 
and  a  sensitive  galvanometer,  suitably  shunted  if  it  goes  too  far. 
Test  cells  with  Voltmeter,  and  deduct  a  couple  of  volts  to  allow  for 
the  '  back  E.M.F.  of  polarization  '  in  the  '  liquid.' 


§788] 


RESISTANCE 


639 


Standardize  the  galvanometer,  as  so  shunted,  by  using  one  cell 
and  a  5000-ohm  coil ;  it  thus  becomes  your  '  Ammeter.' 

Very  high  resistances,  of  many  Megohms,  such  as  the  insulation 
resistance  between  the  wiring  of  your  house  and  the  earth  ought 
to  be,  demand  higher  voltages,  often  supplied  by  a  diminutive 
hand-magneto  which  gives  a  steady  600  volts,  and  micro-ammeterh 
to  measure  the  small  current  it  forces  through. 

And  so  on,  with  increasing  sensitivity,  to  higher  and  higher 
resistances  still. 

In  insulation  resistances  it  is  essential  to  record  also  the  applie<l 
voltage,  the  time  it  has  been  kept  on,  and  the  temperature.  The 
results  enable  an  observer  of  experience  to  judge  whether  the 
insulation  will  be  adequate  for  the  purpose  in  hand,  but  it  is  no 
use  calculating  from  them  by  Ohm's  law. 

§  788.  Low  Resistances,  on  the  other  hand,  can  be  dealt  with  by 
using  galvanometer  as  micro-voltmeter,  and  a  big  ammeter  in  the 
main  circuit. 

They  cannot  be  compared  at  all  well  by  the  Wheatstone  Bridge, 
because  the  resistance  of  their 
connecting  wires  inevitably  adds 
in  with  them  (which  is  why 
Fig.  350  shows  you  short 
thick  wires  there). 

The  Kelvin  Double  Bridge 
overcomes  this  difficulty,  Fig. 
352.  In  its  simplest  practical 
form  this  consists  of  a  pair  of 
thinnish  (No.  28)  eureka  wires 
stretched  along  a  metre  scale, 
with  a  twin  slider,  between 
the  two  contacts  of  which 
the  galvanometer  is  connected. 

Metres  of  No.  20  copper  wire 
connect  their  ends  with  the 
'  potential '  points  BC  and  DE 
on  unknown  and  known  low  re- 
sistances, say  a  length  of  bar 
iron  and  a  001 -ohm  standard  : 
the  contacts  must  be  good. 

An  accumulator  sends  a  current  round  through  the  circuit 
ABCDEF,  and  the  galvanometer  twin-slider  GG'  is  moved  until 
there  is  no  deflection,  i.e.  no  leak  across. 

Following  the  potential  diagram  round,  you  see  the  gradual 
fall  of  potential  along  the  circuit,  from  +  to  —  side  of  the  battery, 
BC  and  DE  being  now  the  falls  in  the  two  resistances.  The  same 
current  flows  through  both,  therefore  their  resistances  are  propor- 
tional  to  BC  and  DE  ;  and  that,  you  see  easily  enough,  is  the  ratio 
\  in  which  the  slider  divides  the  wires. 


Fio.  352. 


640  MAGNETISM   AND   ELECTRICITY  [§  788 

The  resistances  of  the  four  metre  wires  do  come  in,  but  only  with 
those  of  the  bridge  wires,  of  which  they  simply  form  extensions. 
If  they  are  No.  20  copper  (the  thickness  one  prefers  for  ordinary 
pins)  they  each  count  as  0-5  cm.  of  the  5-ohm  bridge  wires,  so  that 
the  point  shown  60/40  is  accurately  60-5/40-5.  This  is  a  very 
different  matter  from  counting  in  with  CB  or  DE,  either  of  which 
they  greatly  exceed.  (On  the  potential  diagram  it  means  that  the 
little  bits  shown  level  are  really  inclined  about  1/200  the  slope  of 
BE  or  CD.) 

A  Potentiometer  method  for  comparing  Resistances,  also  free 
from  connector  trouble,  is  described  in  §  797. 


EXAM   QUESTIONS,    CHAPTER   XLVIII 

In  Chapters  XLVI,  XLVII  we  have  dealt  with  the  engine  and  the  dash- 
board of  the  car,  as  it  were ;  now  come  the  brakes.  §§  771,  772,  773  are  general 
explanation ;  §  775  goes  rather  into  details ;  §  776  represents  it  in  most  labs. ; 
§  779 — 787  are  laboratory  work.  §  774  will  repay  your  study  before  you  \ 
leave  the  chapter  :  it  shows  how  even  you,  with  a  few  bits  of  apparatus  you 
have  handled  already,  can  make  and  understand  and  calculate  out,  taking 
nothing  on  trust,  all  absolute  measurements  in  Current  Electricity.  Of 
§  778  you  can  read  little  or  much ;  §  788  describes  a  pretty  contrivance  which 
you  may  happen  to  use. 

6.  State  Ohm's  Law,  and  define  Electrical  Resistance.  How  does  the 
resistance  of  a  wire  depend  on  its  temperature  ?  How  would  you  measure 
the  variation  ? 

7.  Briefly  describe  the  method  of  measuring  by  substitution  the  resistance 
of  a  conductor. 

8.  State  Ohm's  Law,  and  the  conditions  of  its  application. 

A  240-volt  lamp  filament  showed  120  ohms  by  cell  and  galvanometer 
method;  but  took  only  0-6  ampere  when  running  on  the  mains.  Compare 
the  two  resistances,  and  reconcile  them.     (  X  2) 

9.  What  is  meant  by  a  temperature  coefficient  of  electrical  resistance  ? 
What  do  you  know  of  its  magnitude  for  platinum,  manganin,  and  carbon  ? 

10.  A  metal  spiral  is  heated  by  5  amp.  to  250°  C,  the  voltage  being  20. 
When  this  is  raised  to  40  the  current  becomes  6-7  amp.  and  the  temperature 
420°  C.     Calculate  the  coefficient. 

11.  Define  Resistivity,  or  Specific  Resistance,  and  describe  a  method  of 
measuring  it.  A  wire  1  m.  long  and  0-6  mm.  diam.  had  a  resistance  1-16 
ohms ;  calculate  the  resistivity  of  the  metal.     (  X  2) 

12.  A  10-ohm  coil  is  to  be  constructed  of  wire  of  diameter  0-46  mm.  and 
specific  resistance  50  microhms.     What  length  will  be  required  ? 

How  would  you  make  the  coil  and  test  its  accuracy  ? 

13.  Compare  the  resistance  of  1  yard  of  copper  wire  1  mm.  diameter  with 
that  of  106  cm.  of  mercury  thread  1  sq.  mm.  cross-section. 

14.  How  do  you  calculate  the  resistance  of  conductors  in  parallel  ?     A  j 
piece  of  wire  of  resistance  10  ohms  is  made  into  a  circular  ring.     Current  is  j 
led  in  at  A  and  taken  out   at  the  opposite  point  B.     What  is  the  effective 
resistance  between  A  and  B  ?     How  would  it  be  affected  by  gradually  moving 
B  round  towards  A  ?     (  x  2) 


RESISTANCE  641 

15.  Part  of  a  circuit  consists  of  two  equal  wires  of  the  same  metal  in  parallel. 
What  change  will  be  made  in  the  resistance  if  the  wires  are  clamped  so  that 
a  point  one-fifth  of  the  length  from  the  end  of  one  wire  is  in  conta^-t  with  a 
point  three-fifths  of  the  length  from  the  corresponding  end  of  the  other  wire  ? 

16.  A  box  contains  three  coils  of  3  ohms  each.  What  different  resistances 
can  be  obtained  by  coupling  up  any  or  all  of  them  in  various  ways  ? 

17.  What  length  of  manganin  wire  0-253  mm.  diam.,  resistivity  45,  must 
be  shunted  across  1  03  ohm  to  reduce  it  to  1  ohm  ? 

18.  A  dynamo  lights  a  group  of  lamps  at  a  distance.  \Mion  one  only  is 
alight,  the  P.D.  between  the  dynamo  terminals  is  220  volts,  but  when  50  are 
alight  this  has  to  be  raised  to  230  volts  to  keep  them  as  bright  as  the  one. 
Explain  this. 

19.  ABCD  is  a  square  of  wires  each  1  ohm ;  across  the  diagonal  AC  is  a 
wire  of  2  ohms  containing  a  2-volt  cell.  What  is  the  potential  difference 
between  A  and  C,  what  does  it  become  when  A  and  C  are  joined  also  by  a 
1-ohm  wire;   or  when  instead  B  and  D  are  joined  by  a  l-ohm  wire ? 

20.  Define  the  practical  units  of  Current,  E.M.F.,  and  Resistance.  Three 
wires,  cut  from  the  same  reel,  1,  2,  and  3  m.  long,  are  joined  in  parallel;  what 
length  of  wire,  of  twice  the  diameter,  of  the  same  metal,  would  have  the  same 
resistance  as  this  system  ?     (  X  2) 

21.  Describe  the  action  of  a  shunt.  A  circuit  is  made  up  of  a  cell  of  internal 
resistance  1  ohm  and  E.M.F.  1  volt,  a  resistance  of  1  ohm  and  a  galvanometer 
of  8  ohms,  which  should  not  carry  more  than  0-1  amp. 

What  shunt  would  you  use  ? 

22.  A  270-ohm  galvanometer  has  a  shunt  of  30  ohms.  What  cuiretnt 
would  be  sent  through  it  by  a  cell  of  4-5  volts  E.M.F.  and  3  ohms  internal 
resistance,  when  there  are  70  ohms  in  the  rest  of  the  circuit  ? 

23.  When  a  steady  voltage  is  applied  to  50  ohms,  and  1  ohm  in  series. 
004  amp.  passes.  If  a  galvanometer  of  10  ohms  resistance  is  placed  in  parallel 
with  the  50-ohm  resistance,  what  current  will  pass  through  the  galvanometer  ? 

24.  Describe  experimental  procedure,  and  give  the  theory  of  the  balanced 
Wheatstone  bridge. 

Three  of  the  conductors  are  AB  2,  AC  3,  and  BD  10  ohms.     The  bridge 
balances  when  AB  is  shunted  by  20  ohms ;  what  is  the  value  of  CD  ? 
How  is  the  method  modified  for  electrolytes  ?     (  X  5) 

25.  Describe  the  use  of  the  Post  Ofiice  Box  for  measuring  large  and  small 
resistances. 

26.  Describe  the  construction  and  method  of  calibration  of  an  electrical 
thermometer  suitable  for  measuring  temperature  at  a  distance.     (  X  3) 


PRACTICAL  QUESTIONS 

Find  the  resistance  of  a  metre  of  wire,  and  its  resistivity. 

Compare  the  resistivities  of  two  wires,  using  a  metre  bridge. 

Measure  a  resistivity  by  P.O.  Box ;   or  the  diameter  of  a  wire. 

Measure  resistances  in  parallel,  and  compare  with  calculation. 

Find  the  thickness  of  a  strip  of  lead  foil,  by  resistance  motisureraont. 

Measure  the  coefficient  of  increase  of  resistance  with  toniiK^raturo  of  a  coil 
of  wire  [in  a  test  tube,  which  you  put  into  ice,  and  boiling  liquids] ;  or,  measure 
one  of  these  boiling  points. 

Measure  the  resistance  of  a  lamp  [probably  by  ammeter  and  vollmotorj. 

Y 


CHAPTER  XLIX 


ELECTROMOTIVE  FORCE 


§  791.  As  has  been  stated  in  Chapter  XLV,  Difference  of  Electric 
Potential  plays  the  same  part  in  Current  Electricity  as  does  difference 
of  Temperature  in  the  conduction  of  Heat,  or  difference  of  level 
in  the  flow  of  water.  The  electricity  flows  from  the  place  of  higher 
to  that  of  lower  potential ;  Difference  of  Potential  may  be 
regarded  as  the  driving  force,  and  is  usually  alluded  to  as  Electro- 
motive Force. 

The  unit  in  terms  of  which  this  is  measured  in  Current  Electricity, 
the  Volt,  has  been  defined  in  §  752  as  being  produced  in  a  con- 
ductor which  is  cutting  a  hundred  million  unit  magnetic  lines  per 
second.  This  definition  rather  suggests  a  hasty  scramble  after  an 
elusive  unit ;  nevertheless,  there  are  small  magneto  machines 
which,  turned  by  hand  at  a  fair  speed,  furnish  a  self -regulated 
E.M.F.  of  600  volts  quite  steady  enough  for  insulation  testing. 

Fortunately  it  is  found  that  very  steady  and  reliable  electro- 
motive forces  arise  during  certain  Chemical  Actions,  to  be  dealt 
with  in  Chapter  LII,  and  nowadays  the  Volt  is  realized  in  almost 
as  portable  and  handy  a  form  as  the  Ohm.  The  potential  difference 
between  the  terminals  of  the  little  '  standard  cadmium  cell '  to  be 
described  in  §  872  is  1-019  volt  at  15°  C,  with  a  very  trifling  correction 
for  change  of  temperature. 

The  Ohm  Coil  and  the  Cadmium  Cell  are  the  workaday  standards 
in  a  good  many  laboratories,  Ohm's  law.  Volts  =  Amperes  x  Ohms, 
affording  the  connecting  link.  The  Daniell  cell  is  usable  as  a  rougher 
standard.  It  makes  not  the  least  difference  what  size  a  cell  happens 
to  be,  so  long  as  the  same  chemical  substances  are  present,  at  the 
same  concentration  and  temperature  :  the  electromotive  force 
on  '  open  circuit '  {i.e.  when  ready  to  send  a  current  but  not  actually 
doing  so)  is  the  same  whether  the  cell  be  made  up  in  drops  on  the 
bench  or  in  a  bucket. 

Methods  of  Comparing  the  Electromotive  Forces  of 
Voltaic  Cells,  etc. 

§  792.  Electromotive  forces  are  differences  of  potential,  therefore 
the  Electrometers  mentioned  in  §  735  of  Electrostatics  are 
available  for  comparing  them.  They  must,  however,  be  of  sensitive 
construction,  for  it  takes  300  volts  to  make  one  electrostatic  unit 
of  potential,  and  Electrostatic  Voltmeters  only  begin  to  be  useful 
thereabouts  ;  although  they  always  have  the  advantage  of  taking 
no  current. 

642 


§  793]  ELECTROMOTIVE   FORCE  643 

One  pattern  looks  very  like  the  familiar  variable  condeniicr  of 
the  radio  set :  the  moving-plate  system  is  delicately  pivote<l,  and 
is  attracted  in  among  the  fixed  plates,  against  the  control  of  a  Kpiral 
balance  spring,  as  the  voltage  increases. 

Attraction  increasing  as  (voltage)*,  it  soon  becomes  practicable 
to  use  light  metal  plates,  like  electrophorus  plates,  directly  attracting 
each  other  (at  a  safe  distance)  against  springs  or  weights,  up  to 
50,000  volts. 

Higher  voltages,  such  as  those  in  JT-ray  apparatus,  are  often  de- 
duced from  the  sparking  distance  between  great  knobs,  §  895. 

§  793.  Applying  Ohm's  law,  that  Electromotive  Force  =  Current 

X  Resistance,  we  can  see  that 

(1)  To  keep  the  current  constant  the  resistance  in  a  circuit 
must  be  proportional  to  the  E.M.F.  acting,  or 

(2)  In  a  circuit  of  constant  resistance  the  current  will  be  pro- 
portional to  the  E.M.F. 

Thus,  to  compare  two  battery  E.M.F.*s,  a  circuit  is  made  up  as  in 

Fig.  353  with  battery  B,  galvanometer  G,  re- 
sistance box  R,  and  a  plug  key  K. 


The  circuit  is  made  and  resistance  r  is  un-      ,r-=- 


^-oa^Qy 


plugged  in  R  until  the  galvanometer  stands  at 
some  conveniently  large  reading  ^.  B  is  removefl 
and  replaced  by  the  other  battery  B',  then 

(1)  The  resistance  is  altered  to  the  r'  which 
is  found  to  cause  precisely  the  same  reading  of 
the  galvanometer,  and  ^°-  ^^' 

E.M.F.  of  B/E.M.F.  of  B'  =  r/r'. 

Or  else  (2)  r  is  left  unaltered  and  the  new  reading  g'  of  the 
galvanometer  is  observed.  Then  if  c  and  c'  are  the  relative  values 
of  current  represented  by  g  and  g'  {e.g.  for  a  tangent  galvanometer 
c/c'  =  tan  gr/tan  g')  we  have 

E.M.F.  of  B/E.M.F.  of  B'  =  c/c', 

Reading  §  875,  in  which  is  described  an  experiment  only  too  apt 
to  be  confused  with  the  present  one,  you  will  perceive  a  iK>»«iblc 
defect  in  this  method  of  comparing  battery  electromotive  forces. 
For  the  two  batteries  may  not  have  the  same  Internal  Resistances, 
and  in  any  case,  if  these  are  comparable  with  r  and  r',  the  ratio 
given  above  becomes  only  a  rough  approximation. 

e.g.  r  for  an  *  accumulator '  was  10  ohms,  r'  for  a  I.«'lanch<J  was 
5  ohms,  hence  E.M.F.'s  apparently  as  2:1.  But  with  a  more 
sensitive  galvanometer  r  was  1000  ohms  and  r '  700  ohms,  E.M.F.  a 
as  2:1-4.  The  discrepancy  was  due  to  the  Leclanchd  having  an 
internal  resistance  of  2  ohms,  while  that  of  the  accumulator  wa« 
insignificant ;  this  made  the  actual  ratio  of  resistances  in  cuncuit 


644  MAGNETISM   AND   ELECTRICITY  [§  793 

in  the  first  case  10  :  7,  while  in  the  second  case  1000  :  702  does  not 
differ  appreciably  from  the  accepted  1000  :  700. 

A  way  of  dodging  this  difficulty  is  to  keep  both  batteries  in  circuit 
always.  At  first  they  are  connected  properly  in  series,  head  to  tail, 
but  subsequently  the  weaker  one  is  turned  round  in  its  place  and 
connected  in  opposition.  Weak  as  it  may  be,  however,  its  E.M.F. 
almost  always  rises  in  wrath  at  being  driven  backwards,  and  upsets 
your  scheme,  and  destroys  even  its  educational  value. 

§  794.  Undoubtedly  the  better  plan  is  to  swamp  variations  in 
battery  resistance  by  using  a  galvanometer  with  which  is  incor- 
porated a  high  constant  resistance.  This  combination  forms  a 
Voltmeter;  it  is  method  (2)  above  in  portable  form. 

A  Voltmeter  is  a  High -Resistance  Galvanometer  with  a  scale 
graduated  to  read  volts  pressure  between  the  terminals,  instead 
of  the  magnitude  in  amperes  of  the  current  passing  through,  just 
as  a  balance  in  a  butcher's  shop  may  be  graduated  to  read  prices, 
instead  of  weight  in  pounds  avoirdupois. 

Quite  likely,  nowadays,  it  has  a  dainty  little  '  movement '  {i.e. 
'  works  ')  which  a  fiftieth  of  an  ampere  will  send  right  across  the 
scale  ;  its  coil  of  a  few  dozen  turns  of  thin  copper  wire  may  have 
a  resistance  of  perhaps  5  ohms  ;  see  Fig.  354,  left  hand,  below,  now 
follow  the  arrow. 

Suppose  it  is  wanted  for  testing  single-cell  accumulators  :  its 
maker  will  pack  in,  in  series  with  it,  a  double-wound  (§775)  coil 
of  eureka  wire  enough  to  bring  up  the  total  resistance  to  150  ohms, 
and  then  graduate  the  scale  up  to  3  volts.  The  anticipated  2  volts 
will  then  drive  a  current  2/150  =  2/3  of  1/50  amp.,  i.e.  will  push  the 
pointer  2/3  the  length  of  the  scale. 

Having  had  good  money  spent  on  it,  a  good  thing  should  be  willing 
to  give  you  a  good  return.  Pack  in,  again  in  series,  another  re- 
sistance coil  of  600  ohms,  and  bring  its  end  out  to  a  third  terminal, 
so  that  the  total  resistance  is  now  5  times  what  it  was.  Five  times 
the  voltage  will  be  necessary  to  drive  the  same  current  through, 
so  put  another  line  of  figures  along  the  scale,  0 — 15  volts,  and  now 
you  can  test  your  12-volt  car-battery. 

Yet  again,  pack  in  9  times  the  750  ohms  you  have  already, 
and  bring  it  to  a  fourth  terminal,  7500  ohms  in  all :  stick  O's  on 
the  last  row  of  figures,  and  test  your  H.T.  battery  up  to  150  volts. 

Perhaps  you  have  an  insulation- testing,  or  wireless,  magneto, 
running  up  over  600  volts,  and  you  want  to  keep  an  eye  on  it  : 
any  maker  will  be  happy  to  supply  a  separate  well-ventilated  and 
-insulated  resistance  coil,  of  no  inconvenient  size,  of  4  x  7500  ohms  ; 
and  now  if  you  use  that  in  series  with  the  first  pair  of  terminals, 
the  instrument  reads  up  to  600  volts,  or  if  with  the  whole  lot, 
5  X  7500  ohms  in  all,  it  reads  up  to  750  volts. 

And  of  course,  as  it  isn't  taking  even  a  whole  fiftieth  of  an  ampere, 
a  car  battery  would  never  notice  if  you  left  it  on  all  day.  A  Volt- 
meter is  a  Pressure  Gauge,  and  you  don't  want  it  to  run  away  with 


796] 


ELECTROMOTIVE   FORCE 


•4ft 


current   any  more  than  you  want  your  tyre  preaauroKaug©  to  rob 
you  of  the  last  dozen  pumpfuls  of  air  you  have  just  put  in. 

You  never  put  a  Voltmeter  into  the  main  circuit.  Alwavi  it  it 
just  touched  on,  at  the  last  moment,  at  the  two  point*  between 
which  you  want  to  know  the  P.D.,  the  E.M.F.,  the  Voltage. 

§795.  The  Ammeter.  That  selfsame  little  galvanometer  can 
just  as  well  be  made  up  as  an  Ammeter.  Alone,  seeing  that  O-OS 
ampere  sends  it  right  over,  the  scale  would  Ik*  graduate<l  in  20  jwirU, 
each  of  which  would  be  0-001  amp.,  and  it  would  Ix*  a  Milliammctcr! 

If  it  is  to  be  available  for  direct  currents  up  to  2  ampi^res.  99% 
of  the  current  must  be  shunted  past  it,  or  bv-passed,  (larmiciiiily, 
in  a  strip  of  manganin  which  has  99  times  the  conductance  of  the' 
little  galvanometer  coil,  or  practically  005  ohm  resistance. 


Fio.  354. 

If,  say,  10,  or  50,  or  2000  amperes  of  *  D.C. '  have  to  bo  measured, 
the  2-arap.  shunt  is  disconnected,  and  others  of  thicker  and  thicker 
manganin  strip,  and  5,  25,  or  1000  times  as  conductive,  are  mib- 
stituted  (one  can't  bother  with  the  odd  1).     *  A.C.*  is  not  so  simple. 

AH  Ammeters  are  put  in  the  main  circuit.  Our  particular  ammeter, 
of  course,  is  [galvanometer  complete  with  suitable  shunt],  not  the 
galvanometer  alone  ;  for  it,  whether  asked  to  measure  high  voltage 
unbacked  by  resistance,  or  heavy  current  unshunted  by  conductance, 
would  *  burn  out '  hastily. 

This  whole  story  of  Little  Milli-Ammeter,  what  will  she  become  f 
is  pictured  in  Fig.  354. 

§  796.  The  best  of  voltmeters  has  its  limitations,  however.  The 
voltmeter  just  described  would  read  only  0*2  volt  if  connecte<l  to  a 
standard  1-019-volt  cadmium  cell.  For  the  cell  has  an  internal 
resistance  of  about  600  ohms,  and  demands  a  far  more  perfect  method 
of  comparing  electromotive  forces  than  those  deecribed  almve. 
Such  a  method  is  afforded  by  the  Potentiometer. 

The  Potentiometer  in  its  simplest  form,  Fig.  355,  consists  of  a 
long  thin  wire  of  resistance  metal,  strctche<l  beside  a  scale  of  equal 
parts.  An  accumulator,  which  is  a  particularly  constant  sort  of 
voltaic  cell,  maintains  a  steady  current  through  the  wire,  using  no 
switch  in  its  circuit.    A  point  on  the  wire  near  the  end  connected 


646  MAGNETISM   AND   ELECTRICITY  [§796 

to  the  4-  terminal  of  the  accumulator  is  at  a  potential  nearly  2 
volts  higher  than  a  point  near  the  other  end.  If  the  wire  is  perfectly 
uniform  the  fall  of  potential  takes  place  perfectly  uniformly  along 
it ;  the  potential  difference  between  two  points  10  cm.  apart  is  the 
same  wherever  the  pair  is  located  on  the  wire,  the  potential  difference 
between  any  pair  of  points  20  cm.  apart  is  twice  as  much,  and  so  on. 

Just  in  the  same  way,  if  Water  were  flowing  along  a  straight 
uniform  channel,  the  water-level  would  fall  uniformly  with  distance, 
the  difference  of  level  between  two  points  3  miles  apart  being 
3  times  that  between  any  two  points  1  mile  apart,  and  so  on. 

At  the  upper  ( + )  end  of  the  long  wire  connect,  through  a  sensitive 
galvanometer,  a  wire  from  the  +  end  of  a  standard  cell  (sc).  The 
—  end  of  the  cell  is  connected  to  a  sliding  contact-maker  lower 
down  the  long  wire,  and  by  trial  there  is  found  for  this  a  position  B, 
such  that  when  contact  is  made,  there  is  no  deflection  of  the  galvano- 
meter. Then  the  difference  of  potential  between  A  and  B  is  equal 
to  the  electromotive  force  of  the  standard  cell. 


Fig.  355. 

[Diagrammatically,  a  voltaic  cell  is  represented  by  the  thin  + 
non-wasting  '  copper '  plate,  and  the  short  thick  dissolving  — 
'  zinc.'] 

For,  returning  to  the  water  analogy,  it  is  as  if  the  standard 
cell  were  a  weir,  in  a  back-water  represented  by  the  wires  joining 
it  to  A  and  B.  No  current  flows  in  the  back-water,  it  is  at  one 
stagnant  level  above  the  weir  and  at  a  lower  stagnant  level  below 
the  weir.  Evidently  these  levels  are  those  of  A  and  B  on  the 
stream,  or  else  water  would  flow  in  or  out  of  the  back-water  there. 
Hence  the  sudden  difference  of  level  at  the  weir  is  the  difference 
of  level  between  A  and  B. 

Now,  substituting  (as  suggested  by  the  dotted  lines)  for  the 
standard  cell,  any  one  TC  of  the  cells  to  be  tested,  a  length  AB'  is 
found  on  the  potentiometer  wire,  such  that  again  the  galvano- 
meter remains  undeflected  when  contact  is  made  there, 

y ,  E.M.F.  of  test  cell      _  length  AW 

E.M.F.  of  standard  cell  ~~  length  AB 

and  any  number  of  cells  being  tested  in  turn,  their  E.M.F. 's  are 

proportional  to  the  distances  required  on  the  wire  to  balance  them. 

Since  no  current  flows  in  cell,  connecting  wires,  or  galvanometer, 


§797] 


ELECTROMOTIVE   FORCE 


647 


when  the  desired  position  of  balance  is  attained,  therefore  their 
resistances  are  wholly  without  influence  on  the  result.  The 
standard  cell  of  600  ohms  can  be  accurately  compared  with  a 
dry  cell  of  half  an  ohm,  the  leading  wires  may  be  miles  in  length, 
the  galvanometer  may  be  the  first  sensitive  instrument  that  comes 
to  hand. 

The  potentiometer  wire  is  sometimes  doubled  back  along  the  board, 
so  as  to  be  200  cm.  long  (please  yourself  whether  you  use  all  or  half), 
and  a  nasty  educational  pattern  laps  it  to  and  fro,  but  the  best 
plan  is  that  adopted  in  the  Crompton  and  other  potent iometen*, 
now  in  widespread  use  in  commercial  work  ;  nine-tenths  of  the  wire 
are  wound  into  little  resistance  coils,  only  the  remaining  tenth  is 
stretched  along  the  scale.  Contact  A  is  made  on  one  of  the  studu 
separating  the  coils,  B  is  on  the  wire  ;  there  is  no  disadvantage 
in  this,  it  is  like  using  a  foot  rule  with  only  the  last  inch  subdivided. 
Fig.  356  I  is  reading  0463. 

In  these  instruments  the  main  current  is  adjusted  by  an  external 
variable  resistance  until  a  reading  1019  balances  the  standartl 
cell,  they  then  read  straightaway  in  millivolts  without  any  *  rule  of 
three,'  their  whole  scale  running  up  to  about  1900. 

§797.  Further  uses  of  the  Potentiometer.    The  E.M.F.  to  l>e 

measured  need  not  necessarily  be  due  to  a  voltaic  cell,  it  may  l>e 
that  between  the  ends  of  a  conductor  through  which  a  current  in 


\,A      6     it.     2      O  'S  -■- 


r]^^^S 


Fio.  356. 


flowing,  and  this  enormously  increases  the  usefulneai  of  the 
Potentiometer,  which,  you  see,  is  really  a  Very  Suixjrior  Voltmeter. 
For  suppose  we  want  to  measure  High  Voltage,  greater  than  the 
2  volts  or  so  which  the  driving  accumulator  maintains  lH»t ween  the 
ends  of  the  wire,  say  the  pressure  somewhere  in  a  nominal  240-volt 
D.C.  lighting  system.  Connect  across  the  mains  a  20.(KH)-ohm 
resistance,  select  two  points  on  this  100  ohms  aprt.  the  fall  of 
potential  between  them  is  only  1%  of  the  whole  drop :  lead  wiwi 
from  these  points  to  the  potentiometer,  and  mea«ure  their  ll  volta 
in  terms  of  the  standard  cell.     Fig.  356, 11. 


648  MAGNETISM   AND   ELECTRICITY  [§  797 

Or  if  a  Large  Current  is  to  be  measured,  it  is  sent  through  a 
standard  low  resistance,  say  700  amp.  through  a  broad  plate  of 
manganin  of  0-001  ohm  resistance.  The  fall  of  potential  between 
the  ends  of  this  will  be  0-001  of  that  over  1  ohm,  i.e.  by  Ohm's 
law  0-001  X  700  =  0-7  volt ;  wires  are  brought  from  the  ends  of  this 
low  resistance  up  to  the  potentiometer,  and  the  standard  cell 
supplants  the  ammeter  in  the  measurement  of  current .     Fig .  356 ,  III . 

The  Potentiometer  finds  further  employment  in  the  accurate 
Comparison  of  Resistances,  and  competes  successfully  with  the 
Wheatstone  bridge.  The  two  resist?inces  are  arranged  in  line 
parallel  to  the  potentiometer  wire,  and  joined  in  series  in  a  parallel 
but  entirely  independent  circuit,  Fig.  356,  IV,  fed  by  another 
steady  accumulator  ;  wires  from  the  ends  of  first  one  resistance, 
then  the  other,  are  brought  to  the  potentiometer  ;  the  ratio  of  the 
readings  obtained  is  that  of  the  potential  drops,  and  that  is  the  ratio 
of  the  two  resistances.  The  resistances  may  be  very  much  smaller 
than  can  be  dealt  with  by  the  Wheatstone  bridge,  cf .  §  788. 

Note. — In  the  Laboratory,  apart  from  bad  contacts,  determined 
kicks  all  one  way  suggest  a  cell  connected  the  wrong  way  round ; 
while  plunging  variability  points  to  a  failing  accumulator  :  get  a 
fresh  one.  And  do  not  warm  the  contact  slider  with  your  finger, 
for  that  is  the  weak  point  of  the  Potentiometer,  its  susceptibility 
to  thermo-electric  contact  E.M.F.'s  :  complete  instruments  have 
a  commutator  which  reverses  both  circuits,  to  average  this  out. 

A.C.  potentiometers  are  complicated,  and  little  used. 

§  798.  Thermo-Electricity.  Peltier  discovered  in  1834  that  when 
a  current  is  sent  through  the  Junction  of  two  different  metals,  there 
is  a  local  Heating  which  does  not  depend  on  the  size  of  the  conductor, 
but  only  on  the  current,  to  which  it  is  proportional,  reversing  and 
becoming  a  Cooling  when  the  current  is  reversed. 

It  is  thus  an  entirely  different  effect  from  the  Joule  heating  to  be 
described  in  the  next  chapter,  for  this  takes  place  in  the  one 
conductor,  is  proportional  to  its  resistance,  and  to  the  square  of  the 
current,  therefore  never  becoming  negative. 

It  is  as  if  there  were  a  very  small  difference  of  potential  between 
the  metals  :  the  current  has  to  climb  this  and  deposit  electrical 
energy  as  heat  in  the  one  case,  in  the  reverse  it  is  given  heat  energy 
to  take  away  with  it  as  increased  electrical  energy. 

It  is  therefore  to  be  expected  that  if  we  keep  on  supplying  heat 
from  without  to  the  junction  which  the  current  cools,  and  arrange 
to  take  away  the  heat  it  evolves  at  the  other  junction,  we  may 
keep  a  current  going  by  this  means  alone.  This  talhes  with  Seebeck's 
experimental  discovery  in  1821  that  In  a  circuit  composed  of  two 
different  metals,  when  one  of  the  junctions  is  heated,  and  the  other  kept 
cold,  an  electric  current  is  caused  to  flow  round  the  circuit. 

This  thermo-electric  current  is  usually  very  small,  but  by 
reducing  the  resistance  of  the  circuit  it  may  become  pretty  large. 


§  798]  ELECTROMOTIVE  FORCE  A49 

The  2-in.  ring  of  J-in.  copper  rod  in  Fig.  357  i«  completed  by 
a  short  block  of  eureka  hard-soldered  in  :  when  one  of  the  copp^ 
tails  is  warmed  in  a  flame,  that  copper-eureka  junction  becomes 
hotter  than  the  other,  and  remains  so,  since  the  alloy  is  a  poor 
conductor  of  heat.  The  resistance  of  the  ring  is  leas  than  0001  ohm, 
and  a  current  is  soon  circulating,  which,  as  the  ring  lies  in  a  gutter 
in  an  iron  casting  (broken  away),  converts  this  into  a  *  lifting 
nfcagnet,'  Fig.  325,  capable  of  a  3-  or  4-lb.  pull. 

Twisting  firmly  together  ends  of  these  twocommonest  of  lal>oratory 
wires,  connecting  the  far  ends  to  a  sensitive  galvanometer,  wanning 
the  junction  between  finger  and  thumb, 
then  cooling  it  with  ice,  you  get  deflections 
opposite  ways,  and  by  putting  in  circuit  a 
considerable  resistance  you  can  cut  dou-n 
the  deflection  proportionately,  thus  showing 
that  it  is  a  Thermo-electric  E.M.F.  that  the 
temperature  difference  produces,  and  leaves 
to  send  whatever  current  the  resistance 
permits.  Fio.  357. 

You  have  here  disguised  the  inevitable 
cold  junction,  in  the  brass  terminal  which  connects  eureka  wire  to 
copper  galvanometer  coil.  The  interposition  of  this  brass,  or  of 
solder,  does  not  matter  in  the  least,  but  it  is  better  to  make  a  pair 
of  similar  junctions  complete,  by  twisting  both  ends  of  the  stranger 
wire  up  with  copper,  and  to  keep  the  cold  junction  at  a  definite. 
temperature,  preferably  in  ice. 

Then  with  junctions  at  0°  and  100°,  copper-eureka  gives  0-0037 
volt,  the  current  flowing  from  eureka  to  copper  at  the  hot  junction. 
Copper  and  iron  wires  give  about  000 12  ;  and  iron-eureka,  the 
difference.  Between  copper,  and  lead,  solder,  mercury,  or  platinum, 
the  thermo-E.M.F.  is  very  small ;  between  copper,  phosphor- 
bronze,  and  manganin  it  is  almost  non-existent. 

That  is  why  Manganin  is  preferred  above  all  other  resistance 
metals  for  accurate  work,  stray  differences  of  temperature  in  the 
circuit  cannot  provoke  disturbing  little  E.M.F.*s,  such  as  arise 
only  too  readily  with  eureka. 

The  E.M.F.  is  not  simply  proportional  to  the  temperature 
difference.  This  is  easy  to  show  with  iron  and  copper  wires; 
heating  the  junction  gradually  with  a  match,  the  galvanometer 
spot  first  swings  out  to  the  left,  slows  down  and  stops  at  137"  C, 
returns  to  zero  at  275°  C.  with  increasing  sj)eed,  and  then  rushes 
out  the  other  way  ;  very  much  like  the  motion  of  a  stone  thrown 
up  into  the  air  from  the  edge  of  the  cliff  into  the  sea.  In  fact,  the 
relation  between  E.M.F.  and  temperature  difference  is  a  parabolk; 
one.  and  iron-copper  happens  to  be  near  the  head  of  its  paral)ola. 

Fortunately,  many  pairs  of  metals  are  well  away  doi*Ti  the  long 
leg  of  theirs,  where  it  differs  but  little  from  a  straight  line,  so  that, 
for  them,  E.M.F.  does  become  proportional  to  temperature  difference, 
with  only  a  workable  correction. 


660  MAGNETISM   AND   ELECTRICITY  [§  799 

§  799.  Such  thermo-couples  are  widely  used  for  measuring 
temperature.  Copper  and  eureka  serve  up  to  300°  C. ,  iron  and  eureka, 
with  less  power  but  less  correction,  up  to  800°  C,  other  '  base  metal ' 
alloys  up  to  1100°,  and  platinum-platinum  10%  rhodium  up  to 
1500°. 

All  that  is  necessary  is  the  junction,  of  any  convenient  pattern ; 
the  corresponding  cold  junction,  preferably  in  ice,  but  often  merely 
at  air  temperature,  which  is  allowed  for ;  and  a  sensitive  modern 
voltmeter,  or  potentiometer  for  high  accuracy.  The  scale  is  cali- 
brated by  trial  with  known  temperatures,  §  778,  and  graduated  in 
temperatures  direct. 

For  instance,  an  enamel-insulated  eureka  wire  can  be  run  down 
the  bore  of  a  hypodermic  needle,  and  be  welded  or  hard- soldered 
to  it  at  the  point  :  ground  sharp,  this  can  be  stuck  into  a  patient 
wherever  required,  and  his  temperature  read  at  any  distance,  or 
recorded  continuously  on  an  automatic  recorder.  Indeed,  one  can 
visualize  a  patient  stuck  full  of  these 
couples,  all  brought  into  circuit  in  turn  by 
a  many -way  switch  :  a  modern  St.  Sebastian, 
mild  mart3rr  to  research. 

Or  two  half -inch  squares  of  copper  and 
eureka  are  laid  flat  side  by  side,  hard- 
soldered  together  along  that  edge,  and 
then  rolled  out  in  that  direction  into  a  strip 
0-01  mm.  thick.  This  is  cut  across  into 
narrow  strips,  and  these  are  folded  round  a 
cork,  and  welded  up,  copper  to  eureka,  so  that,  as  in  Fig.  358  left,  a 
line  of  very  perfect  junctions  is  formed  in  front,  and  the  cold  junc- 
tions lie  under  cover  behind.  This  Thermopile  of  couples  in  series 
is  used  in  the  study  of  all  problems  of  Radiation,  §961,  such  as 
the  heat  of  the  stars,  or  the  energy  of  spectrum  lines,  being  put 
at  the  focus  of  the  telescope,  or  shrouded  by  a  slit  and  travelled 
along  the  spectrum  while  the  galvanometer  swings  are  automatically 
recorded. 

Dainty  contrivances  like  this  have  superseded  the  ancient  ther- 
mopile of  a  hundred  bars  of  antimony  and  bismuth,  which  weighed 
a  pound  and  fired  about  one  shot  an  hour.  But  these  old  friends, 
which  produce  nearly  double  the  copper-eureka  E.M.F.,  have 
reappeared,  in  much  more  hopeful  form,  as  Ghny  patches  distilled 
on  to  a  little  strip  of  glass  1  micron  thick,  the  two  patches  joined 
by  one  of  gold,  blackened  by  tellurium  to  make  it  a  good  radiation 
absorber,  Fig.  358,  right. 

§  800.  Theory  of  the  thermo-electric  E.M.F.  In  §  778  we  found 
metalhc  conduction  due  to  wandering  electrons  present  in  the  massive 
solid,  their  number  being  greater  in  a  good  conductor,  the  average 
energy  JmV^  of  every  one  being  that  of  any  gas  molecule,  and 
proportional  to  the  absolute  temperature.  Hence  the  rate  at  which 
they  diffuse  through  metal  is  proportional  to  y'T,  since  it  oc  V. 


801] 


ELECTROMOTIVE   FORCE 


651 


Take  a  copper-constantan  circuit,  Fig.  359  :  diflPusion  of  electrons 
takes  place  across  both  solid  junctions,  in  both  caws  the  copper, 
from  the  superabundance  of  a  good  conductor,  driving  an  excev 
into  the  alloy.  The  excess  is  greater  at  the  hot  junction,  because 
V  is  greater  there.  The  accumulation  of  these 
—  charges  means  differences  of  potential,  the 
thermo-E.M.F. 

Once  in  the  constantan,  they  diffuse  along 
towards  equal  distribution,  carrying  their 
energy  with  them,  i.e.  on  the  whole  a  stream 
both  of  negative  electricity,  and  of  heat,  flows 
from  hot  to  cold.  In  the  copper,  the  hot  end 
is  now  short  of  electrons,  and  the  electronic 
drift  in  it  is  from  cold  to  hot. 

This  appears  as  a  positive  current  circu- 
lating, crossing  from  eureka  to  copper  at  the 
hot  junction,  and  being  maintained  by  the  continued  diffusion  of 
electrons,  until  the  temperature  difiference  is  all  used  up. 

§801.  Some  metallic -looking  minerals,  such  as  iron  p\Tite«, 
develop  immensely  greater  thermo-E.M.F. *8  than  those  among 
metals  alone  :   they  are  awkward  substances  of  poor  conductivity. 

That  is  perhaps  the  reason  for  the  action,  and  the  first  part  of  the 
preceding  theory  may  be  apphed  directly  to  them  ;  and  also  to  the 

once-familiar  CYystal-cat's 
whisker  detector,  and  the 
copper-copner  oxide  Recti- 
fler.  The  flow  of  electrons 
from  the  metal's  abundance, 
into  the  metalloid,  is  enor- 
mously easier  than  any 
reverse  flow,  oscillating  cur- 
rents are  transmitted  fairly 
freely  one  way,  but  can  be 
stopped  the  other,  so  that 
groups  of  radio- fretjuency 
alternations  become  single 
uni-directional  pulls  on  the 
telephone  diaphragm,  or  al- 
ternating  current  is  roi'tificd 
into  D.C.  for  battery  charg- 
ing, wireless,  or  innumerable 
other  purposes. 

It  is  rather  remarkable  that  copper  oxide,  having  been  diligently 
scraped  off  wire-ends,  as  obstructing  dirt,  for  a  century,  now  iiroves 
to  have  this  valuable  rectifying  property  (and  al.no  a  photo-electric 
one,  see  §  984).  Fig.  360  shows  how  in  the  Westinchouse  Rectifier, 
now  in  wide  commercial  use,  coin-like  discs  of  copper,  neat -oxidiiecl  on 
one  side — the  rectification  taking  place  between  metal  and  adherent 


Fio.  360. 


652  MAGNETISM   AND   ELECTRICITY  [§  801 

crystalline  oxide — are  threaded,  alternately  with  soft  lead  discs, 
on  a  central  rod  (from  which  all  are  insulated  by  a  sleeve),  in  size 
and  number  required  for  the  current  and  voltage  to  be  dealt  with 
(from  0-001  to  1  amp.  per  string,  and  up  to  250  v.),  spacing  pieces 
and  cooling  fins  being  also  inserted  to  ensure  that  there  will  be 
air-cooling  enough  to  prevent  the  oxide  being  spoiled  by  any  over- 
heating, due  to  the  20%  of  the  entering  energy  which  is  absorbed 
in  action. 

Underneath  is  a  tracing  showing  how  exactly  the  reverse  halves 
of  the  A.C.  current  waves  are  turned  up  into  D.C.  By  putting 
inductance  into  the  circuit,  e.g.  a  choking  coil,  §  829,  the  pulsations 
of  current  are  smoothed  down  as  in  the  dotted  line. 

Current  such  as  this  can  very  well  be  used  in  electro-medical  treat- 
ment (Galvanism)  instead  of  that  from  heavy  H.T.  batteries. 

An  odd  use  of  these  rectifiers  is  to  smother  an  oscillatory  self- 
inductance  spark  (cf .  §  826)  following  a  break  in  a  motor  circuit, 
e.g.  bus  trolleys,  rattling  along  the  wires,  devastate  all  wireless 
sets  in  the  vicinity,  but  a  rectifier,  passing  current  one  way  and 
refusing  it  the  other,  cures  them  of  this  mischief. 

These  Rectifiers  go  on  working  year  in  and  year  out,  never  needing 
repair  nor  renewal. 

§  802.  Pyro-  and  piezo-electricity.  Dark  crystals,  finger-like, 
of  that  same  Tourmaline  we  have  used  to  polarize  light,  were 
carried  in  the  pocket  of  Mynheer  of  a  former  generation,  not  indeed 
to  efface  the  sky  shine  from  his  Old  Dutch  Masters,  but  to  extract 
the  hot  ashes  from  the  deep  bowl  of  his  old  Dutch  pipe,  for  when 
warmed  the  crystal  becomes  electrified,  oppositely  at  the  two  ends  ; 
it  is  pyro-electric. 

This  action  may  be  due  simply  to  the  temperature  strains  caused 
by  thermal  expansion,  for  it  and  Quartz,  and  Rochelle  Salt,  are 
piezo-electric,  crystal  slices  developing  opposite  electrifications 
on  their  faces  when  pressed  or  pulled,  the  charge  proportional  to 
the  total  force  employed. 

Tourmalines  have  been  used  to  follow  the  course  of  explosions 
in  guns ;  while  yet  another  use  of  first-class  importance  has  been 
found  for  that  versatile  substance.  Quartz,  of  which  little  slices, 
looking  like  broken  bits  of  window-glass,  control  the  frequencies 
of  the  world's  radio,  and  compete  with  the  best  clocks,  see  §§  157, 
451,  and  837. 


ELECTROMOTIVE  FORCE  053 


EXAM   QUESTIONS,   CHAPTER   XLIX 

The  chapter  takes  up  again  the  Potential  of  Chap.  XLV  : 

The  first  part  is  descriptive  of  apparatus  and  methods  you  will  uso  in  the 
laboratory;  §§  798,  799  introduce  a  fresh  way  of  gonorating  oloctrictty.  mad 
its  present-day  uses;  §  800  may  help  you  to  recollett  which  way  it  goes;  I  801 
you  may  likely  have  in  the  house;   §  802  is  referred  to  ebewhere. 

1.  Describe  two  or  three  methods  of  comparing  the  electromotive  forres 
of  cells,  and  discuss  their  relative  advantages.     (  x  2) 

2.  Two  cells  are  joined  in  series  and  give  0-044  amp.  through  a  rc«i«t4Uice. 
One  being  now  reversed  they  give  00 13  amp.  If  one  hn«  E..M.F.  1-08  volt, 
calculate  that  of  the  greater.     (  X  2) 

3.  Explain  the  Potentiometer  method  of  comparing  eleetrorootive  forrw. 
and  point  out  in  what  respects  it  is  preferable  to  the  ordinary  voltmotrr. 

How  can  it  be  used  to  compare  Resistances  too  low  to  be  moomired  accurmtely 
on  a  metre  bridge,  and  also  for  measuring  large  currents  ?     (  X  3) 

4.  Describe  some  form  of  potentiometer  suitable  for  the  comparison  of 
potential  differences  of  the  order  of  1  mv. 

Show  that  a  microammeter  of  resistance  100  ohms  can  be  converted  into 
a  millivoltmeter  by  a  suitable  additional  resistance.  Calculate  this,  and  abow 
how  it  should  be  connected. 

6.  Explain  the  construction  and  action  of  some  form  of  moving-coil  galvano* 
meter. 

By  what  external  devices  would  you  adapt  it  to  measure  (o)  largo  currmtA, 
(&)  high  voltages  ? 

6.  Why  are  galvanometer  indications  not  necessarily  proportional  to  the 
E.M.F.'s  of  cells,  and  why  does  the  proportionality  become  cloeer  the  higher 
the  galvanometer  resistance  ? 

7.  Describe  a  moving-coil  voltmeter,  and  explain  its  action. 

A  cheap  voltmeter  has  an  internal  resistance  of  35  ohms;  what  will  it  read 
when  connected  to  a  Leclanchd  of  1-5  volts,  and  internal  resistance  5  ohms  T 

8.  Describe  a  moving-coil  ammeter. 

An  ammeter  has  a  resistance  of  1  ohm  and  a  range  of  0*15  amp.  What 
length  of  wire  of  diam.  1-22  mm.  and  8po<ilic  resistance  0000034  ohm  era. 
will  make  a  shunt  which  will  increase  the  range  to  1-5  amp.  ? 

9.  Differentiate  between  voltmeter  and  ammeter.  A  lO-volt  voltmeter 
has  a  resistance  R,  what  auxiliary  resistance  will  enable  it  to  be  used  for  IOI» 
volts  ?     Would  the  same  do  for  an  ammeter  ? 

10.  Describe  with  diagrams  how  the  same  moving-coil  sj-stem  may  be 
utilized  in  (a)  an  ammeter,  (6)  a  voltmeter.  If  such  a  lOohm  voilwihcmrry 
safely  001  amp.,  how  can  you  use  it  in  (a)  an  ammeter  reading  up  to  2  amp^ffas, 
(6)  a  voltmeter  to  5  rolts  ? 

11.  What  are  the  requisites  of  a  galvanometer  for  use  as  an  ammeter? 
Having  only  a  5-ohm  milli-ammoter  reading  up  to  6  ma.,  how  would  you 
adapt  it  to  measuring  (a)  up  to  16  ma.,  (6)  volts  up  to  150? 

12.  Give  methods  for  the  measurement  of  high  voltages. 

13.  What  are  the  Joule  and  Peltier  effecU.  how  would  you 
them  and  how  distinguish  between  them  experimentally  ? 


664  MAGNETISM   AND   ELECTRICITY 

14.  How  would  you  arrange  to  generate  a  thermo-electric  current  ?  De- 
scribe the  instruments  you  would  employ  for  its  measurement. 

16.  Describe  the  production  of  a  thermo-electric  force,  and  how  by  its 
means  to  measure  temperature.     (  X  2) 

16.  How  would  you  calibrate  a  thermo-couple,  and  use  it  to  take  the 
temperature  of  an  oven  or  a  sterilizer  ? 

17.  Half  the  length  of  a  long  wire  is  copper,  the  rest  of  nickel  alloy;  a 
strong  current  flows  through  the  whole.  Electrodes,  kept  10  cm.  apart,  and 
connected  to  a  sensitive  galvanometer,  are  brought  up  into  contact  with 
the  wire  (a)  in  the  copper  part,  (6)  bridging  the  junction,  (c)  in  the  nickel 
part.  What  comparative  deflections  of  the  galvanometer  would  you  expect 
in  these  cases,  and  what  alterations  would  be  produced  if  the  current  were 
reversed  ? 

18.  Describe  a  method  whereby  an  observer  can  record  the  temperature  of 
a  patient  in  another  room,  without  visiting  him. 


PRACTICAL  QUESTIONS 

Compare  the  E.M.F.'s  of  two  cells  by  a  direct  deflection  method. 

Compare  cell  E.M.F.'s  by  potentiometer. 

Find  how  the  E.M.F.  of  a  cell  depends  on  the  dilution  of  the  electrolyte. 

Compare  two  resistances  by  potentiometer. 

Find  the  internal  resistance  of  a  voltaic  cell,  in  two  ways. 

Find  the  resistance  of  a  voltmeter,  given  a  standard  resistance. 


CHAPTER   L 
ELECTRICAL  POWER  AND  ENERGY 


§  811.  The  great  utility  of  Electricity  lies  in  the  power  it  gives  u* 
of  doing  work  at  a  distance.  That  is,  an  Electric  Current  carries 
Energy. 

It  was  explained  in  §  734  that  to  raise  a  Quantity  of  Electricity 
through  a  Difference  of  Potential  involved  the  doing  of  an  amount 
of  work  equal  to  the  product  of  charge  and  potential  difference. 
In  that  paragraph,  the  unit  of  electrical  quantity  was  the  electro- 
static unit  defined  in  §721,  and  the  unit  of  potential  waH  such 
that  their  product  was  one  erg  of  energy.  Now,  in  Current 
Electricity,  although  very  different- sized  units  arc  employed, 
the  fundamental  relation.  Quantity  of  electricity  x  potential  dif- 
ference, i.e.,  quantity  x  electromotive  force  =  Energy,  of  course 
still  holds.  The  primary  unit  of  quantity  is  that  carried  by  the 
decampere  in  one  second,  and  to  raise  this  through  the  small  unit 
potential  difference  of  §  752  demands  the  expenditure  of  one  erg 
of  energy. 

The  practical  unit  of  quantity,  the  Coulomb,  carried  by  one  ampire 
flowing  for  one  second,  is  one- tenth,  and  the  Volt  is  one  hundred 
million  times  the  corresponding  primary  electromagnetic  unit, 
§  752  ;  their  product  is  therefore  ten  million  ergs,  the  Joule  of  §  62. 

To  drive  one  Coulomb  of  electricity  against  a  jwtential  dijferenee 
of  one  Volt  requires  one  Joule  of  work  to  be  done. 

Conversely,  when  an  electromotive  force  amounting  to  one  Voli 
between  point  P  and  point  Q  in  a  circuit  has  driven  one  Coulomb 
of  electricity  from  P  <o  Q,  one  Joule  of  work  has  been  done. 

Whether  any  of  this  electrical  work  has  been  converted  into 
useful  mechanical  work,  or  whether  it  has  all  been  dissipated  in 
heat,  depends  on  the  nature  of  the  circuit  between  P  and  Q.  An 
electro-motor  in  PQ  could  give  us  most  of  this  as  mechanical 
energy,  a  mere  resistance  wire  converts  it  at  once  into  heat  with 
perhaps  a  little  light. 

§  812.  To  measure  the  expenditure  of  electrical  energy  we  a<lopt 
the  arrangement  of  Fig.  301.  The  Ammeter  A  measures  the 
current  through  PQ,  the  watch  T  measures  the  duration  of  the  current 
in  seconds,  AT  is  the  number  of  coulombs.  The  Voltmeter  V.  applied 
from  time  to  time  as  a  shunt  over  the  points  PQ.  measure*  the 
electromotive  force  or  potential  difference  lHMw(H*n  them,  in  Volts; 
VAT  Joules  of  work  have  been  expended  in  PQ.  whether  it  be  motor. 
resistance  wire,  lamp,  electrolytic  cell,  or  whatnot.  (Kccollecl 
4-2  joules  make  1  calorie  of  heat.) 

655 


656 


MAGNETISM   AND   ELECTRICITY 


[§812 


Fig.  361. 


Of  course  there  is  in  practice  a  wide  choice  of  quantity-  and 
pressure-measuring  instruments. 

In    experiments    of    high    accuracy    most    observers   nowadays 

would  probably  employ  a  '  silver 
voltameter,'  §  858,  for  coulombs,  and 
a  potentiometer  and  standard  cadmium 
cell  for  volts,  §  872. 

Or,  taking  the  commonest  instance 
of  all,  the  quantity  of  electricity 
entering  your  house  is  measured  on 
the  spot  by  a  meter  such  as  was 
described  in  §  766,  the  pressure  is 
measured  at  the  Electric  Supply  Station 
by  a  voltmeter,  a  little  loss  of  pressure  in  the  mains  is  allowed  for, 
and  you  are  charged  for  the  (coulombs  x  volts)  you  '  consume.' 

§  813.  Your  quarterly  account,  however,  contains  no  mention 
of  either  of  these  things,  but  is  reckoned  on  the  number  of  '  Board 
of  Trade  Units  '  (B.T.  Units,  or  simply.  Units)  at  a  few  pence  each. 
When  you  inquire  of  the  engineer  what  these  may  be,  you  hear 
that  they  are  *  Kilowatt  hours.' 

The  Watt  is  the  unit  Power  adopted  in  electrical  measurements, 
i.e.  the  unit  rate  of  doing  work. 

A  Power  of  one  Watt  doss  one  Joule  of  Work  per  second,  §  66, 

Watts  X  seconds  =  Joules. 

One  Horse-Power  =  746  watts  ;  one  Kilowatt  =  1000  watts 
=  about  IJ  h.p. 

That  is,  the  B.T.  Unit,  the  Kilowatt-hour,  is  the  amount  of  work 
done  by  1 J  h.p.  working  for  an  hour. 

1  B.T.  Unit  =  1000  X  (joule/sec.)  x  3600  sec.  =  3,600,000  joules 

Also  Watts  =  Volts  x  Amperes 

Multiplying  together  the  readings  of  voltmeter  and  ammeter  in 
Fig.  361  gives  the  Power  in  Watts  in  PQ  at  the  moment  (while 
of  course  dividing  gives  its  resistance). 

§  814.  Wattmeters  combine  the  two  instruments,  and  read  Power 
straightaway.  They  are  moving-coil  instruments  of  the  A.C.  type 
of  §  766 ;  the  main  current  flows  in  the  fixed  coils,  Fig.  362,  and 
produces  a  field  proportional  to  itself,  while  the  shunt  current 
of  Fig.  361,  which  is  proportional  to  the  voltage,  is  separately  sent 
through  to  the  moving  coil,  the  deflection  of  which  therefore  reads 
the  product.  Watts. 

In  Power  Stations,  the  Wattmeter  is  master  ;  especially  in  A.C. 
supply,  where,  in  consequence  of  volts  and  amperes  getting  '  out 
of  phase,'  a  variable  '  power-factor  '  discounts  their  full  product ; 
§  829. 


§  815]  ELECTRICAL   PO\VER   AND   ENERGY  «57 

The  ironless  Wattmeter  of  the  A.C.  Laboratory  ifl  a  verj-  accurate 
instrument  indeed.  Commercial  instruments,  of  the  build  of  Fig. 
362,  are  often  filled  in  with  laminated  iron  to  get  ample  workiiur 
force.  or  •* 

Power  being  the  product  of  Volts  and  Amperes,  a  large  current 
at  a  low  voltage  carries  no  more  power  than  a  quite  small  current 


Fig.  362. 

at  a  high  voltage  ;  J  amp.  from  a  3o-volt  dry  battery  runs  a  pocket 
lamp,  J  amp.  on  a  240- volt  supply  runs  a  60-watt,  50  candle-power 
lamp  ;  373  amp.  at  2  volts  would  be  necessary  to  drive  the  1-h.p. 
motor  that  IJ  amp.  drives  at  600  volts,  1/16  amp.  at  12,000  volt* 
or  1/180  amp.  at  132,000  volts. 

Fig.  363  is  the  Ohm-meter  of  §  787,  put  here  for  contrast,  as 
giving  V/A  instead  of  VA. 

§  815.  Heat  produced  in  a  resistance.  In  the  particular  east 
of  PQ  being  simply  a  resistance  of  some  sort  obeying  Ohma  law, 
we  can  find  another  expression  for  the  energy-  expended  in  it,  now 
in  the  form  of  Heat  (including  light). 

For  volts  =  amperes  x  ohms 

.*.  Energy  VAT  =  ARAT  =  A*RT  joules 

which  expresses  Joule's  Law  that  The  dissipation  of  energy  as  heat 
in  a  resistance  is  proportional  to  the  resistancty  the  time,  and  the 

square  of  the  current. 

Watts  =  (amperes)*  x  ohms 
Joules  =  (amperes)*  x  ohms  x  seconds 

and  since  4-2  joules  =  1  calorie,  §  254 

Heat  produced,  in  calories  =  (ampires)^  X  ohms  x  seconds  -i-  4-2. 

To  test  this  law  experimentally,  an  open  coil  of  eureka  wire 
(the  resistance  of  which  does  not  appreciably  change  with  torn- 
perature)  is  wound,  and  its  resistance  measured.  It  is  immenicd 
in  paraffin  oil  in  a  calorimeter,  the  total  water-equivalent  of  which 
is  known,  and  the  current  from  a  strong  battery  of  several  celU 
is  sent  through  for  a  definite  time.  The  current  is  regulated  by 
adjustable  resistance  and  measured  by  an  ammeter. 


658  MAGNETISM   AND   ELECTRICITY  [§816 

§  816.  Heating  can  be  localized  in  a  circuit  by  introducing 
short  pieces  of  high  resistance,  and  this  local  heating  is  made  the 
greatest  possible  use  of,  as  you  know. 

*  Hot-wire  '  Ammeters  are  actuated  by  the  expansion  of  a  fine 
wire  which  is  heated  by  the  current  to  be  measured,  either  con- 
tinuous or  alternating.  The  wire  sags,  a  thread,  attached  to  its 
middle,  and  wound  round  the  axle  of  the  pointer,  is  pulled  back 
by  a  spring,  and  the  pointer  gives  the  true  power-value  of  the  current, 
however  irregular  the  latter  may  be. 

Resistances  of  coiled  iron  wire  or  ribbon,  used  for  regulating 
considerable  currents  {e.g.  for  starting  motors),  have  to  be  well 
ventilated.  On  the  other  hand,  electric  car-  and  room-heaters, 
ovens,  flat-irons,  warming-pans,  kettles,  saucepans,  etc.,  etc., 
are  designed  to  make  the  best  use  of  the  heat  generated  in  wires, 
or  strips,  of  '  nichrome  '  or  other  non-corroding  resistance  metal, 
usually  embedded  in  insulating  covering,  and  forming  part  of  their 
walls. 

The  surgeon's  electric  cautery,  and  the  car  cigarette  lighter, 
alike  depend  on  a  short  piece  of  thin  wire  heated  to  white-heat 
by  the  current  from  a  few  cells. 

The  Electric  Incandescent  Lamp  was  at  first  also  a  fine  platinum 
wire,  enclosed  in  a  vacuous  glass  bulb.  The  platinum  wire  was 
soon  superseded  by  a  carbonized  vegetable  fibre  (Edison),  or  (Swan) 
a  filament  obtained  by  carbonizing  a  squirted  thread  of  what  years 
later  reappeared  as  artificial  silk,  and  then  precipitating  a  glossy 
coating  of  graphite  on  it  by  heating  it  to  redness  in  a  hydrocarbon 
vapour. 

In  1890,  twelve  Court  dressmakers  sewed  round  an  electric  lamp, 
which  must  be  good,  because  while  it  gave  the  light  of  eight  whole 
candles  it  dropped  no  grease  on  their  work.  Their  grandmothers 
were  vastly  better  off  with  Fig.  244. 

Incandescent  lamps,  as  bought,  are  marked  to  '  take  '  W  watts 
at  voltage  V  of  the  mains.  This  means  that,  striking  a  balance 
between  cost  of  power  and  cost  of  renewals,  the  best  economy  will 
be  obtained  by  using  that  particular  voltage. 

The  power  consumption,  in  '  Watts  per  candle  '  VA/c.p.,  is  too 
great  at  lower  voltages,  for  the  lamp  may  then  csiTiy  quite  half 
the  current  and  be  only  dull  red  hot ;  it  diminishes  rapidly  as  the 
voltage  is  increased,  but  the  life  of  the  lamp  shortens  faster. 

The  carbon  filament  lamp  glowed  yellow-hot  at  3-5  watts  per 
candle  for  an  average  life  of  1000  hr.,  or  as  dazzling — and  as  efficient 
— as  any  modern  flood-light,  for  half  a  minute.  For  you  will  read 
in  Chapter  LVI  how  the  output  of  light  from  an  incandescent  solid 
increases  as  more  than  the  fifth  power  of  its  absolute  temperature, 
an  immense  effect,  easy  to  test  in  your  own  room  at  night,  a  white- 
hot  pocket-lamp  filament  no  bigger  than  this  S  giving  more  light 
than  a  whole  glowing  red-hot  fire.  Skipping  the  troubles  of  §  478, 
this  is  a  single-crystal  thread  of  tungsten,  which  can  be  drawn  down 
to  wire  1/1200  in.  thick,  twice  as  strong  as  steel,  spider- stretched 
in  vacuo  and  run  at  2600°  A.,  or  coiled  and  double-coiled  in  argon 


§816]  ELECTRICAL   POWER   AND   ENERGY  650 

just  dense  enough  to  keep  it  from  disinteerating  and  blackening 
the  bulb,  and  run  at  from  3000°  A.,  at  which  40  watts  gives  32  c.p. 
to  the  100  candles  of  a  100-watt,  and  the  1360  c.p.  of  a  1000-watt 
flood-light  bulb,  at  3300°  A. 

Now  that  stainless  materials  have  provided  us  with  rcflcctoni 
that  stay  bright,  a  great  improvement  in  illumination  has  been 
made  possible,  as  you  see  in  every  well-lit  street,  by  catching  and 
reflecting  light  formerly  wasted  towards  the  back.  Consequently 
more  attention  is  now  paid  to  the  all-round  brilliance  of  the  lamp, 
its  '  mean  spherical  candle-power,'  and  it  has  become  customary 
to  quote  its  output  in  *  Lumens  per  Watt.' 

A  1-c.p.  lamp  emits  1  lumen  into  unit  solid  angle,  i.e.  into  1  sq. 
m.  of  the  surface  of  a  surrounding  sphere  1  m.  radius,  which  contains 
47r  sq.  m. — '  all  round '  is  solid  angle  4:: ;  think  of  a  blackberry  with 
12  fat  drupels  and  a  little  one.  Thus  an  all-round  efficient  radiant 
has  lumens  per  watt  =  4::  x  c.p.  per  watt ;  an  arc,  which  lighu 
up  scarcely  1/4  the  sphere,  would  have  l/w.,  about  3  c.p./w. 

[Selective  radiation  in  the  visible  spectrum  can  be  more  efficient 
than  this  full  radiation  due  to  temperature,  see  §  975  :  neon  tubes 
take  only  J  watt  per  candle,  and  flame  arcs  and  mercury  tubes 
a  third  of  a  watt.] 

Lamps  are  always  arranged  in  parallel  between  the  mains,  and 
the  highest  candle-power  lamps  have  the  highest  conductance«, 
i.e.  the  lowest  resistances  ;  for  V  being  constant,  A  is  proportional 
to  the  conductance,  and  so  therefore  is  the  power  consumed, 
VA,  or  A^K. 

The  Electric  Furnace  is  a  trough  packed  at  start  with  a  con- 
ducting mixture  of  coke,  ore,  etc.  Several  hundred  volts  is  applied 
between  carbon  blocks  at  the  ends,  current  starts,  warms  the 
mass  and  increases  its  conductance ;  VA  rapidly  increases,  and 
partly  by  pure  resistance,  partly  by  arc  formation,  the  whole 
contents  are  presently  boiling  somewhere  between  2000°  and 
3500°  C.  Such  furnaces  are  employed  in  stainless  steel  manufacture 
in  this  country,  and  even  in  iron-smelting  in  Norway ;  at  Niagara 
coke  and  sand  yield  the  intensely  hard  carborundum,  from  which 
the  highest  temperatures  distil  everything  to  leave  pure  soft 
graphite. 

One  is  sure  to  hear  it  asked,  '  Why  cannot  an  electric  current  be 
regulated,  like  water  or  gas,  by  partly  turning  off  a  tap  ?  '  The 
sharp  constriction  in  a  half -closed  tap  causes  violent  eddies  in  the 
fluid,  and  it  is  the  dissipation  of  energy  in  these  that  absorbs 
driving  pressure  and  slows  the  stream,  §  124.  No  such  eddies  are 
produced  in  the  electric  stream,  and  the  whole  resistance  of  a 
short  sharp  constriction,  say  the  points  of  contact  in  a  nearly 
opened  switch,  is  but  small.  Further,  a  stream  of  fluid  of  great 
heat-carrying  capacity  conveys  away  the  frictional  heat  generated 
in  the  tap,  but  the  electric  stream  cannot,  and  the  constricted  jMirt 
burns  out.  . 

Fuses  are  short  bits  of  thin  wire  introducwl  into  a  circuit :  being 
short  they  normally  absorb   no   energy   worth    mentioning,  but 


660  MAGNETISM   AND   ELECTRICITY  [§816 

when,  from  a  *  short-circuit '  or  an  '  overload,'  a  current  too  great 
for  safety  begins  to  flow,  the  fuse  wire  melts  and  stops  it. 

§817.  The  dangerous  high  voltages  that  the  Electric  Supply 
Companies  adopt  are  a  consequence  of  Joule's  law ;  they  are 
imposed  by  economic  considerations  of  the  loss  of  energy  in 
the  mains.  As  an  instance,  suppose  it  is  desired  to  drive  a 
150-h.p.  motor  half  a  mile  away.  At  the  old-fashioned  supply 
pressure  of  110  volts  the  motor  requires  1000  amp.,  for  150  h.p.  is 
about  110  kilowatts.  A  copper  conductor  1  sq.  in.  cross-section 
is  usually  allowed  for  1000  amp.,  go  and  return  mains  would 
therefore  contain  63,360  cu.  in.  of  copper,  weighing  20,000  lb. 
and  costing  £300  or  more.  Their  resistance  would  be  0-04  ohm, 
and  to  drive  1000  amp.  through  this  absorbs  1000  X  0-04  =  40  volts, 
by  Ohm's  law.  That  is,  150  volts  pressure  must  be  maintained 
by  the  generator  to  keep  110  at  the  motor  ;  the  power  represented 
by  1000  amp.  driven  by  the  difference,  40  volts,  (about  50  h.p.) 
being  absolutely  wasted  in  merely  warming  up  the  mains. 

But  at  5500  volts  the  required  110  kilowatts  would  be  carried 
by  20  amp.,  for  5500  X  20  =  110  X  1000.  A  copper  wire  only 
l/50th  sq.  in.  section  need  be  allowed  for  this  ;  weighing  only  400  lb., 
costing  £6  for  copper,  and  rather  more  for  its  share  in  the  oily 
brown-paper  insulation  of  the  cable. 

Its  resistance  would  be  50  times  as  much  as  before,  i.e.  2  ohms  ; 
to  drive  20  amp.  through  this  takes  2  X  20  =  40  volts.  This  is 
as  great  a  fall  of  pressure  on  the  way  as  before,  but  now  it  is  only 
an  insignificant  addition  to  5500  volts — less  than  1% — only 
40  X  20  =  800  watts  =  about  1  h.p.  is  now  wasted  in  the  mains. 

Thus  the  economy  of  transmission  improves  about  proportionally 
to  the  voltage. 

Long-distance  transmissions  are  working  at '  extra-high-pressures ' 
to  which  loss  by  sputtering  off  into  the  air  sets  a  practical  limit. 
In  the  British  '  Grid  '  this  has  been  raised  to  132,000  volts  by  using 
steel-cored  aluminium  cables,  which  are  thicker  than  copper  of  the 
same  conductance,  i.e.  are  of  less  curvature,  and  can  therefore  be 
pushed  to  a  higher  potential  before  '  point-discharge,'  §  895,  becomes 
prohibitive.  In  the  350-mile  Merano-Rome  transmission  oil-filled 
cables  are  working  at  200,000  volts. 

§  818.  The  following  brief  analysis  of  the  outgoings  of  the  County 
of  London  Electric  Supply  Co.,  which  does  a  thoroughly  mixed 
business,  may  be  of  interest  as  showing  where  your  money  goes  : 


Coal,  oil  and  water 

Wages  and  salaries 

Repairs  .... 

Depreciation 

Rent,  rates,  taxes  and  insurance 

Advertising,  legal  and  official  . 

Interest  on  capital 


£2f. 

£31-  millions 

13-3 

15-50/0 

9 

8-75 

7 

7-75 

18-2 

20-5 

9 

8-5 

2-5 

2-5 

41 

36-5 

§818]  ELECTRICAL   POWER   AND   ENERGY  661 

The  average  selling  price  of  the  energy  was  0.88d.  per  Unit  in 
1933;  and  0.68od.  m  1934,  when  a  quarter  of  the  output  of  a 
thousand  million  units  was  sold  to  '  the  Grid,'  which  of  courw 

imposed  its  own  charges  later. 

The  cleanliness  and  efficiency  of  electric  lamps  bring  them  into 
use  for  indoor  lighting  at  much  higher  prices  than  thin,  and  tho 
ready  convenience  of  electro-motors  will  always  find  them  employ- 
ment about  the  home,  but  the  use  of  electricity  for  Heatinir  is  another 

story.  ^ 

A  pound  of  ordinary  good  coal  burns  with  the  production  of 
12,000  British  Thermal  Units,  §  228,  and  accordingly  it  takes  Hi  lb 
to  produce  1  Therm,  100,000  B.Th.U..  costing,  with  coal  Bi  60s. 
per  ton,  twopence-farthing. 

The  Gas  Companies  charge  about  five  times  as  much,  and  one 
doesn't  see  many  gas  fires  roaring  away  lavishly  under  wide-open 
flues  :  users  are  apt  to  content  themselves  with  a  much  more  re- 
stricted ventilation,  and  to  ignore  a  good  deal  in  the  way  of  gassy 
fumes. 

The  B.T.U.  of  electrical  energy  is  only  3,600,(X)0  ~  4-2  calories 
■^  252  =  3416  B.Th.U.  of  heat  =  003416  Therm,  so  that,  even 
if  you  are  lucky  enough  to  get  your  electric  power  supply  at  a  penny 
a  unit,  the  Therm  costs  you  half-a-crown,  almost  three  times  as 
much  as  gas. 

In  the  light  of  the  analysis  above,  and  of  the  consideration  that 
heat  engines  inevitably  waste  the  greater  part  of  the  fuel  energy 
supplied  them,  §294,  it  looks  unlikely  that  electrical  units  will 
be  vastly  cheaper,  though  a  certain  amount  of  *  dumping  '  at 
off  times  goes  on,  at  merely  the  cost  of  keeping  the  plant 
running,  so  much  of  it  having  to  be  installed  to  meet  *  peak  load  * 
conditions. 

Hence,  although  electric  heaters  are  all  100%  efficient  (and 
very  inoffensive  unless  they  get  dusty  ;  roasted  dust  smells  horribly), 
one  finds  their  activities  carefully  localized — flat-irons,  closed 
toasters,  '  built-in  '  boiling  elements,  focussed  bowl-fires,  etc.,  etc. — 
and  although  cooking-stoves  are  well  lagge<l,  so  as  to  cook  the  dinner 
without  cooking  the  cook,  their  use,  and  that  of  adequate  room 
heaters,  is  reckoned  beyond  the  means  of  the  majority. 

Yet  when  you  find  yourself  in  practice,  and  dealing  winter  by 
winter  with  bronchial  cases  among  good  housekeepers,  addicted, 
as  so  many  of  them  are,  to  overmuch  worship  at  the  shrine  of  the 
gas-cooker,  and  possibly  to  warming  their  bedrooms  by  a  little 
gas-stove  too  small  to  maintain  its  owi\  draught  up  an  otherwise 
cold  chimney  (it  may  be  all  right  if  the  dining-room  flue  goes  up 
alongside),  I  do  adjure  you  to  get  them  away  from  the  sulphurous 
gas  fumes,  which  are  as  mikindly  to  their  lungs  as  to  their  indoor 
pot-plants,  and  to  persuade  them  that  '  all-electric  *  is,  in  the  end, 
not  so  costly  as  it  looks. 

Tliat  is,  unless  you  can  stir  up  the  Oas  Company  to  inMal  the 
process  for  removing  all  sulphur  from  their  gas. 


662 


MAGNETISM   AND   ELECTRICITY 


[§819 


Fig.  364.      Fig.  365. 


§  819.  The  earliest  power-transmission  systems  were,  of  course, 
the  telegraph  lines.  Telegraphy  is  now  far  too  highly  specialized 
to  discuss  here,  but  something  may  well  be  said  of  its  more  domestic 
representative,  the  Telephone. 

The  many  patterns  of  Carbon  Microphone  transmitter  all  depend 
on  the  fact,  which  you  have  probably  made  use  of  in  adjustable 
carbon-plate,  or  carbonized  cloth,  Rheostats,  in  the  laboratory, 
that  the  conductance  of  the  contact  between  carbon  surfaces  is 
roughly  proportional  to  the  force  squeezing  them  together. 

In  Fig.  364  an  aluminium  diaphragm  (shown  black  and  thick), 
clipped  in  the  rebate  of  the  casing  by  a  spring  ring,  carries  on  its 
middle  a  metal  block  faced  with  a  smooth 
disc  of  carbon,  and  this  faces  a  similar  disc 
firmly  fixed  in  the  casing,  but  insulated  from 
it.  Enclosing  the  two  discs  is  a  rather  larger 
box,  lined  with  insulating  paper  inside,  and 
nearly  filled  with  granules  of  hard  carbon 
(shown  piled  between  the  two  carbon  plates, 
and  accumulated  in  the  space  at  the  bottom). 
Mica  and  soft  felt  washers  prevent  any 
getting  out  past  the  front  disc  as  it  vibrates 
with  the  diaphragm.  The  ample  spare  supply 
of  loose  granules  ensures  that  this  instrument 
is  always  ready  for  use  in  any  position. 
There  is  an  additional  thin  front  plate  cover- 
ing in  the  diaphragm  except  for  a  central 
hole  :  this  virtually  increases  the  stiffness  of  the  diaphragm  and 
raises  its  natural  Chladni  notes  well  away  from  interference  with 
ordinary  speech. 

An  older  pattern,  with  granules  among  felt  pads,  appears  in  Fig. 
366  ;   and  in  the  specially  sensitive  microphones  which  are  palmed  ' 
off  on  poor  deaf  people  at  prices  suggestive  of  surgical  fees,  the 
granules  are  sometimes  carbonized  poppy-seeds  :    a   pocket  lens 
will  show  you  why,  next  time  you  stroll  through  a  cornfield. 

In  Fig.  366,  two  or  three  wet  or  dry  Leclanches*  locally,  or  as  ; 
many  accumulators  at  the  exchange,  supply  current :  the  resistance 
of  the  microphone  is  roughly  50  ohms,  and  ordinary  speech  sends 
at  most  100  milliwatts  to  line.  This  low  voltage,  adopted  so  as 
not  to  burn  the  carbon  contacts,  is  ill  adapted  to  a  long  transmission 
line,  and  the  fluctuating  current  therefore  goes  into  a  little  3  : 1  step- 
up  Transformer,  which  sends  out  a  faithful  8-volt,  10-milliamp.  copy 
of  it.  Leakage  losses  on  a  long  line  reduce  the  power  entering  the 
Receiver  to  an  average  of  0-01  milliwatt,  100  ergs  per  second,  of 
which  it  reproduces  a  minute  fraction  as  sound  (for  this  is  the  full 
power  of  ordinary  speech,  §  420).  This  scarcely  suggests  efficiency, 
but  then  the  very  smallest  2000  squeak  audible  to  a  keen  ear  under 
good  conditions  has  lately  been  measured  as  only  a  four  hundred 
millionth  of  an  erg.  (People  '  a  little  hard  of  hearing  '  probably 
want  10,000  times  as  much,  which  is  why  these  much -advertised 
deaf -aids,  amplifying  100  or  so,  are  frequently  a  failure). 


§819]  ELECTRICAL   POWER   AND   ENERGY  663 

In  a  Condenser  Microphone,  Fig.  365,  a  thin  aluminium  allov 
diaphragm  is  stretched  tightly  l/IOOOth  in.  in  front  of  a  solid 
metal  plate,  from  which  it  is  insulated  by  a  peripheral  ring.  The 
two  form  an  air  condenser,  which  is  kept  charged  up  to  2o6  volu 
by  an  H.-T.  battery  connected  through  a  20-megohm  resiatance. 
Audio-frequency  vibration  of  the  diaphragm  alters  the  capacity. 
b/47r^  too  rapidly  for  any  particular  flow  in  this  circuit,  and  the 
fluctuating  current  emitted  is  fed  through  a  001-mfd.  condenser 
to  the  grid  of  an  amplifying  valve  ;  for  while  it  is  understandable 
that  this  thin  free  diaphragm  follows  the  motion  of  the  air  far  more 
delicately  than  does  a  carbon  slab  with  a  heap  of  coke  piled  against 
it,  it  is  only  a  hundredth  as  sensitive,  and  its  use  is  consequently 
confined  to  broadcasting,  and  talkie  studios.  Set  in  a  block  of 
marble,  and  lightly  slung,  it  escapes  mechanical  interferences. 


Fio.  366. 


Fio.  367. 


The  telephone  line  is  double,  for  an  earth  return  causes  disturbing 
noises,  and  if  alongside  telegraph  wires  the  go  and  return  wires 
twist  round  each  other  every  four  spans,  to  prevent  them  picking 
up  signals  inductively.  The  wire  is  40  lb.  bronze ;  high-grade 
porcelain  insulators  are  used  in  England  because  boys  shy  at  glass 
ones,  glass  throughout  America  because  they  have  bugs  which  build 
inside  porcelain  ones  and  spoil  the  insulation. 

In  cables,  each  wire  is  spirally  wrapped  in  dry  paper ;  1200  pairs 
of  these  7-lb.  wires  fill  a  3-in.  lead  sheath. 

Long  trunk  lines  employ  valve  repeaters  amplifying  100:  1,  at 
intervals.     Submarine  cable  must  be  lapped  with  mumetal,  §  833. 

The  whole  system  is  laid  out  for  frequencies  between  250  and  2500, 
bass  not«s,  and  s's,  which  demand  6000,  are  impracticable,  and  at 
best  the  sound  distortion  is  considerable. 

In  the  Head-phone  Receiver  the  current  traverses  coils  of  fine 
wire  wound  round  the  small  soft  iron  pole-pit'ces  of  a  steel  magnet, 
often  ring-shaped,  Fig.  367,  and  weakens  the  pull  of  the  latter  on 
a  thin  iron  plate  about  1/100  in.  in  front  of  the  poles,  and  so  sets  it 
in  vibration.  It  has  its  own  chief  frequency  of  IKX),  but  this  .seldom 
intrudes,  §  453,  the  resistance  of  the  coils  is  70  ohms,  and  im])cilance. 
at  800  cycles,  250  ohms  :  high-resistance  coils  are  for  *  crystal ' 
circuits. 

The  pull  on  the  diaphragm  is  proportional  to  magnet  |>ole  and 
to  the  pole  it  induces  in  the  plate,  i.e.  to  (magnet  jwle)*,  M*  l)ecome« 
(M  —  m)2  =  M2  —  2  Mm(-}-  w*,  negligible),  the  difference  in  pull 
.*,  oz  M,  hence  the  necessity  for  a  strong  permanent  magnet.     A 


664  MAGNETISM  AND   ELECTRICITY  [§  819 

demagnetized  receiver  is  dumb,  or  else  tries  to  talk  in  unintelligible 
double  frequency. 

Moving-iron  Loud  Speakers  suffer  from  the  defective  magnetic 
properties  of  iron.  Moving-coil  loud  speakers  have  the  strongest 
possible  cobalt-steel  magnet  with  a  central  mushroom-head  pole 
entirely  surrounded  by  the  opposite  pole,  rather  like  Fig.  325,  so 
that  the  intense  field  is  everywhere  radial  across  the  narrow  annular 
crevasse.  In  this  moves  the  pill-box-shell-shaped  coil,  pumping 
up  and  down  at  right  angles  to  both  field  and  wire,  §  748,  and 
directly  coupled  to  the  diaphragm. 


EXAM   QUESTIONS,    CHAPTER   L 

The  greatest  value  of  an  electric  current  at  the  present  day  lies  in  the  Work 
it  can  do  for  us  at  a  distance  :  the  story  of  the  Slave  of  the  Lamp  has  become 
a  commonplace  in  this  era  of  Electric  Power. 

This  chapter  introduces  the  units  and  methods  of  power  measurement, 
and  then  devotes  itself  to  a  discussion  of  your  everyday  uses  of  electricity. 

1.  Define  ampere,  volt,  ohm,  watt,  Board  of  Trade  unit.  Define  the  c.g.s. 
unit  of  electromotive  force  and  state  how  it  is  related  to  the  volt. 

2.  A  wireless  enthusiast  charges  his  accumulator  (3  cells,  each  of  2-5  volts 
when  on  charge)  from  the  210-volt  D.C.  lighting  circuit,  using  a  lamp  as  a 
resistance.  Sketch  a  circuit,  and  specify  the  most  suitable  lamp  for  a  half- 
ampere  current.  Calculate  the  cost  of  a  charge  of  40  ampere-hours  if  the 
energy  costs  Qd.  per  unit. 

3.  Find  an  expression  for  the  power  dissipated  in  a  circuit  in  terms  of  the 
current  and  the  resistance. 

An  electric  lamp  takes  60  watts  on  a  200-volt  circuit.  Find  (a)  its  resistance, 
(6)  the  current,  (c)  the  time  taken  to  use  1  kilowatt-hour.     (  x  2) 

4.  What  is  meant  by  the  luminous  efficiency  of  an  incandescent  filament 
lamp  ?     How  is  it  measured  ? 

Give  a  diagram  showing  the  apparatus  necessary,  and  the  connections 
required,  in  a  determination  of  it. 

5.  Calculate  the  cost  of  lighting  a  house  with  20,  1-watt,  50-c.p.  lamps, 
per  100  hr.  at  5d.  per  B.T.U. 

6.  What  is  a  joule  ?  A  station  maintains  550  volts  on  a  trolley -wire  of 
0-55  ohm  per  mile,  and  the  current  returns  by  rails  of  0-05  ohm  per  mile; 
if  there  is  only  one  car  on  the  line,  how  far  out  is  it,  to  be  using  35  amp.  at 
500  volts  ? 

7.  How  would  you  measure  the  heat  generated  by  a  current  in  a  given 
time  ?     Show  how  to  deduce  the  value  of  the  mechanical  equivalent  of  heat. 

8.  Derive  Joule's  Law  of  the  production  of  heat  in  a  circuit.  A  heater  of 
resistance  55  ohms  carries  2  amperes  for  1  hr. ;  what  is  the  necessary  voltage, 
and  how  much  heat  is  produced  ?     (  x  2) 


ELECTRICAL    POWER   AND   EXEKGY  666 

9.  The  tungsten  filament  of  a  lamp,  00054  cm.  diam..  in  hoat«d  by  0-5  amp 
Find  the  heat  radiated  per  sec.  per  sq.  cm.  of  surface  of  filamont.     [Specifie 

resistance  69  microhms.] 

10.  What  factors  control  (a)  the  rate  of  ri«e  of  Unnperature  of  a  lamp 

filament,  (6)  its  highest  temperature  ? 

IL  A  wire  of  resistance  0-67  ohm  per  metro  carriwi  1-5  amp6nM.  If  tha 
emission  of  heat  is  002  calorie  per  second  per  dogroo  difTeronce  of  t«mperatur« 
between  the  wire  and  its  surroundings,  which  aro  at  15"  C,  find  the  ateady 

temperature  of  the  wire.     (  X  2) 

12.  Describe  a  wattmeter. 

Eighty  joules  are  supplie<l  to  a  machine  each  second ;  thia  energy  ia  uaed 
in  two  ways  :   (o)  in  generating  current,  and  (6)  in  overcoming  friction. 

If  the  friction  produces  5  caloriee/sec.,  what  is  the  current  availahjo  at  SO 

volts  ? 

13.  How  is  electrical  energy  calculated  ?  At  throe- halfpence  a  unit,  what 
is  the  cost  (a)  of  a  joule,  (6)  of  heating  your  100  litroe  of  bath-water  from 
10°  to  40°  C.  ?     How  much  energy  does  a  bath  roproeont  ? 

14.  An  electric  stove  employs  a  thin  metal  ribbon;  tlie  iiaer  moves  to 
another  district  where  the  voltage  is  doubled.  With  what  length  of  ribbon 
of  what  width  must  the  stove  be  rewound  to  give  the  same  amount  of  heat 
at  the  same  brightness  ? 

15.  How  would  you  measure  the  supply  of  electrical  energy  ?  What 
units  are  used  ? 

A  40-c.p.  lamp,  on  100  volts,  is  immersed  in  600  c.c.  of  water.  The  tero^ 
perature  is  raised  15°  C.  in  10 J  min.  What  current  ia  flowing  through  the 
lamp,  and  how  would  you  express  its  illuminating  efficiency  ?     (  X  4) 

16.  Two  wires  have  the  .same  dimensions,  but  the  specific  resistance  of  A 
is  twice  that  of  B.  Find  the  ratio  of  the  heat  generated  in  A  to  that  in  B 
when  the  wires  are  connected  (a)  in  parallel,  (6)  in  series  across  the  200- volt 
mains.     (  X  2) 

17.  Compare  the  heat  produced  in  three  lamps  in  series,  each  of  100  ohms; 
and  two  in  parallel,  each  of  500  ohms;  both  systems  being  supplied  by  a 
100- volt  circuit.     (  X  2) 

18.  A  battery  of  4  ohms  internal  resistance  supplies  current  to  a  10-ohm 
coil ;  with  what  resistance  must  thia  coil  be  shunted  to  reduce  the  heat  pro- 
duced in  it  to  a  quarter  ?     (  X  2) 

19.  If  two  wires  are  in  parallel,  prove  that  more  heat  ia  developed  in  the 
thicker  wire. 

20.  Lamps  {Aggregating  1  ohm  resistance  aro  supplied  through  leads  of 
002  ohm  from  a  source  at  51  volts.  The  voltage  ia  subaoquently  raised  to 
250  and  the  lamps  replaced  by  high- voltage  lamps  conauiuing  the  same  total 
energy.  Ciilculate  the  saving  per  thousand  hours  at  fourponco  per  kilowatt 
hour. 


PRACTICAL  QUESTION 

Find  the  mechanical  equivalent  of  heat  by  electrical  heating;  or,  find  the 

specific  heat  of  a  liquid. 


CHAPTER   LI 


ALTERNATING  CURRENT 


§  82L  From  §  751,  and  your  own  experiments  in  the  laboratory, 
you  have  found  that  pushing  a  magnet's  pole  towards  a  coil,  of 
50  yds.  or  so  of  wire,  will  cause  a  deflection  in  a  low- resistance 
galvanometer  connected  to  it,  showing  that  a  current  is  circulating 
in  the  coil,  as  the  lines,  moving  with  the  magnet,  cut  through  the 
wire. 

The  direction  of  the  current  is  most  easily  found  by  recollecting 
that  it  will  always  oppose  the  motion  [Lenz].  Thus,  facing  the 
coil  and  pushing  N  pole  towards  it,  the  current  circulates  against 
the  clock,  giving  the  face  of  the  coil  N  polarity  so  as  to  oppose  the 
oncoming  pole.  The  current  continues  until  the  magnet  is  half-way 
through,  when  the  lines,  now  parallel  to  the 
magnet,  cease  to  be  cut.  Then  an  equal 
reverse  current  flows  as  the  S  pole  passes 
through. 

Now,  this  might  have  been  an  electro- 
magnet, and  then,  instead  of  moving  it,  it 
could  be  magnetized  in  position  by  sending 
a  current  round  it.  The  effect  of  this  is  a 
rapid  spreading  of  magnetic  lines  as  the 
magnetism  strengthens,  the  weak  Fig.  368, 
W  changing  to  Fig.  368,  S.  These  spread- 
ing lines  cutting  the  coil  induce  in  it  a 
current  resisting  the  magnetizing  process, 
just  as  previously  it  resisted  the  coming 
of  the  magnet.  When  the  magnetizing 
current  is  stopped,  the  lines  collapse  again 
on  the  failing  electromagnet,  and  cutting 
the  coil  as  they  move,  now  induce  in  it 
an  equal  direct  current  hindering  the 
demagnetization . 

And  the  effect  will  be  the  same,  though 
weaker,  if  there  is  no  iron  present  at  all.  Fig.  321.  So  that  sending, 
or  stopping,  a  current  in  a  coil  of  wire,  induces  in  a  neighbouring 
coil  transient  currents  opposing  starting  or  prolonging  running. 


1/ 


/ 


\ 


Fig.  368. 


§  822.  The  Alternating-current  Transformer  is  developed  from 
this  pair  of  coils.  In  a  typical  transformer  there  is  a  '  core  '  of 
'  laminated  '  iron,  on  which  is  wound  a  coil  of  thick  insulated 
wire.     Around  this  coil  and  insulated  from  it  is  wound  another 

666 


822] 


ALTERNATING   CURRENT 


667 


coil  containing  a  greater  length  of  wire,  which  may  be  correepondingly 
thinner.  Sometimes  the  core  is  straight  but  more  generallj^-  it  formi 
a  closed  ring  of  iron  ;  Fig.  370  illustrates  the  straight  pattern. 

In  Fig.  369,  on  the  left,  is  Faraday's  original  transformtT,  two 
coils  separately  wound  on  a  ring  coil  of  iron  wire,  and  preserved 
at  the  Royal  Institution  since  1805.  The  next  figure  is  its  modem 
equivalent;  the  core  is  built  up  of  square<U-8hapc<l  stampingii 
of  thin  sheet  iron,  §  824,  stuck  in  alternately  right  and  left,  into  the 
coils  ready- wound  in  the  lathe. 

The  third  figure  employs  E -shaped  stampings,  the  magnetic 
circuit  being  completed  round  their  backs.  It  is  shown  wound  bb 
an  '  auto -transformer,'  one  continuous  coil  with  tapping  points ; 
if  100  volts  A.C.  be  applied  to  the  two  left-hand  terminals,  various 
voltages,  from  a  very  few,  to  240  over  all,  can  bo  drawn  from 
selected  terminals,  the  higher  the  farther  apart.  Without  a 
secondary  connection  it  is  a  '  choking  coil,'  §  829. 


Fia.  369. 

The  right-hand  figure  is  the  H.T.  Transformer  described  in  f  915, 
used  for  working  X-ray  outfits  nowadays  in  place  of  its  *  open- 
magnetic  circuit '  brother.  Fig.  370. 

Beneath,  you  have  the  sign  and  symbol  of  all  transformers. 

Taking  Fig.  370  as  an  example,  when  a  current  is  sent  in*^  the 
inner  coU  the  magnetic  lines,  starting  as  rings  round  the  individual 
wires,  speedily  fuse  into  elongated  loops  like  those  of  hig.  311. 
The  inner  straight  sides  of  these  magnetic  loops  pack  together  by 
thousands  in  the  iron  core,  many  hundre<l  times  more  permeable 
to  them  than  is  air,  the  outer  sides  bulge  out  rapidly,  cutting 
through  the  wires  of  the  second  coil  as  they  spread.  When  the 
current  is  stopped,  all  these  lines  shrink  back  on  to  the  wire,  and 
now  if  a  reverse  current  is  sent,  the  system  spreads  out  again,  witn 
each  magnetic  Une  reversed  in  direction  ;  this  continues  the  current 
in  the  second  coil  induced  by  the  stoppage  of  the  direct  current 
in  the  first.  In  the  closed  iron  circuit  transformers  of  Fig.  36Utho 
lines  do  not  spread  far  afield,  but  run  round  in  the  iron  :  the  effect 
is  the  same  and  the  efficiency  greater.  .      u-  w   ;- 

Thus,   when   an  Alternating  Current,   t.e.   a  current   which   la 
reversed  50—100  times  a  second,  is  sent  into  the  prtmiry  coii 
another  alternating  current,  flowing  nearly  in  opposiUon  to  tbe 
first,  can  be  drawn  from  the  secondary  cou. 


668  MAGNETISM   AND   ELECTRICITY  [§  822 

If  the  Secondary  coil  contains  very  many  turns  of  wire,  the  rate 
of  cutting  of  lines  and  wires,  and  therefore  the  electromotive  force 
in  the  circuit,  is  high,  and  the  transformer  enables  us  to  *  step  up ' 
a  large  low-pressure  alternating  current  to  a  small  high-pressure 
alternating  current,  much  more  suitable  for  economical  transmission 
to  a  distance,  §  817.  A  miniature  transformer  of  this  sort  converts 
the  3-volt-pressure  current  in  a  telephone  into  a  high-pressure  cur- 
rent, capable  of  negotiating  several  miles  of  line  without  such  loss, 
while  distant-power-transmission  systems  employ  transformers 
of  all  sizes  :  each  generator  of  §  755  feeds  the  132,000-volt  line 
through  one  of  120  tons,  containing  43  miles  of  wire. 

Per  contra,  when  the  high-pressure  current  is  supplied  to  the 
Secondary  coil,  a  large  low-pressure  alternating  current  can  be 
drawn  from  the  comparatively  few  turns  of  the  Primary  coil. 
Transformers  are  therefore  used  to  *  step  down  '  from  the  dan- 
gerous voltage  of  the  transmission  line  to  the  240 — 100  volts 
safe  for  domestic  use,  and  thence  to  the  huge  currents  used  in 
electric  welding,  at  very  low  voltage,  or  to  the  few  volts  necessary 
for  electric  bells  and  signals  of  all  sorts,  which  run  just  as  well  on 
A.C.  as  on  the  direct  current  from  voltaic  batteries. 

Little  1  :  1  transformers  are  useful  in  radio-sets  because,  while 
the  current  apparently  goes  through  them  unchanged,  there  is 
really  complete  insulation  between  the  two  windings.  In  treating 
a  patient  with  alternating  current  electro-medically — '  sinusoidal,' 
etc. — it  is  safer  to  put  in  a  transformer  simply  to  insulate  him  from 
the  mains,  on  which  accidental  '  earths  '  might  cause  him  harmful 
shocks. 

On  the  other  hand,  in  '  auto-transformers  '  the  two  coils  run 
right  on  as  one  coil,  '  tappings  '  being  connected  at  different  dis- 
tances along  :  this  is  cheap  and  often  good  enough,  Fig.  369,  No.  3. 

It  is  the  extreme  flexibility  conferred  on  Alternating  Current  by 
the  use  of  Transformers  which  permits  its  extensive  use,  and  has 
made  long-distance  power-transmissions  practicable. 

Transformers  are  called  '  Static  Transformers '  because  they 
have  no  moving  parts,  but  stand  quietly  humming  in  jars  or  tanks 
of  insulating  oil,  like  Ali  Baba's  Forty  Thieves,  except  that  care  is 
taken  that  the  oil,  heated  by  the  copper  (resistance)  and  iron 
(hysteresis)  losses  in  the  transformer,  shall  not  be  '  boiling.'  Often 
scores  of  hollow  '  jug-handle  '  pipes  are  welded  on  to  the  tanks  ; 
down  these  thcr  hot  oil  can  flow,  and  be  cooled  by  the  air,  down  to 
the  bottom  of  the  tank  again. 

§823.  If  Direct  Current  is  wanted,  the  output  A.C.  has  to  be 
run  into  a  moving  machine,  either  a  Rotary  Converter,  which  is 
practically  a  commutator  running  at  synchronous  speed — i.e.  the 
A.C.  frequency — or  a  complete  Motor  Generator,  two  distinct 
machines  coupled  together  ;  or  else  into  some  type  of  Rectifier, 
§§  801,  865,  891.  In  the  '  Underground  '  sub-stations  you  can  espy 
the  transformers  along  the  gallery,  stepping  down  the  11,000  volts, 


§  825]  ALTERNATING   CURRENT  669 

for  the  big  rotary  converters  occupying  the  floor,  which  convert 
it  into  600  volts  D.C.  on  the  side  rail. 

D.-C.  motors  are  so  very  miicli  quicker  off  the  mark,  and  mor« 
tractable  under  variable  loads  and  speecls,  that  it  paya  to  supply 
direct  current  to  the  line  :  400  volts  for  public  road  aervices, 
750  volts  for  third-rail  and  1500  for  overhead  on  railwavH.  arc 
becoming  usual.  The  momentum  of  the  heavy  converter  armature 
reaches  the  train,  and  assists  its  starting,  just  as  does  that  of  the 
car-engine  flywheel. 

§  824.  Laminated  iron.  If  solid  iron  were  used  with  alternating 
currents,  '  eddy  '  currents  would  of  course  be  induced  to  circulate  in 
the  readily  conducting  mass,  which  would  always  oppose  everything, 
and  would  also  heat  the  iron  (like  the  copper  rings  of  Fig.  333)  and 
cause  destruction.  Therefore  all  iron  in  transformers,  and  in  arma- 
tures (which  are  rotating  in  magnetic  field),  is  stamped  out  of  thin 
sheets,  which  are  bolted  together  with  thin  insulating  paper  l)etween 
them.  The  iron  is  really  a  soft  silicon-steel  of  special  quality, 
such  as  'Stalloy,'  made  highly  resistant  to  these  currents  by  the 
silicon,  yet  otherwise  almost  pure,  magnetically  permeable  and  of 
the  lowest  possible  Hysteresis,  §  670.  By  1925  it  was  reckoned 
that  the  U.S.A.  had  saved  the  whole  cost  of  the  Panama  Canal,  on 
its  fuel  bill,  by  using  these  special  steels  in  all  electromagnetic 
machinery,  instead  of  '  best  charcoal  iron.' 

§  825.  The  E.M.F.  produced  in  the  secondary  of  the  '  step-up  ' 
transformer  is  proportional  to  the  rate  at  which  the  magnetic 
lines  cut  the  wires.  Suppose  therefore  we  could  instantaneously 
stop  the  primary  current ;  the  lines  would  travel  in  at  enormouB 
speed  (the  speed  of  light),  and  at  first  sight  it  seems  that  a  practically 
unlimited  voltage  would  result.  But  on  trying  the  expriment, 
say  by  snatching  away  the  supply  wire  from  the  binding  screw 
of  the  primary  coil,  the  secondary  voltage,  though  high,  will  seldom 
be  found  able  to  drive  a  spark  'through  half  an  inch  of  air.  The 
reason  is  seen  in  the  primary  break,  a  flash  of  light  1/4  in.  long 
or  more  follows  the  snatched-away  wire,  through  this  flash  the  current 
continues  to  flow,  and  its  stoppage  is  by  no  means  the  utterly 
abrupt  one  intended. 

Whence  this  flash  ?  ... 

As  the  current  dies  away  in  a  coil,  and  the  wide  .nagnetic  Irncj* 
shrink  down  into  little  rings  round  the  individual  wirea.  each 
has  had  to  cut  a  number  of  neighbouring  wires,  i.e.  a  laro  amount 
of  cutting  of  lines  and  wires  has  gone  on  in  the  coil  itself.  There- 
fore a  current  has  been  induced  in  the  coil  itself,  and  this  current 
opposes  what  is  being  done,  it  is  a  direct  current  deUying  the 
dying  awav.  -it 

Moreover,  the  quicker  we  attempt  to  do  away  with  the  ciirreni. 
the  quicker  is  this  cutting,  and  the  higher  the  electromotive  force. 
which  becomes  quite  able  to  drive  the  dying  current  acroM  a  aliort 


670 


MAGNETISM   AND   ELECTRICITY 


[§825 


air  gap  after  the  retreating  wire.  DonH  call  this  an  '  extra  current 
at  break.' 

Thus  not  only  is  there  *  Mutual  Inductance  '  between  two  coils, 
but  every  coil  possesses  *  Self-inductance  '  of  its  own.  With  an 
iron  core,  enabling  very  many  magnetic  lines  to  be  formed,  this 
self -inductance  is  large  ;  a  regular  flame  appears  on  breaking  the 
circuit  of  a  big  electro-magnet. 

Thus  Inductance  confers  on  Current  an  '  Electromagnetic  Inertia.' 

The  measure  of  an  Inductance,  Self  or  Mutual,  is  the  number  of 
linkages  of  unit  magnetic  lines  and  current  when  1  ampere  is  flowing. 
The  practical  unit  is  the  Henry,  which  is  10^  linkages. 

Joseph  Henry  was  an  American  contemporary  of  Faraday's ; 
having  no  apparatus,  he  judged  by  the  shocks  he  got. 

§  826.  The  Induction  Coil  or  Spark  Coil  is  a  step-up  transformer 
in  which  the  production  of  exceptionally  high  electromotive  forces 


Fig.  370. 


is  specially  aimed  at.     One  is  shown  in  section  in  Fig.  370.     It 
possesses — 

(1)  A  long  stout  core  of  laminated  soft  iron  well  magnetized  by 
a  large  battery  current  in  the  '  Primary  '  winding,  of  two  layers  of 
thick  copper  wire.  This  Primary  Coil  is  connected,  through  a 
'  break,'  with  the  terminals  on  the  left. 

(2)  A  Secondary  Coil  containing  an  enormous  number  of  turns 
of  (necessarily)  thin  wire.  This  coil  must  be  extremely  well 
insulated ;  usually  a  tube  of  ebonite  \  in.  thick  (black  in  figure) 
separates  it  from  the  primary,  and  it  is  built  up  of  many  flat  coils 
sandwiched  by  ebonite  discs.  The  ends  are  led  out  to  the  little 
terminals  on  top. 

(3)  Some  contrivance  for  breaking  the  primary  current  with  great 
rapidity. 

The  commonest  is  a  Spring  Hammer  Break.  A  vertical  spring 
stands  up  from  the  base-board,  and  holds  a  soft-iron  hammer-head 


§826]  ALTERNATING   CURRENT  671 

just  opposite  the  end  of  the  iron  core.  When  the  core  is  magnetized 
it  attracts  this  hammer,  and  in  so  doing  draws  the  spring  away  from 
contact  with  a  platinum-tipped  screw,  carried  hy  a  second  upright 
on  the  base-board.  As  this  screw  and  spring  form  part  of  the  primary 
circuit  (the  continuous  black  line),  this  breaks  the  current,  the  coro 
loses  its  magnetism,  lets  the  iron  spring  back  into  contact  with  the 
screw  again,  the  primary  current  restarts,  and  so  on.  It  in  an 
ordinary  Electric  Bell  mechani&m. 

But  the  moving  apart  of  spring  and  screw  is  not  vtry  ouick, 
and  sparking  at  the  break,  due  to  self-induction,  would  roi*  the 
contrivance  of  almost  all  its  value  but  for  a  Condenser,  such  as 
described  in  §  738,  which  is  placed  as  a  shunt  across  the  break. 
The  dying  current,  instead  of  driving  a  lengthening  spark  across 
the  gap,  flows  into  this  condenser,  charges  it,  and  comes  to  a  stop 
comparatively  quickly — for  a  blazing  path  of  nitrogen  burning  in 
oxygen,  once  established,  offers  very  little  resistance.  Metal 
Rectifiers  can  also  serve  here. 

This  '  buzzer  '  break  is  noisy  and  irregular,  for  the  sparking  cannot 
be  entirely  quenched  and  the  contacts  burn  rough.  However,  it 
serves  well  enough  on  small  coils.  Break  and  condenser  (below) 
are  shown  in  broken  lines  in  Fig.  370. 

A  much  better  contrivance  is  the  modem  motor  Mercur>'  Break, 
in  which  the  primary  current,  flowing  through  mercury,  is  broken 
perhaps  20  times  a  second  in  an  atmosphere  of  coal  gas,  ver>- 
abruptly. 

Wehnelt  introduced  a  noisy  but  powerful  interrupter.  The 
mains  current  passes  between  two  lead  plates  in  a  tank  of  dilute 
sulphuric  acid.  One  plate  is  inside  a  wide  glass  test-tube,  and  the 
current  has  to  pass  through  a  small  hole  in  this  tube.  Bubbles  of 
electrolytic  gas  and  steam,  forming  at  the  hole,  rapidly  interrupt 
the  continuity  of  the  liquid,  and  therefore  of  the  current. 

Induction  coils  of  all  sizes  are  stock  articles  of  commerce,  from 
the  little  medical  shocking  coil  worked  by  a  dry  cell,  and  caiwible 
merely  of  inflicting  harmless  torment,  and  from  the  small  sjwirkmg 
coils  of  petrol  motors,  to  great  coils  giving  sparks  a  foot  or  two  m 

length.  ,     .     ,  •        *     *u 

In  the  'sledge'  coils  used  for  physiological  experiments  the 
secondary  can  be  slid  off  from  the  primary  coil.  This  very  much 
weakens  the  induced  current,  and  permits  adjustment  of  the  eUx-tric 
stimulus  given  to  the  nerve  under  observation. 

Another  regulator,  in  use  on  seaside  shocking  coils,  is  a  copiK-r 
tube  sliding  in  between  primary  and  secondary  :  induced  currents 
circulate  in  it,  and  shield  off  the  secondary.  . «  i     •     i 

There  is  also  a  current  induced  in  the  secondary  at  Make  oi 
primary  current,  but  its  E.M.F.  is  seldom  sufficient  to  cause  a  revcnjc 
spark.  The  discharge  from  the  Induction  Coil  therefore  con.Histi. 
of  a  succession  of  rushes  of  small  quantities  of  electricity  at  high 
pressure  (about  8000  volts  per  cm  spark)  all  >Vim*;Z^•olI^  "^ 
A  moderately  large  coil  will  give  2  milhamps.  at  200.U00  %oIt». 


672  MAGNETISM   AND   ELECTRICITY  [§  826 

All  this  hammering  and  hurry  is  a  confession  of  weakness,  as  it 
always  is  :  greater  strength  would  push  the  nail  in  quietly  with 
much  more  efficiency.  These  coils  are  now  superseded  for  X-ray 
work  by  high-tension  static  transformers,  in  oil,  which  deal  with 
ten  times  the  power.  And  million- volt  transformers  run  on  1000 
h.p. — but  Sparking  Coils  helped  on  high-tension  research  long  ago, 
on  a  hundredth  of  that. 

§  827.  The  Ignition  Magneto.  Fig.  335  showed  the  rapid  change 
of  magnetic  flux  through  the  laminated-iron  shuttle-armature,  with 
its  thick  wire  winding,  of  a  Low-tension  Magneto.  This  flux-change, 
of  course,  induces  a  quick  rush  of  current  round  the  circuit,  which  is 
broken  somewhere  near  maximum,  on  wiper-blades,  inside  the 
engine  cylinder,  separated  by  the  same  mechanism  that  jerks  the 
shuttle  over  from  1  to  3  ;  and  the  heavy  self-inductance  flash  fires 
the  compressed  mixture.  This  cheap  and  simple  contrivance  is 
fitted  on  slow- speed  engines  on  millions  of  farms. 

The  more  familiar  High-tension  Magneto  breaks  its  primary 
circuit  at  the  '  mag.  points,'  which,  as  you  have  found  out,  sometimes 
want  smoothing  into  good  contact  again,  because  they  have  got 
burnt  with  sparking,  in  spite  of  their  size,  and  of  a  shunting  paper- 
insulated  condenser  packed  away  in  the  armature.  This  stoppage 
of  the  primary  current  induces  a  high  E.M.F.  in  the  multitudinous 
thin  wires  of  the  well-insulated  Secondary  winding  wound  over 
the  Primary  on  the  armature,  and  the  H.T.  current  is  led  off  to  the 
distributor  and  the  sparking  plugs.  The  mag.  is  put  out  of  action 
by  short-circuiting  the  break.  It  is  a  magneto  generator  and 
sparking  coil  rolled  into  one. 

Now  that  cars  all  have  starting  batteries  standing  full  of  charge, 
this  expensive  little  independent  current  generator,  which  of  course 
is  not  at  its  best  at  starting  speed,  is  often  replaced  by  simple 
buzzer  spark-coils,  which  are.  This  is  not  altogether  new,  for  in 
191 1  we  had  '  dual  ignition,'  and  switching  on  a  stand-by  accumulator 
and  four  coils  quite  often  started  a  warm  engine  :  if  it  didn't,  and 
we  got  out  in  haste  without  '  retarding  the  ignition,'  and  cranked 
up,  we  broke  our  wrists. 

§  828.  It  will  have  occurred  to  you  that  there  are  two  ways  of 
moving  the  electrons  in  a  conductor  :  first,  by  a  mass-motive  force 
which  moves  everything  ;  second,  by  an  electromotive  force  which 
drives  the  electrons  only ;  and  in  the  presence  of  a  magnetic  field 
the  one  involves  the  other. 

You  play  shove-halfpenny,  and  the  gliding  copper  cuts  the  mag- 
netic lines  of  the  Earth's  vertical  component  field,  §  696,  as  they 
dive  down  towards  the  south  polarity  of  the  Arctic  :  apply  Ampere, 
and  Lenz,  and  you  find  the  electrons  pushed  off  to  one  side  of  the 
coin — positive  charges  to  the  left,  electrons  to  the  right. 

We  said  the  electromotive  force  drove  the  electrons  only ;  so  it 
does,  along  a  wire,  but  what  if  they  come  to  the  side  and  can't  get 


828] 


ALTERNATING   CURRENT 


67S 


i^ 


wl 


\    L 


further  ?  Then  you  may  find  how  tightly  they  cling  to  their 
home  :  hghtnmg  striking  a  tree  produces  this  Hidewavs  push  and 
strips  of  bark  are  rent  off:  Kapitza,  producing,  inside  a  'coir 
just  big  enough  for  the  top  of  your  tliumb,  a  field  of  350,000  mum 
by  a  current  around  it  lasting  001  sec.,  had  to  lap  it  rouncf  with 
metal  tape  capable  of  withstanding  a  bursting  force  of  many  tons. 

Now,  this  determination  to  move  off  sideways  is  exactly  that 
of  a  spinning-top,  §  91,  and  suggests  that  magnetic  lines  are  a  state 
of  spin  of  something  that  pushes  and  pulls  electrons,  i.e.  either  of 
other  electrons,  or  of  lines  of  electric  force  from  them. 

Let  us  take  a  case,  and  let  us  drop  any  talk  of  positive  charges, 
for  in  a  metal  it  is  negative  electrons  'that  carry  the  current — 
backwards— the  two  negatives  cross  out,  so  that  will  not  worr>'  us. 

Let  PP  be  copper  and  LL  laminated  iron,  Fig.  371.     Along  P 
enters   a   stream    of    electrons.     An 
electron  E  in  the  iron  is  repelled  by  p  p 

the  leader,  whom  he  of  course  checks      -• -^  -  ~  ^^-^ 

— back  E.M.F.  of  self-induction.  "^  '  ^ 

The  file  pushes  on ;  the  leader, 
advancing  to  2  and  3,  repels  e  to  j: 
and  g,  the  laminations  keep  him 
from  drifting  far  away  (eddy  cur- 
rent), and  he  has  to  spin,  as  shown. 
There  you  have  a  current  magnetiz-  Fio.  37  L 

ing  iron,  the  magnetic  lines  perpen- 
dicular to  the  paper  are  the  axes  of  spin  of  electrons. 

There  is  now  a  continuous  stream  of  electrons  above,  with  no 
head  to  accelerate  e  :  that  does  not  matter,  magnets  stay  magnetized 
of  themselves.  But  soft  iron  loses  its  magnet i.sm  when  the  current 
stops  ?  Yes,  e  puts  all  his  energy  into  one  accelerating  kick  at  the 
last  electron  of  the  file;  and  there  is  your  self-inductive  K.M.F. 
at  '  break,'  and  the  demagnetization  of  the  iron. 

Now  let  SS  be  a  secondary  circuit  of  copper  :  jis  he  runs  through 
g,  ^,  g,  etc.,  e  pushes  the  movable  electrons  in  it  towards  the  left. 
and  there  is  the  reverse  current  in  the  Secondary  induced  by  the 
rise  of  current  in  the  Primary  of  the  Transformer,  e  has  no  energy 
to  spare  unless  the  primary  current  increases  again,  so  resistance 
soon  stops  the  secondary  current — the  momentary  throw  at  *  make  * 
of  direct-current  in  primary. 

[Don't  push  this  explanation  far ;  it  claims  little  but  plausibility.] 

Is  there  no  escape  from  Resistance,  nothing  like  the  eternal 
spin  in  a  magnet  ?  A  complete  coil  of  Lead  wire  was  hung  in  licjuid 
Helium  by  two  wires,  to  a  galvanometer,  and  when  the  teniiKTnturc 
was  below  6°  A.,  a  near-by  nuignet  was  taken  awav,  which  must  have 
induced  a  current  in  the  coil,  ^exi  day  the  leati  wire  between  the 
two  connectors  was  broken  by  a  fish-hook,  so  that  any  current  must 
now  come  out  and  through  the  galvanometer,  and  of  course  die  of 
resistance.  It  swung  aside,  showing  a  current  7/8th«  as  strong  as 
z 


674  MAGNETISM   AND   ELECTRICITY  [§  828 

the  immediate  one  at  ordinary  temperature  ;    all  that  time  the 
Lead  had  been  in  a  condition  of  *  Superconductivity.' 

One  conclusion,  at  any  rate,  can  be  drawn  from  this  paragraph, 
that  if  you  take  an  electron  with  his  long  straight  tail  of  Electric 
Force,  and  spin  him,  so  that  it  becomes  a  long  rotating  helix  or 
ringlet,  you  have  made  a  Line  of  Magnetic  Force. 

Instances  of  electrons  curving  and  revolving  round  this  will 
be  found  in  Chapters  LIII  and  LV. 

§  829.  When  a  current  c  is  flowing  through  a  circuit  of  '  ohmic  ' 
resistance  R,  Ohm's  law  tells  us  that  an  E.M.F.  cR  suffices  to  main- 
tain the  flow.  But  the  current  had  to  be  started  somehow,  and  there 
had  to  grow  up  with  it  the  magnetic  field  belonging  to  it,  the  whole 
system  of  magnetic  lines  interlinked  with  it.  Per  ampere  of  current, 
these  lines  number  L,  the  self-inductance  of  the  circuit,  and  there 
may  be  an  additional  M  of  mutual  inductance  if  another  circuit 
is  near  by. 

If  this  lot  grew  up  in  1  sec,  they  acted  as  a  back  E.M.F.  of  L 
milli-microvolts,  if  in  1 /100th  second,  of  100  L,  and  so  on.  If  L  is 
reckoned  in  henries,  each  10^  lines,  the  E.M.F.  is  in  volts. 

That  is,  additional  E.M.F.  had  to  be  employed  in  starting  the 
current,  just  as  extra  effort  is  required  in  starting  any  heavy  load. 
Sometimes  the  effort  is  trifling,  as  when  switching  in  a  lamp,  and 
the  current  jumps  to  full  value  quicker  than  most  instruments  can 
tell ;  sometimes,  as  in  energizing  a  big  field-magnet,  the  ammeter 
needle  crawls  very  reluctantly  across  the  dial. 

But  the  starting  and  running  E.M.F.'s  are  not  just  added  together  ; 
one  is  wanted  before  the  other,  and  dies  off  as  the  other  grows, 
which  is  an  economy,  and  if  it  were  a  question  of  wanting  the  extra 
E.M.F.  only  for  a  fraction  of  a  second  once  or  twice  a  day,  it  would 
not  be  worth  our  while  considering ;  but  an  ordinary  Alternating 
Current  starts  and  stops  100  times  a  second  all  day  long,  and  con- 
sider it  we  must. 

The  E.M.F.  actually  engaged  at  any  instant  in  driving  the  electrons 
through  the  copper,  apart  from  making  alterations  in  the  magnetic 
field  all  round,  is,  as  usual,  R  times  the  current  at  that  instant. 
But  the  A.C.  is  perpetually  varying  ;  we  will  make  the  simplest 
possible  assumption,  that  it  varies  simple -harmonic ally  (as  the 
engineer  always  tries  to  make  it),  and  the  mathematician  would 
write  Ohmic  E.M.F.  =  Re  sin  {pt  —  0),  where  c  is  the  maximum 
value  of  the  A.C. 

We  are  not  mathematical,  so  we  turn  again  to  Fig.  119,  and  from 
it  build  up  Fig.  372.  Make  Re  the  radius  of  the  pecked  circle, 
the  crank-arm,  or  clock-hand,  and  set  it  rotating  with  the  '  cycle 
frequency  '  of  the  A.C,  50  per  second  in  '  the  Grid,'  60  in  America. 
Then  the  height  of  the  crank-pin  above,  or  depth  beneath,  the  centre 
IX — III  line,  is  the  ohmic  E.M.F.  at  any  instant.  Fill  in  your 
own  hours. 


829] 


ALTERNATING   CURRENT 


•75 


fK^if'i^  are  we  to  put  in  the  rotating  *  vector  '  arm  to  represent 
the  *  fie  d-producing  '  E.M.F.,  and  how  long  is  it  to  be  ?  "'P"*^"^ 
It  will  be  straight  up,  at  its  maximum  height,  puHhina  hardeut 
when  the  current  is  being  increased  f^ustest,  f e /when  the  cunTm 
VlTl  '"^oZ\y^  vertically  upwards,  through  IX  o  clock,  for  from 
V  111  .  du  to  IX  .  30  IS  nearly  straight  up. 


Fio.  37:2. 


Then  when  Re  is  at  its  maximum  XII,  and  has  ceased  to  grow, 
the  '  field -increasing  '  E.M.F.  must  be  at  III,  on  the  zero  line. 
Inset  on  the  right  is  the  rectangle  tume<l  through  thi«  three  houm, 
up  on  end. 

That  is,  it  is  always  90° -in  phase  ahead  of  Ke. 

Its  maximum  value  in  volts  is  L  henries  x  rate  of  inrre««»  of 
amps. ;  the  Re  arm  is  going  round  n  times  per  sec.  and  it*  tracing 
point  therefore  movies  a  peripheral  distance  27rn  .  Rr  in  the  sccoml. 
it  passes  IX  o'clock  travelling  straight  up  at  this  spcetl  :  hence, 
just  leaving  out  R,  the  maximum  rate  of  growth  of  r  in  the  circuit 
is  2Ttnc.  Therefore  the  field-producing  E.M.F.  is  given  by  a  crank- 
arm  90°  ahead  of  Re  and  of  value  2Knc  .  L. 


676  MAGNETISM   AND   ELECTRICITY  [§  829 

These  combine  into  a  resultant  E,  the  E.M.F.  that  must  be 
applied  to  the  circuit  to  keep  it  going  ;  its  value  given  by 

Applied  E  =  c  X  \/(^^  +  ^T^'^nHJ).     (Pythagoras.) 

The  quantity  under  the  -y/  is  the  Impedance  of  the  circuit,  and 
takes  the  place  in  A.C.  circuits  that  Resistance  held  in  Direct 
Current.  Notice  at  once  that  it  depends  on  the  speed  of  alternation  ; 
and  reduces  to  R  for  zero  speed,  i.e.  for  D.C. 

The  current  lags  behind  E  in  phase  by  an  Angle  of  Lag  the  tangent 
of  which  is  27t7iL/R.     The  name  Reactance  is  given  to  2nnL. 

The  time-graph  shows  the  applied  E  in  solid  line,  and  the  current 
lagging  by  this  angle. 

The  Watts  of  Power,  in  any  circuit,  at  any  moment  =  volts  x 
amps.,  got  by  multiplying  the  simultaneous  values ;  without 
attempting  this  calculation  we  can  see  at  once  that  the  power  is 
less  than  it  would  be  if  there  were  no  lag,  because  then  maximum 
would  multiply  maximum,  and  there  would  be  no  places  like  Q 
where  the  product  is  — ,  and  actually  subtracts. 

The  engineer  has  to  design  his  machinery  to  deal  with  simultane- 
ous maxima,  and  expresses  its  power  in  KVA,  kilovolts  x  amperes, 
and  he  devoutly  hopes  that  his  customers'  circuits  will  not  be  ex- 
cessively inductive,  or  their  Lag  will  pull  his  '  Power-Factor  '  below 
70%,  i.e.  the  actual  Kilowatts  in  circuit,  by  wattmeter  §  814,  will 
be  less  than  70%  of  the  KVA  readings  on  volt-  and  am-meter  ;  a 
waste,  not  indeed  of  coal,  but  of  capital  cost. 

The  figure  as  drawn  would  be  about  right  for  the  coil  of  a 
laboratory  tangent -galvanometer  made  of  50  turns  of  ordinary 
copper  wire,  and  supplied  at  mains  frequency. 

For  a  lamp,  which  has  a  high  R  and  a  very  small  and  weak  mag- 
netic field,  the  current  increases  to  practically  its  D.C.  value,  and 
the  lag  is  negligible,  the  pecked  curve  moves  into  coincidence  with 
the  applied  E.M.F. 

Suppose  we  get  a  length  of  thin  wire-rope,  or  some  iron  fence- 
wire,  and  wind  it  through  and  through  the  coil,  facilitating  the 
passage  of  magnetic  lines,  and  thereby  soon  increasing  L  to  double, 
or  more  ;  tan  (lag)  doubles,  and  you  arrive  at  the  pecked  line  of 
the  lower  diagram,  the  current  is  much  more  out  of  step,  and  is 
also  reduced,  because  E  and  R  are  both  fixed  in  value. 

If  we  went  on  packing  in  iron  wire,  we  should  arrive  at  a  Choking 
coil,  or  Choke. 

Choking  coils  are  describable  as  Transformers  with  only  one 
coil,  or  any  transformer  with  its  secondary  out  of  action,  i.e.  carrying 
no  load  ;  they  have  very  large  L,  the  diagram  goes  even  farther,  into 
the  dotted  shape,  where  c  can  be  only  very  small  (since  R  remains 
unchanged)  :  the  inductance  dams  back  the  current,  instead  of 
resistance  wasting  it  in  heat.  Hence,  built  with  movable  iron 
cores,  they  are  an  economical  way  of  regulating  A.C,  for  theatre- 
lighting,  and  all  sorts  of  purposes  ;  the  lag,  already  75°,  can  approach 
90°,  and  the  power-factor  shrinks  towards  zero,  with  '  wattless 
current.* 


§  831]  ALTERNATING   CURRENT  677 

Suppose  you  are  helping  turn  a  winch  :  if  you  putth  while  the 
crank  goes  away,  and  pull  while  it  is  coming  towanin  you,  your 
wmch-motive  force  is  in  phase  with  the  flow  of  the  motion!  and  your 
power-factor  is  nearly  1,  that  is  the  upper  diagram.  Hut  if*  you 
are  only  shamming,  you  wait  until  the  handle  U  far  away  and  tien 
push  desperately,  wait  again  until  it  is  close,  and  pull  hanl.  you  keep 
almost  90°  out  of  phase,  and  your  power-factor  is  negligible ;  you 
are  following  the  dotted  lower  line. 

A  large  choking  coil  finds  employment  as  a  Smoothing  Coil 
{e.g.  in  a  '  mains  '  radio  set),  for  while  a  continuous  current  builds 
up  gradually,  and  flows  through  it  unchecked,  little  sudden  variationt 
have  to  meet  the  full  value  of  the  inductance,  and  are  smoothed 
from  quick  kicks  to  a  low  quiet  swell  such  as  causes  no  noi«e  in  a 
telephone. 

§  830.  We  can  now  dispose  of  the  A.C.  Transformer.  The  complete 
crank-diagram  of  what  happens  in  it  is  beyond  us,  but  a  simple 
dodge  will  carry  us  as  far  as  we  are  expected  to  go,  and  the  diagram 
is  already  to  hand. 

With  the  Secondary  coil  open,  doing  nothing,  we  have  just  called 
the  Primary  a  Choke  coil,  and  have  seen  that  the  transformer 
'carrying  no  load'  'takes  very  little  magnetizing  current,*  it  it 
the  dotted  lower  line  of  Fig.  372.  Now  '  put  a  load  on  the  trans- 
former,' i.e.  close  the  secondary  circuit  through  lamps,  apparatus, 
etc.  ;  we  have  seen  already,  §  822,  that  an  A.C.  will  flow,  and  that 
it  will  be  in  opposition  to  the  primary  current  (Lenz). 

It  will  set  up  its  own  magnetic  field  in  the  iron  of  the  transformer, 
opposing,  i.e.  reducing  the  flux  of,  magnetic  lines  ever^nrhere. 
That  is,  practically,  it  reduces  L,  and  thereby  the  reactance  2irfiL. 

The  dotted  Reactance  in  the  lower  figure  diminishes,  and  as  it 
is  one  side  of  a  rectangle  (hence  the  constructional  semicircle), 
the  other  side  is  bound  to  increase,  the  short  dotted  ohmic  K.M.F. 
vector  growing  into  the  solid  Rr,  i.e.  the  inflow  c  into  the  primar>* 
circuit  increases,  and  the  lag  reduces — in  the  figure  from  75  to  49  , 
and  then  to  30°. 

In  a  Tran.sformer  R  is  kept  small ;  unloadtnl,  the  vector  triangle 
is  flattened  almost  to  nothing,  the  transformer  wastes  Uttle 
'  magnetizing  current,'  and  is  economical  at  all  loads. 

§  831.  The  Alternating-Current  Motor.  The  direct -current  motor 
of  §  750  reverses  its  rotation  upon  reversal  of  the  current  either 
through  the  field-magnets  or  through  the  brushes,  as  in  train  motors, 
but  if  both  be  reversed  at  once  the  two  negatives  make  a  positive, 
and  the  motor  runs  on.  Therefore  such  motors  can  nm  on  A.C., 
but  as  the  field-magnet  current  is  now  alternating  rapidly,  all  its 
iron  must  be  laminated,  or  eddy  currents  will  soon  make  it  re<l  hot. 
Meccano  motors,  vacuum  cleaners,  etc.,  are  of  this  build,  to  run 
on  A.C.  or  D.C.  indifferently. 

In  larger  sizes,  however,  the  impe<iance  of  the  great  magnet 
circuit  becomes  enormous,  and  it  is  difficult  to  get  enough  current 


G78 


MAGNETISM   AND   ELECTRICITY 


[§831 


into  it  to  secure  a  strong  field  economically.  A  low  frequency, 
such  as  25  cycles,  gives  more  time  (reduces  the  impedance,  §  829), 
but  this  frequency  is  disliked  for  public  supply,  because  its  inter- 
mittent heating  makes  everybody's  lamps  flicker  annoy ingly. 
Further,  commutators  are  costly,  and  difficult  to  insulate  for  more 
than  1500  volts.  The  result  is  that  A.C.  Commutator  Motors  are 
little  used,  especially  considering  that  A.C.  Induction  Motors  with 
a  Squirrel-cage  Armature  need  no  armature  connections  whatever  : 
Fig.  373  shows  one  reduced  to  its  simplest,  and  with  only  the  bars 
on  the  near  side  laid  bare. 

The  cage  consists  of  copper  rods  welded  into  stout  copper  rings ; 
and  the  outer  ring  of  it  is  filled  with  laminated  iron.  All  rods  one 
side,  the  end  rings,  and  all  rods  the  other  side,  form  a  circuit  which 
acts  exactly  like  the  copper  ring  of  Fig.  333  ;  repelled  straight  away 
from  the  A.C.  field  magnet  it  may  rattle  in  its  bearings,  but  has 
no  reason  to  turn  either  way  ;   it  cannot  start. 


Fig.  373. 


Fig.  374. 


Fig.  375. 


I 


Suppose,  however,  that,  as  in  Fig.  374,  we  arrange  a  second  field- 
magnet  so  that  it  pushes  a  field  at  right  angles  to  the  first ;  and 
further,  that  by  inserting  great  impedance  into  its  circuit  (choking 
coil,  below  it)  we  make  its  field  lag  60 — 90°  in  phase  behind  the 
first,  §  829.  With  resistance  in  the  main  magnet  circuit,  to  keep 
the  two  starting  currents  roughly  equal,  we  now  get  Fig.  121  A, 
the  two  fields  combining,  not  into  one  which  merely  alternates,  but 
into  one  which  rotates.  First  magnet  pushes  field  VI-XII  o'clock, 
a  quarter-period  later  second  magnet  pushes  field  III-IX  ;  current 
rising  to  —  maximum  in  first  now  pulls  XII-VI,  and  then  second 
pulls  IX-III ;  the  field  rotates,  with  but  little  change  of  strength, 
with  cyclic  frequency. 

This  field  sweeps  through  the  copper  bars,  and  induces  currents 
in  them  as  it  passes,  all  resisting  and  dragging  on  it,  by  Lenz.  It 
wins,  of  course,  and  drags  the  cage  round  after  it,  with  only  '  slip  ' 
enough  to  enable  the  cutting  of  field  and  conductor  to  go  on,  and 
so  keep  up  the  armature  current  (more  slip  on  heavy  loads). 


§  832]  ALTERNATING   CURRENT  679 

But  once  spun  up,  the  auxiliary  field-magnet  can  be  cut  out, 
either  by  hand  or  by  a  centrifugal  switch,  and  the  motor  nin*  on. 
(Choke  and  resistance  are  usually  the  other  way  about  in  prattice, 
but  this  way  is  a  little  simpler  to  follow.)  For  the  impedance  of  the 
armature  circuit  ABCD,  surrounding  it«  mass  of  iron,  is  coniiiderable, 
so  that  it  has  now  turned  througli  a  fair  angle  before  itii  U^ng 
current  has  risen  to  its  full  value,  magnetizing  the  armature  to  have 
poles  at  PP  which  perpetually  repel  the  field-magnet  jwles  sideways. 
This  is  a  Single-phase  Motor.  To  start,  the  linkeci  switches  are 
moved  to  the  left  as  shown  ;   to  run,  to  the  right. 

If  two  currents  are  supplied  from  the  generating  station,  90* 
different  in  phase,  along  two  circuits,  into  the  two  field-magnets 
of  Fig.  374,  this  constitutes  a  Two-phase  Motor,  which  is  self- 
starting,  without  any  special  gadgets,  and  runs  just  like  an  ordinarj' 
locomotive  engine,  with  its  two  cranks  at  right  angles.  The  two 
return  wires  can  be  combined  into  one  rather  thicker  one. 

Finally,  if  you  take  two  equal  simple  harmonic  curves  one-third 
wave-length  apart  {i.e.  120°  different  in  phase),  and  add  them  to- 
gether by  the  principle  of  Fig.  120,  you  will  find  that  the  resultant 
is  a  third  sine-curve  exactly  like  either,  and  occupying  the  other 
120°  position.  That  means  that  by  widening  out  the  phase- 
difference  between  the  two  currents  to  120°,  the  return-wire  is  now 
occupied  by  an  A.C.  exactly  like  either.  So  why  not  now  pull  a 
bit  off  each  of  the  two  magnet  coils,  and  combine  them  into  a  thini 
one,  on  the  return  wire,  and  set  them  round  at  120',  for  symmetry. 
Fig.  375,  and  you  have  the  self-starting,  free  armature,  Three- 
phase  Motor,  with  its  Three-Wire  supply,  which  you  see  on  the  pylons 
of  the  Grid.  Generated  in  three  sets  of  coils  on  the  turbo-alternator, 
stepped-up  by  three  separate  tran.sformer8,  or  one  with  three 
separate  circuits,  carried  on  three  cables,  transformed  down  again 
on  a  triple  transformer  and  fed  to  three  sets  of  circuits  on  the  motor, 
giving  the  equable  turning  effect  of  a  triple-expansion  marine  engine, 
or  that  of  the  three-cylinder  Flying  Scotsman  ;  it  is  the  best  means 
yet  devised  for  the  wholesale  transmission  of  power. 

§  832.  We  have  seen  that  an  A.C.  carries  power  just  as  well  as 
a  D.C.,  with  the  additional  advantage  of  being  transformable  into 
an  excellent  high-jumper  when  the  hedges 

are  stiff  :    here  follow  some  curious  and  I  ":i_ 

important  activities  of  its  own  : —  nj"^  ^ 

Circuit    with    Capacity.     If    a    steady         <  \ 

voltage  V  were  applied  to  the  circuit  in      o>  \       ^    ^ 

Fig.  376,   left,   a   minute  current  would  S      1 ^  y 

flow    momentarily    into    the    condenser     ^.<  J  -v 

Capacity    C,    charging    it    once    for    all,        'Ir^j^ y  -. 

with   CV,  and   then    the   whole   of   the  |'  K 

current     V/R    would    flow    by    way    of  Fio.  376. 

resistance  R.  , ,     .  a-    u...— 

Not  so  an  alternating  voltage,  which  would  charge,  disctiargc, 

and  reverse-charge  C  with  cyclic  frequency  n.  alwa>Ti  inducing 


680  MAGNETISM   AND   ELECTRICITY  [§  832 

equal  and  opposite  charges  in  the  lower  plate,  so  that  a  current 
nCV  alternates — or  '  an  A.C.  flows  ' — by  way  of  C,  as  well  as 
V/R  through  non-inductive  R. 

The  two  branches,  however,  are  not  in  phase,  for  when  V  has 
reached  its  maximum  the  current  down  R  is  flowing  its  fastest, 
but  the  current  into  the  now  fully-charged  C  has  stopped. 

Now,  as  the  voltage  falls  towards  its  reversal,  the  ohmic  current 
decreases  with  it,  but  the  condenser  returns  its  +  charge  into  the 
circuit,  becoming  empty  and  then  rapidly  reloading  — ,  as  V  swings 
through  zero. 

This  unexpected  reflux  evidently  delays  the  establishment  of 
the  expected  —  voltage  at  a,  and  the  —  outflow  from  the  lower 
plate  delays  the  +  voltage  at  the  lower  end. 

A  Submarine  Cable  consists  of  a  few  strands  of  copper  wire  thickly 
insulated  with  gutta-percha  (the  one  thoroughly  water-proof 
flexible  insulator),  round  which  is  the  necessary  armouring  of  wet 
hemp  and  steel  wire,  and  sea -water.  It  is  a  long  Condenser,  §  732, 
with  distributed  capacity,  as  suggested  in  Fig.  376,  right,  all  these 
capacities  having  to  be  filled  and  emptied  as  the  flickering  signal 
current  passes.  The  result  is  that  it  takes  a  second  before  signals 
begin  to  ooze  out  at  the  American  end,  and  that  then  they  are 
blurred,  more  than  4  or  5  per  second  run  hopelessly  into  one  another  ; 
while  even  in  a  short  cross-Channel  cable  the  2000  demanded  by 
telephony  flatten  out  into  nothing  at  all. 

§  833.  Consider,   however,   an  Inductive   circuit   with  Capacity, 

Fig.  377.     The  current  through  L  lags  behind  the  voltage  applied 

at  its  ends,  only  rising  to  its  full  rush  when  this  has  already  fallen 

towards  reversal :   instead  of  this  rush  now  causing 

a  drain  on  the  main  circuit  it  is  abundantly  sup- 

^  ^"V»  plied  at  a  by  the  backrush  from  C,  while  at  the 

lower  end  of  L  the  stream,  instead  of  flooding  on 

along   the  main  circuit,  meets   and  is   cancelled 

by  the   outpouring — from  the  discharging  lower 

plate  ;    and  by  a  suitable  choice  of  L  and  C  the 

voltage  at  both  ends  in  left  free  to  rise  and  fall  as 

if  the  circuit  were  a  mere  bit  of  resistance  wire. 

Fig.  377.  [Notice  the  similarity  to  primary  and  condenser 

in  a  sparking  coil.] 
Round  the  gutta-percha  insulation  of  a  modern  submarine 
cable  is  lapped  one  layer  of  a  very  thin  tape  of  mumetal,  a  non- 
corroding  alloy  of  enormous  magnetic  permeability,  §  667.  This 
encourages  the  formation  in  itself — increases  the  flux — of  the  cir- 
cular magnetic  lines  of  Fig.  317,  i.e.  increases  L,  the  linkage  of 
lines  with  circuit,  up  to  the  value  necessary  to  compensate  the 
inevitable  C. 

Without  appreciable  addition  to  its  considerable  cost,  a  long 
cable  signals  ten  times  as  fast ;  over  the  Channel  cables  come  speech 
and  perfect  concert  music. 


1 


§  834]  ALTERNATING   CURRENT  681 

§834.  Inductive    Circuit,    High    Frequeney.    Look    again    at 

E  =  c  X  \/{R^  +  4tnhi^L^)   and  let  us  increase   n  from  50  to  a 
million. 

R  is  swamped ;  and  we  can  write  very  approximately  K  ^  e 
X  2TznL  ;  at  high  frequency  the  ohmic  resistance  of  a  circuit  mattem 
almost  nothing,  the  Inductance  takes  complete  controL 

Also,  if  E  remains  unchanged,  the  current  drops  to  a  fraction  of 
a  ten-thousandth  of  what  it  was ;  the  choking  effect  i»  enormou». 
To  maintain  a  few  milliamps.  we  must  either  reduce  L  by  uncoiling 
wire,  or  increase  P]  immensely. 

To  see  this  in  the  simplest  form,  bend  a  stout  copper  wire  into 
a  12-cm.  circle,  all  but  a  2-mm.  gap  (forming  a  circuit  with  a  natural 
frequency  of  several  millions)  and  apply  the  long  ends  to  dijwharge 
a  leyden  jar.  More  often  than  not  the  discharge  jump«  the  gap 
instead  of  travelling  round  the  inductive  ring,  see  this  do!<B. 
This  is  the  most  elementary  Choking  Coil. 

The  circuit  of  Fig.  377  set  to  work  to  supply 
itself,  let  us  cut  it  and  straighten  it  out  :  i.-.:";;**^ 

Circuit  with  inductance  and  capacity  in  series.     — 
Oscillating  circuit.    Let  C  and  C  be  given  equal 
and  opposite  charges,  and  then  the  circuit  left       L^ 
to   itself.     This  can   be  done   by   having  outer            g 
plates  at  the  two  ends,  charging  them,  and  then           $ 
sparking  between  them,  Figs.  378,  379.  r^'V..*'"^ 

A  current  starts  to  flow  through  L,  and,  as  * 

it  increases,  builds  up  an  interlinked  magnetic  *^»°-  '^*- 

field,  Fig.  321,  etc.     Presently  the  circuit  is  left 
with  C  and  C  quite  empty ,'^  but  a  maximum  current  occupying 
L ;   this  current  is  now  maintained  in  diminuendo  by  the  return  of 
the  interlinking  lines,  until  all  have  disappeared  and  current  has 
ceased,  but  has  filled  C  and  C  with  reversed  charges. 

These  charges  now  swing  back  again,  and  the  whole  procem 
goes  on  repeating  itself,  'pendulum  fashion,'  until  by  resistance 
or  other  losses  it  gradually  dies  away. 

Of  course,  its  oscillations  can  be  killed  by  havmg  too  much 
brake  on,  i.e.  too  much  resistance  in  circuit,  and  that  is  why  vm 
circuit  is  usually  taken  care  of  in  mwlern  apparatus,  and  is  Imked 
only  by  induction  to  those  which  have  to  send  away  their  energ.v. 

This  is  the  Oscillating  Circuit,  which  is  the  basis  of  high  fm|uency, 
diathermy,  etc.,  and  of  modern  wireless  telegraphy  and  teleplumy. 

'  Pendulum  fashion;  so  let  us  apply  the  i>endulum  formula  oj 
S  84  to  find  the  Periodic  Time  of  Oscillation  of  this  circuit 

lnt  =  2nV{llg)  the  length  is  what  gives  the  ixMi<lulum  (or  umt 
mass)  its  power  to  carry  on  in  mid-swing ;  g  is  the  «^f^^J^*J 
checks  unit  mass  from  going  higher,  and  sends  it  back  at  the  end 

of  its  swing.  .       .  .       •*         u-«-.  .*««  thmt 

To  the  electric  charges  movmg  in  our  circuit  we  haveww  tnat 

L  gives  momentum,  the  power  of  carrying-on  in  ^fV^J^.J^ 

current ;  while  the  force  that  pulls  it  up,  and  forces  it  back,  is  tlie 


682 


MAGNETISM   AND   ELECTRICITY 


[§834 


voltage  rising  against  it  as  it  squeezes  into  small  capacities  at  the 
end  of  the  swing.     This  is  1/C,  being  greater  the  less  room  there  is. 

/.     t  =  ^T^J-  transforms  into  t  =  2Tt  J  YJn  =  StuVLC. 

Your  mathematical  friend  can  give  you  a  fuller  investigation,  with 
the  same  result. 

§  835.  The  apparatus  of  Fig.  379  produces  electromagnetic 
oscillations  of  High  Frequency.  Current  is  sent  from  a  battery  B 
through  a  key  K  and  a  break  of  some  sort,  into  an  induction  coil  O, 
which  at  every  '  break  '  overcharges  the  inner  coatings  I  I  of  two 
ley  den  jars  C  C.  They  discharge  through  the  spark  gap  shown. 
This  circuit  I  I  has  very  little  self -inductance,  and  the  spark  is  a 
single  short  discharge  :  various  means,  such  as  burnishing  the  knobs 


Fig.  379. 


and  enclosing  them  in  an  inert  atmosphere,  are  employed  to  ensure 
its  abruptness. 

The  charging  of  I  I  with  -|-  and  —  involves  the  charging  of  the 
outer  coatings  C  C  with  —  and  +  charges,  which  travel  round 
through  L,  a  coil  of  a  few  wide  turns  of  stout  copper  wire. 

After  the  spark,  the  charges  on  C  C  are  left  in  the  circuit  CLC, 
which  possesses  capacity  in  C  C,  and  self-inductance  in  coil  L  ; 
consequently  they  oscillate  to  and  fro  in  an  alternating  current 
of  high  frequency  through  L. 

A  development  of  this  apparatus  is  the  High-Frequency  Furnace, 
of  which  a  diminutive  example  was  noted  in  §  108,  where  the 
'  plate  '  and  other  conductive  contents  of  a  wireless  valve,  held 
inside  the  coil  L,  were  rapidly  raised  to  red -heat  by  the  high-fre- 
quency eddy  currents  induced  in  them.     The  same  thing  is  carried 


ALTERNATING   CURRENT  683 

out  in  metal-melting  furnaces,  of  sizes  running  into  tons,  when-  it 
IS  desired  to  avoid  the  introduction  of  any  impurities  from  electro<ieii  • 
the  coil  L  becomes  a  wrapping  of  stout  copper  tubing,  cookxl  bv 
circulating  water,  round  the  clay  walls  of  the  furnace,  and  supplied 
with  many  hundred  kilowatts  at,  say,  30,000  cycles. 

The  'High-Frequency'  apparatus  of  the  medical  electrician 
adds  to  the  coil  L  a  prolongation  R,  usually  a  winding  of  a  few 
dozen  turns  of  bare  wire  on  a  varnished  wooden  cage  jHThaps  20  cm 
diam.  and  50  cm.  high.  This  long  coil  'resounds  '  electrically  to 
the  oscillation  in  L,  just  like  a  resonance  pipe  to  a  whistle,  *and 
high-frequency  discharges  of  varying  intensity  can  be  drawn 
from  different  parts  of  it,  masses  of  sparks,  or  pretty  brushes  and 
aigrettes  inches  long. 

High-frequency  discharge  of  less  spark-length  but  greater 
quantity  is  obtainable  from  any  part  of  coil  L,  as  the  charge  surges 
to  and  fro,  spilling  a  little  out  every  time,  adding  up,  from  the  great 
frequency,  into  fierce  white  sparks.  This  is  the  original  form  of 
Diathermy  apparatus. 

High-frequency  discharge  to  the  body  alternates  much  faster 
than  the  ions  in  the  nerve-endings  can  follow  it,  consequently  they 
can  send  no  message  to  the  brain,  and  H.F.  is  perfectly  painlesM. 
But  of  course  '  one  '  short  spark  ifrom  L  is  really  a  concentrated 
storm,  and  stings  like  a  hornet,  burning  a  hole  in  the  skin,  yet.  taken 
on  the  end  of  a  door-key  which  you  grip  firmly,  there  is  nothing  to 
feel. 

§  836.  From  this  arises  its  great  value  in  Diathermy  Treatment : 
by  an  applied  pad  wet  with  conductive  saline  the  H.-F.  discharge 
from  L  is  distributed  into  the  patient,  the  rapidly  oscillating  currents 
penetrate  painlessly  to  some  depth  and  dissipate  their  energ>' — often 

3  or  4  h.p as  heat  in  the  resistance  of  the  tissues,  warming  them 

up  deeply  without  injury  to  the  skin,  such  as  might  follow  ex- 
posure to  radiant  heat,  or  hot  applications. 

The  heating  can  be  concentrated  by  a  small  electrode  to  the  extent 
of  coagulating  and  baking  morbid  tissue  ;  or  from  a  small  liietaJ 
point  there  issues  the  little  unquenchable  flame  of  the  Cold  Cautery, 
searing  an  aseptic  and  bloodless  way,  preferred  by  many  surgeons 
to  the  knife. 

The  built-up  experimental  parent  apparatus  of  Fig.  379  gives 
too  little  current  at  needlessly  high  voltage  ;  the  modem  Diatherniy 
Machine  has,  in  place  of  C,  an  ordinary  transformer  taking  probably 
10  amp.  at  250  volts  A.C.  and  stepping  up  only  10:  1,  which 
suffices  to  produce  swarms  of  sparks  only  1 — 2  mm.  long  l)ctwcen 
flat  tungsten  plates  about  3  cm.  diam.  the  chill  of  the  metal,  and 
the  strong  field  in  the  gap,  prevent  sparks  blazing  a  trail  anywhere, 
and  quench  them  abruptly,  the  electrodes  keep  clean  and  smooth. 
and  every  spark  has  to  take  its  full  jump.  C  and  C  are  built  as 
multi-plate  condensers  from  sheet  metal  and  small  slabs  of  plate- 
glass,  embedded  in  wax,  L  lies  flat  beneath  the  top  plate  with  its 


684 


MAGNETISM   AND   ELECTRICITY 


[§836 


regulator  switches.  Your  Electro-Medical  Department  will  show 
you  these  machines  in  action. 

Practically  the  same  machine,  in  smaller  power,  forms  the 
Quenched-spark  Transmitter  of  the  stand-by  wireless  on  ships. 

The  latest  Diathermy  machine  is  run  by  a  Radio-transmitter 
Valve,  and  can  be  '  tuned-in  '  to  the  patient.  Very  high  frequencies 
may  be  employed. 

§  837.  Another  instance  of  an  oscillating  circuit  is  that  which 
maintains  the  Quartz  Plate  Standard  of  Frequency,  of  §§157,  451, 
and  802.     The  plate,  1/16  in.  thick  for  a  frequency  of  3,150,000, 

lies  betT^een  brass  plates  on 
the  left  of  Fig.  380  and  starts 
to  vibrate  in  thickness  imme- 
diately the  350  volts  from 
D.-C.  generator  and  steadying 
condenser  reaches  it  through 
the  valve.  The  grid -leak  and 
choke  circuit  is  impassable  to 
these  oscillations,  and  is  only 
a  means  of  slowly  removing 
unwanted  charge  accumula- 
ting in  the  valve.  Amplified 
current  of  this  frequency 
therefore  reaches  the  circuit  on  the  right,  which  has  to  be  tuned,  by 
varjang  the  capacity  of  its  condenser,  to  near  the  frequency  of  the 
quartz,  by  which  it  is  then  rigidly  controlled,  any  mistuning  that 
may  gradually  arise — through  coil  and  condenser  warming  up,  for 
instance,  or  through  the  mutual  inductance  with  the  next  circuit 
being  switched    in — affecting    only    the  amplitude,   and  not  the 


Fig.  381. 


frequency.     A  Quartz  Plate  is  the  Referee  for  the  radio -frequencies 
of  Europe. 

In  Fig.  381  the  main  circuit  of  Fig.  380  is  its  own  master,  to 
sufficient  accuracy  for  practical  wireless  transmission,  and  is  con- 
trollable in  frequency  over  a  wide  range  by  its  variable  condenser. 
What  was  the  quartz  master  circuit  is  now  only  drawing  oscillation 


§  838J  ALTERNATING   CURRENT  685 

from  the  main  inductance,  and  so  ensuring  that  the  circuit  keeps 
oscillatmg  vigorously— acting  as  valve  gear  to  the  engine,  ho  to 
speak.  The  middle  circuit,  coupled  through  the  inductance,  and 
tuned  m  by  its  variable  condenser,  applies  a  copy  of  these  oscillationj* 
to  the  grid  of  the  250-watt  power  valve,  which  feeds  a  50-tinu»i* 
amplified  current,  from  the  generator  below,  into  the  main  power 
circuit  on  its  left,  again  tuned  to  about  the  correct  frequency— 
but  not  so  closely  that  supply  to  the  aerial  is  going  to  react 
appreciably  on  the  master  circuit — and  thence  power  is  inductively 
transmitted  to  the  aerial  on  the  left,  just  as  from  L  to  Q  in  Fig.  379. 

Evidently,  audio-frequency  variations  of  the  grid  current  on  the 
right,  as  caused  by  inserting  on  the  right  the  secondary  of  a  trans- 
former, the  primary  of  which  is  fed  by  a  microphone,  will  be  copitni 
and  amplified  through  the  circuits,  and  finally  sent  out  from  the  aerial. 

Having  thus  sufficiently  illustrated  a  Radio  Transmitting  Circuit, 
the  infinite  variety  of  Receiving  Circuits  shall  be  left  to  you,  who 
can  probably  draw  them  by  the  dozen.  A  crystal,  gradually 
whittled  away,  has  kept  the  peace  of  this  house  for  seven  yean 
past,  and  we  are  not  looking  to  the  profits  of  this  book  to  replace  it. 

§  838.  Electro-magnetic  Waves.  Suppose  opposite  electric  chargcft 
are  moving  up  and  down  in  the  conductor.  Fig.  382.  As  two  op- 
posite charges  separate  and  move  off  to  charge  the  ends  oppositely, 
lines  of  electric  force  spread  out  between  them.  And  the  move- 
ment of  -f  electricity  downwards  and  —  upwards  is  of  cour«e 
equivalent  to  a  double  current  flowing  down,  and  sends  forth 
circular  lines  of  magnetic  force.  Thus  at  any  external  point  there 
will  be  an  electric  force  in  the  plane  of  the  wire  and  a  magnetic 
force  at  right  angles  to  it,  and  to  the  paper. 

During  the  return  swing  this  electro- magnetic  system  is  gradually 
withdrawn  and  replaced  by  a  reversed  system.  But  if  we  make 
the  oscillations  very  rapid,  there  comes  about  a  remarkable  change. 

Suppose  a  piston  is  being  worked  up  and  down  in  an  open 
cylinder.  The  air  near  by  moves  to  and  fro,  its  motion  is  not 
perceptible  10  ft.  away.  But  let  the  piston  move  a  few  hundred 
short  strokes  a  second,  and  strong  sound-waves  are  '  radiated  out,* 
and  can  affect  the  ear  or  other  detectors  at  long  distances. 

Similarly,  when  the  electric  oscillations  become  very  rapid. 
the  electromagnetic  lines  no  longer  quietly  return  to  the  wire  to 
be  replaced  by  a  reversed  system,  but  are  driven  out  and  away 
at  great  speed  as  successive  waves,  each  wave  bearing  in  its  front 
an  electric  force  parallel  to  the  conductor,  and  a  magnet  ic  force  at 
right  angles  to  it,  and  in  its  back  equal  reversed  forces.  Each  pair 
of  oscillating  charges  originates,  per  wave,  a  pair  of  closed  loops 
of  electric  force,  formed  as  in  Fig.  382. 

On  the  left  is  a  series,  at  eighth-period  intervals,  showing  two  ions 
in  sole  possession  of  the  wire  in  which  they  are  oscillating,  and  lieing 
abruptly  reflected  at  the  ends.  You  can  imagine  them  jwurin^ 
out  a  line  of  force  between  them,  like  two  firework  *  flying  pigeons 


686 


MAGNETISM   AND   ELECTRICITY 


[§838 


sliding  on  a  wire  pouring  out  trails  of  sparks,  the  line  always  leaving 
perpendicularly  to  the  wire,  and  everywhere  travelling  at  right 
angles  to  itself  at  the  great  speed  to  be  discussed  shortly  :  conse- 
quently it  is  always  semi-circular. 

As  they  pass  each  other,  the  line  is  nipped  off,  and  the  free  crescent 
loop  flies  outwards,  ever  growing,  while  the  two  ions  now  make  a  new 
one,  with  force  reversed. 

With  the  countless  crowd  of  ions  actually  present,  reflection 
at  the  ends  is  a  more  gradual  squeeze,  the  sharp  cusp  becomes 
rounded,  and  the  lower  figure  shows  the  great  loops  of  force  spreading 
from  a  transmitting  aerial.  Their  lower  halves  have  disappeared 
in  the  conducting  land  or  sea,  as  do  all  lines  of  force  in  a  conductor. 


Fig.  382. 


Each  pair  of  loops,  the  direct  and  the  reversed,  form  one  electric 
wave.  At  right  angles  to  them  are  the  waves  of  magnetic  force, 
indicated  by  the  arrows  circling  on  the  ground ;  for  these  nobody 
has  yet  succeeded  in  finding  a  use. 

Fig.  382  is,  of  course,  only  a  one-side  section  of  the  complete 
system,  which  spreads  in  rings  all  round.     Cut  a  big  onion  in  halves 
at  ground  level,  and  then  cut  a  quarter  out,  and  this  section  of  the ! 
overlapping  bulb-scales  gives  you  Fig.  382. 

You  see  that  there  is  an  alternating  vertical  push  and  pull  on 
the  electrons  in  the  surface  layer.  This  is  not  perfectly  conductive, 
and  causes  more  or  less  drag  on  the  spread  of  the  passing  wave  : 
the  result  is  that  the  wave  overhangs  slightly,  and  therefore  beats 
down — just  exactly  as  does  the  sound-wave  down  wind  in  Fig.  147, 
and  actually  penetrates  a  little  distance  beneath  the  imperfectly 
conducting  surface,  so  that  radio  signals  can  be  picked  up  by  a 
submarine  at  15  ft.  depth,  or  deeper  in  tunnels  and  mine  galleries. 

Very  poorly  conducting  dry  soil  exaggerates  the  distortion  and 
drains  away  the  energy  of  the  system  :  over  the  desert  the  range 
may  be  only  a  sixth  as  much  as  over  sea. 

The  overhead  Ionosphere  is  another  story,  see  §  884. 


§840]  ALTERNATING    CURRENT  687 

§839.  The  electrostatic  and  the  electromagnetic  measure  of 
quantity  of  electricity.  You  saw  a  very  natural  unit  of  electrical 
charge,  or  quantity,  in  §  721  ;  and  again  a  unit  current  defined  very 
naturally  in  §  749,  which,  flowing  for  a  second,  of  course  carriwi 
a  unit  quantity  of  electricity,  of  which  the  Coulomb  is  one-tenth. 

Surely  there  must  be  some  natural  connection  between  thene 
units  ;  how  do  their  sizes  compare  ? 

Take  a  parallel-plate  air  condenser,  and  calculate  its  capacity 
SI^Tit,  §  731.  Then  wire  it  to  a  contact  attached  to  a  vibrating 
tuning-fork,  which  connects  it  to  an  ordinary  H.-T.  batter>'  n  timc« 
per  second,  and  discharges  it  at  the  other  end  of  the  swing  into  a 
Wheatstone  bridge  circuit :  it  is  in  effect  a  Conductance  (n  x  itii 
capacity  in  e.m.  measure),  and  this  the  bridge  soon  measures  for  you. 

It  turns  out  that  3  X  10l^  thirty  thousand  million,  electrostatic 
units,gotomake  one  electromagnetic  unit  of  quantity — the  difference 
between  a  speck  of  coal-dust  and  a  comfortable  winter  supply  of 
15  tons  of  coal.  (And  the  ratio  of  units  of  Capacity  is  this  number 
squared.) 

§  840.  Now  the  question  can  be  put  in  another  way,  and  the  answer 
reveals  that  this  value  is  no  mere  accident  of  numbers,  but  has  a 
physical  meaning  of  interest.  Instead  of  asking  how  vast  a  horde 
of  electrostatic  units  must  be  driven  past  a  given  point  in  a  second, 
give  a  single  unit  a  centimetre  length  of  circuit  all  to  itself — say 
a  ring  1  cm.  circumference — and  give  it  the  task  of  imitating  l-cm. 
length  of  unit  (electromagnetic)  current — of  producing  the  same 
magnetic  effect  as  10  amps,  flowing  round  the  ring.  How  fast 
must  it  move  ? 

Evidently  it  must  pass  a  given  point  3  x  10*'  times  per  sec., 
consequently  its  speed  must  be  3  X  10"  cm.  per  sec. 

The  perfectly  natural  way  in  which  this  number  has  arisen 
suggests  that  this  actually  is  the  speed  of  free  movement  of  an 
electric  charge  in  a  conductor.  This  disagrees  badly  with  the  slow- 
moving  electrons  of  §  778.  But  compare  the  slow-moving  air 
particles  and  the  rapid  sound  wave  they  pass  on  ;  each  had  its 
momentum,  and  handed  it  on;  now  each  charge  has  its  line  of 
electric  force  and  hands  it  on  :  lines  stand  out  perpendicularly 
to  the  conductor  and  move  along  it  at  this  same  speed. 

The  lines  of  Fig.  382  as  they  spread  are  moving  at  right  angles 
to  themselves  at  this  speed  :  it  is  the  Speed  of  Travel  of  any  Electro- 
magnetic Wave.    Turn  to  §  952,  it  is  the  Speed  of  Light 

Since  the  length  of  a  wave  is  the  product  of  the  time  taken  in 
generating  it  and  the  speed  with  which  its  front  travels,  the  length 
of  the  electromagnetic  waves_radiated  from  an  aerial  fed  by  the 
circuit  of  §834  will  be  27cVLC  x  3  X  10"  cm.sec.,  where  both 
L  and  C  are  in  electromagnetic  measure.  But  if  we  now  meoiiure 
the  capacity  in  the  S/^izt  electrostatic  way  it  counts  9  x  10»  times 
larger,  and  we  can  write  : — 

Wave-length  =  2kVLC  cm.. 


688  MAGNETISM   AND   ELECTRICITY  [§  841 

where  L  is  the  number  of  unit  magnetic  lines  linked  in  the  oscillating 
circuit  when  10  amp.  flows  in  it,  and  C  is  the  capacity  in  electrostatic 
measure,  §  729. 

These  waves,  now  so  familiar,  were  discovered  as  a  by-product 
of  his  induction  experiments  by  Hughes  in  this  country  in  the 
eighteen-sixties,  but  the  discovery  was  not  followed  up,-  and  they 
were  actually  employed  in  America  by  Loomis,  who,  believing  in 
layers  of  electricity  in  the  air,  stretched  a  high  fishing-wire  up  from 
his  apparatus,  and  carried  another  into  a  pond,  and  found  he  could 
transmit  signals  to  some  distance  ;  but  they  were  not  understood 
until  their  existence  was  forecast  by  Clerk  Maxwell  from  his  equations 
in  1874,  and  they  had  been  transferred  from  mathematics  to  physical 
reality  by  young  Heinrich  Hertz  in  1888. 

His  oscillating  circuit  consisted  of  a  metre  of  straight  rod,  broken 
by  a  spark-gap  in  the  middle,  and  with  a  30-cm.  zinc  disc  on  each 
end  ;  these  were  directly  wired  to  the  spark  coil,  L  was  about 
600  and  C  5,  so  that  his  wave-length  was  about  350  cm.  His 
detector,  or  resonator,  was  a  metre  circle  of  wire  held  parallel  to 
the  oscillator,  and  having  a  very  minute  spark  gap  in  it. 

With  this,  or  much  smaller  apparatus,  down  to  oscillating 
circuits  which  were  more  short  lines  of  silver  on  glass,  giving  3-mm. 
waves,  waves  were  diffracted,  reflected,  focussed,  refracted  in  prisms 
of  brimstone,  and  pitch,  shown  to  be  polarized,  etc.  ;  thus  possessing 
the  general  characters  of  light  waves  of  exaggerated  size,  see  further 
Fig.  408.  Their  speed  of  travel,  also,  has  been  verified  by  direct 
measurement  as  identical  with  that  of  Light. 

Hertz  did  not  long  survive  his  discovery,  and  it  fell  to  Marconi 
to  make  the  greatest  advance,  when  he  turned  the  horizontal  ap-  ■ 
paratus  up  on  end,  so  that  the  waves  now  dance  along  over  the 
ground  instead  of  wriggling  away  all  their  energy  on  it,  worm- 
fashion,  within  the  hundred  yards. 

§  841.  Insulator  and  Conductor.  You  see  through  glass,  and  not 
through  copper ;  current  passes  through  copper,  not  through  glass. 
Unrelated  effects,  say  you  ? 

Make  your  glass  the  dielectric  of  a  condenser,  and  in  §  832  it 
readily  '  transmits  '  a  high-frequency  A.C.  to  which  in  §  834  copper 
can  prove  almost  impassable. 

What  are  Light-waves  but  an  alternating  electric  stress  of  fre- 
quency about  500  billion?  In  the  Dielectric,  §  737,  electrons, 
swinging  at  their  atomic  anchorages,  are  displaced  proportionally 
to  it,  and  proportionally  to  the  S.I.C.,  =  k,  so  that  Ijlc  is  a 
kind  of   '  elastic  modulus,'  and  the  waves  travel  on  with   speed 

V  =  VE/D,  §  395,  .-.  oc  Ijy/k. 
Also,  §  407,  V  oc  I/jx,  hence  Dielectrics  are  transparent  and  should 

have  refractive  index  \l  =  ^Jk. 

But  in  conductors,  §  778,  electrons  slip  their  moorings,  and  while 
offering  a  plastic  resistance,  can  give  no  elastic  kick,  E  collapses, 

V  is  zero,  light  cannot  pass.  Conductors  are  opaque. 


§  841]  ALTERNATING   CURRENT  680 

Both  these  you  dispute  :  the  short  lists  in  §§  485,  783,  do  not 
support  (X  =  ^Jk,  and  ebonite  is  black,  battery-acid  clear. 

Spare  a  thought  for  frequency  :  those  values  of  k  were  measured 
at  almost  zero,  fx  at  5  x  10»*  :  Chapter  XXXVIII  shows  there  iji 
room  for  discrepancy.  Using  short-electric-wave  methodi*  to 
measure  both,  it  almost  disappears. 

Or  as  a  first  step  towards  that,  try  infra-red  :  ebonite  traniimit« 
radiant  heat  freely,  §  965,  battery-acid  hardly  at  all. 

And  anyway,  paper-thin  ebonite  is  no  more  opaque  than  gold- 
leaf  1000  times  thinner,  while  battery-acid,  §  777,  conducU  only  a 
millionth  as  well  as  copper. 


EXAM  QUESTIONS,   CHAPTER  LI 

Alternating  Current,  as  such,  is  but  newly  put  in  the  •yllabus,  ao  that  the 
questions  below  are  few.  They  are  bound  to  increase,  for  most  pow«r  ayatMm 
now  supply  A.C.,  and  it  has  very  distinctive  propMtie«  of  it«  own. 

This  chapter  continues  Chap.  XL VI,  developing  a  simple  ezperiment 
really  belonging  to  it  into  the  all-important  Transformer,  with  «  gUnce  at 
the  Sparking  Coil,  now  dislodged  from  its  pride  of  pla<o  as  the  maker  of  X- 
rays,  and  in  turn  dislodging  the  Magneto  from  its  prodominanco  among  car 
engines.  §  828  contains  an  effort  which  you  may  or  may  not  find  helpful; 
thus  far  the  questions  go. 

§  829  begins  the  natural  history  of  the  Alternating  Current,  and  is  worth 
while  struggling  with — it  is  only  the  expression  of  what  you  know  alrtady — 
§  830  applies  it  to  the  Transformer,  making  its  action  much  more  definttv. 
§  831  is  for  the  mechanical  mind,  §§  832.  833  lead  up  to  the  OaciUating  Cirruit. 
basis  of  diathermy  and  radio-waves,  with  which  the  chapter  conclude*.  This 
treatment  seems  to  me  fairly  to  fulfil  our  intentions :  Figs.  S80,  S8I  woiiKI 
not  be  expected  of  you,  nor  are  very  searching  questions  liluly. 

1.  What  are  the  principles  or  laws  of  electromagnetic  induction  ?  Describe 
the  induction  coil,  explain  its  action  in  producing  high  tension  discluuges, 
and  mention  some  of  its  uses.     (  x  2) 

2.  Why  is  it  important  that  the  current  in  the  primary  should  be  intemiptod 
rapidly  ?  How  is  this  effected  in  practice  ?  What  is  the  action  of  the  coo* 
denser  ?     (  X  2) 

3.  Describe  an  induction  coil,  explaining  why  it  yields  a  brief  current  of  h|^ 
E.M.F.     How  do  the  '  make '  and  '  break '  induced  currents  diCer  ?    (  X  2) 

4.  State  the  principles  or  Faraday's  laws  of  Electromagnetic  Inductioo* 
Show  how  an  alternating  current  is  produced  by  mechanical  mrana.  On 
what  does  its  E.M.F.  depend  ?     {  X  2) 

5.  What  tests  would  you  apply  to  determine  whether  a  houM*h«>ld  electric 
supply  was  alternating  or  direct  current  ?  Describe  some  iiistrunwnt  suitable 
for  measuring  A.C. 

6.  You  are  provided  with  a  battery,  a  key,  and  an  electromagnet.  Describe 
how  the  current  varies  on  depressing,  and  on  raising,  the  key. 

7.  State  the  laws  of  electromagnetic  induction  and  describe  the  elliscts 
of  self  and  mutual  inductance. 

8.  Why  are  high  potential  currents  employed  in  transmitting  aosffy  to  a 
distance  ?  Explain  the  use  of  transformers  in  such  cases,  and  point  out  the 
sources  of  waste  of  energy. 

9.  What  conditions  determine  whether  the  discharge  of  a  coodsnaersball 
be  oscillatory,  or  merely  a  diminishing  direct  current  ?  Show  how  tha  eirraii 
is  arranged  to  yield  high-frequency  current  for  treatment. 


^ELECTRICITY 


CHAPTER   LTI 


THE  TRANSPORT  OF  ELECTRICITY  THROUGH  LIQUIDS      J 


§  85L  When  the  electrons  streaming  though  the  metal,  which, 
as  we  saw  in  §  778,  obstructs  them  no  more  than  a  forest  does  its 
flies,  arrive  at  the  end  of  a  wire  dipping  into  a  liquid,  what  happens  ? 
The  material  is  still  about  as  dense  as  ourselves,  it  contains  atoms 
fairly  closely  packed,  but  now  possessing  more  mobility  :  do  the 
electrons  plunge  in  and  swim  ? 

No,  a  wire  passes  no  current  into  oil,  or  alcohol,  or  purest 
water. 

Let  us  provide  boats,  the  molecules  of  sugar  and  things  that 
were  so  effective  in  Osmosis.  No,  the  current  will  not  pass  through 
wine  or  tea. 

But  there  were  strange  hyper-effective  substances  which  gave  us, 
in  §  379,  too  much  osmotic  activity,  KCl  1-8  times,  MgClg  or  Ca(N03)2 
2-5  times  or  more,  molecules  that  broke  into  pieces  ?  Stir  in  a 
spoonful  of  salt ;  and  current  passes  at  once. 

Do  these  broken  fragments  carry  some  attraction  that  induces 
the  electrons  to  venture  on  board  ?  It  can  only  be  electric  charge  ; 
are  they  charged,  some  +,  other  correspondingly  —  ?  If  so,  and 
we  put  +  and  —  wires  into  the  solution,  won't  the  two  kinds  be 
attracted  opposite  ways,  and  produce  differences  of  some  kind 
round  the  wires  ? 

Taste  a  drop  of  the  salt  water  from  near  either  wire  ;  one  is  bitter 
and  the  other  is  sharp  ;  one  gives  blue  with  litmus,  the  other  reddens 
it,  or  sometimes  bleaches  ;  but  the  intervening  bulk  of  the  solution 
remains  just  salt  water. 

So  there  were  carriers,  called  Ions  {r.iov,  going),  some  of  which 
were  short  of  an  electron,  or  perhaps  two,  and  were  therefore 
+  ,  or  +  -fj  charged;  and  drew  up  to  the  landing-stage  where 
electrons  were  waiting  to  embark,  and  were  able  by  their  attraction 
to  overcome  that  of  the  main  bulk  of  the  metal  behind,  so  that 
the  electrons  came  on  board.  These  were  cat-ions,  and  that 
electrode  (6So?,  a  threshold)  is  the  cathode  (xara,  down  from),  by 
which  the  conventional  positive  current  leaves  the  liquid. 

Other  carriers  had  an  electron  too  many,  and  were  —  charged 
(there  were  no  —  — ,  unless  perchance  you  used  Epsom  salts), 
and  these  anions  drew  up  to  the  anode  (ava,  up),  and  unloaded 

690 


§853]  ELECTRICITY   IN   UQUIDS  691 

their  electrons  upon  it,  to  become  the  electronic  current  in  solid 
metal  again. 

From  the  simple  chemical  tests  suggested,  you  find  that  the  cations 
are  the  +  sodium  half  of  the  NaCl  (or  the  +  -f  Mg  half  of  the 
MgS04),  which  attacked  the  water  as  soon  as  they  had  taken  aboard 
electrons,  and  made  bitter  alkali ;  and  the  anions  are  the  —  CI 
or  —  —  SO4,  which  seized  hydrogen  from  the  water  and  became 
acid,  after  they  had  unloaded  their  one,  or  two,  electrons ;  setting 
free  oxygen,  '  the  acid  producer,'  which  you  may  see  as  a  film  of 
bubbles  on  the  anode. 

§  852.  But  how  could  free  sodium  atoms  be  wandering  aljout  in 
water  ? 

We  have  seen  already,  §§  737,  778,  that  an  atom  is  an  elaborate 
system  of  a  central  +  nucleus  defended  by  rings  of  flying  electrons. 

'  Chemical '  actions  never  penetrate  beyond  the  outer  defences. 

Sodium  happens  to  be  a  very  clear  case,  it  normally  possesses 
11  electrons,  and  of  these  the  odd  one  is  in  an  outer  orbit  by  itself  : 
robbed  of  that  one  {=  +  charged),  as  it  is  while  wandering  in  the 
solution,  it  presents  only  the  closed  ring  of  8  electrons  characteristic 
of  its  next  lower  neighbour  in  the  Periodic  Table,  the  utterly  inert 
Neon  :   so  in  that  condition  it  cannot  attack  anything. 

Chlorine  is  equally  clear  :  normally  it  possesses  17  electrons, 
and  its  outer  ring,  of  7,  is  just  one  short  of  its  next  higher  neighlx)ur. 
Argon,  which  again  has  the  complete  8  ;  so  there  again,  CI  loaded 
with  an  extra  electron  is  inert. 

§  853.  Are,  then,  these  salts,  which  make  the  transport  of  electricity 
possible  in  this  electrolyte  solution,  by  themselves  Ijecoming 
electrolysed  (Xuao>,  Xuaw,  unloose),  spontaneously  dissociated  in  this 
way,  or  ionized,  in  solution ;  or  is  that  actually  caused  by  the 
electric  field  we  set  up  between  the  electrodes  ? 

The  ionization  is  spontaneous,  for  it  accounts  for  the  action  of 
§379,  where  no  electrified  plates  are  about.  Also,  once  we  get 
an  electric  current  into  the  electrolyte,  we  find  that  Ohm's  Law  is 
obeyed,  and  that  means  that  energy  is  being  lost  merely  by  f rid  ion  ; 
and  not  by  any  manufacturing  process  which  might  be  reversed 
with  the  current,  and  betray  itself  by  an  active  voltage. 

Why  should  mere  mixing  with  water  break  up  NaCI  in  this 
strange  fashion  ? 

Well,  to  start  with,  it  breaks  up  the  hard  crystals  much  finer  than 
your  best  efforts  with  pestle  and  mortar  1 

From  §  737  it  can  be  seen  that  in  a  medium  of  a|H»cific  inductive 
capacity,  K,  the  electric  force  between  two  charges  is  K  times  leas 
than  in  air ;  because  K  times  as  much  charge  must  be  put  on  to 
get  up  the  same  unit  P.D.,  whereat  you  measure  capacity,  and 
P.D.  =  force  x  distance  apart  of  charges,  and  the  latter  remams 

unchanged.  ,         ,.     ,  ..j    •    oc     / 

From  the  Table  §  733  the  S.I.C.  of  carbon  disulphide  is  2-5,  ol 

alcohol  27,  and  of  water  80. 


692  ELECTRICITY  [§  853 

Now,  it  is  almost  impossible  to  make  up  any  conducting  solution 
with  CS2  ;  and  solutions  of  salts  in  water  are  much  better  conductors 
of  electricity  than  equally  strong  solutions  in  alcohol.  Evidently 
in  NaCl  the  solitary  outer  electron  of  Na  has  fitted  itself  into  the 
vacant  place  in  the  outer  ring  of  CI,  but  when  the  force  holding  it 
to  Na  has  been  reduced  to  1/80,  the  two  systems  break  apart, 
leaving  the  —  charge  still  attached  to  the  CI ;  and  there  are  your 
two  ions. 

§  854.  Are  there  any  difficulties  about  the  embarkation  of  the 
electrons,  and  their  going  ashore  ? 

Indeed  there  are  :  we  saw  in  §§  798,  800  how  an  E.M.F.  arose  at 
the  junction  of  two  metals  ;  here  there  is  a  similar  thing,  but  the 
E.M.F.  or  Contact  Potential  Difference  is  half -a- volt  or  more,  between 
metal  and  solution,  and  there  are  contact  P.D.'s  between  different 
solutions.  Sometimes  they  oppose,  and  sometimes  help,  the  current ; 
they  build  up  the  Polarization  E.M.F.  of  §  860,  and  the  E.M.F.'s 
of  Voltaic  Cells. 

To  investigate  conduction  in  the  electrolyte  one  must  break 
through  these  '  quayside  formalities,'  and  that  is  done  by  using  a 
rather  high -voltage  testing  current  which  is  small  and  rapidly 
alternating,  and  gives  no  time  for  appreciable  electrolysis  to  take 
place.  To  §  785  it  only  remains  to  add  that  the  testing  cell  is  usually 
a  small  beaker  into  which  two  platinum  plates  dip  from  a  cross-bar  ; 
one  doesn't  measure  dimensions,  but  standardizes  the  cell  once  for 
all  by  measuring  its  resistance  when  filled  with  Normal  KCl,  which 
has  resistivity  10-59  ohms/cm.^  at  18°  C. 

By  working  with  different  current -strengths  it  has  been  shown 
that  Ohm's  Law  is  obeyed,  to  1  part  in  10,000,  when  once  inside 
the  polarization  E.M.F.  barrier  ;  hence  Resistivities  can  be  quoted, 
as  in  the  Table  §  777,  although  the  electro-chemist  usually  prefers 
to  speak  of  Conductivities,  in  mhos/cm.^,  as  in  the  last  column. 

Quite  unlike  metals,  the  Resistivity  diminishes,  or  Conductivity 
increases,  with  rise  of  Temperature,  largely,  about  *2-5%  per  °  C. 

This  is  the  rate  of  increase  of  mobility  (diminution  of  viscosity, 
Fig.  107)  of  water,  with  temperature  ;  so  it  appears  that  Electrolyte 
Resistance  is  just  the  fluid  frictional  hindrance  to  the  movement 
of  particles  of  molecular  size. 

§  855.  By  working  with  different  concentrations  of  solution,  it 
appears  that  whereas  the  conductivity  is,  for  quite  Dilute  Solutions, 
proportional  to  the  concentration,  the  rate  of  increase  falls  off 
as  the  solution  gets  strong  :  the  simple  explanation  is  that  all  the 
salt  is  ionized  in  dilute  solutions,  but  only  a  diminishing  fraction  of 
it  in  stronger,  a  recombination  process  setting  in  as  the  ionic 
population  becomes  denser. 

Thus  HCl  and  HNO3  are  about  0-995  ionized  at  milli-normal 
concentration,  and  0-86  at  iV/2  ;  NaHO  0-97  at  iV^/1000  and  0-80 
at  NI2,  and  KCl  is  much  the  same.     Weak  acids  and  bases  show  the 


§  856]  ELECTRICrrY   IN    UQUIDS  691 

effect  much  sooner — in  fact  it  is  imposHiblo  to  get  them  anything 
like  completely  ionized  at  any  useful  strength  ;  e.g.  acetic  neiil 
0126  at  A7IOOO  and  00()6  at  N j'l,  carbonic  acid  0-017  and  0-4M)lW. 
ammonia  0141  and  OOOOS.     They  will  not  come  out  and  fight. 

In  alcohol  the  falling-off  is  five  times  faster. 

Weakest  of  all  ionizations  is  that  of  Pure  Water  itself,  nomething 
like  1  part  in  ten  millions. 

By  comparative  measurements  with  many  salts,  Specific  Ionic 
Mobilities  may  be  made  out  for  the  ions,  which  show  their  com- 
parative speeds  of  travel,  i.e.  the  part«  they  usually  play  in  carr>'ing 
the  current. 

Then  further,  by  calculation,  or  by  direct  experiment,  the  actual 
slow  Speeds   of   Ionic    Travel  are  obtainable,  and  for  a  {Miti>ntial 
gradient  of  1  volt  per  cm.,  these  are,  in  cm.  per  sec., 
H  0000325,  Na  0000043,  Ag  0000054,  Cu  00(KH>4<)  anu.ng  lation.H 
OH  0000174,  CI  0000066,  NO3  0000062,  SO4  0(KX)061>  among  anions 

You  see  that  the  light  H  ions  do  most  of  the  transport  work, 
travelling  a  whole  centimetre  in  an  hour. 

If  you  were  the  electron,  H  would  be  a  100-ton  high-speed  craft. 
and  Ag  an  11,000-ton  cargo  liner. 

§856.  Examples   of  Electrolysis.    A  few  of  the  many  actual 

electrolytic  processes  are  outlinetl  below. 

I.    With  no  complex  secondary  actions. 

Dipping  arc-lamp  carbons,  wired  to  a  batter>'  of  a  few  accumula- 
tors, into  a  crucible  of  fused  lithium  chloride,  the  choking  smoll 
of  chlorine  arises,  and  the  cathode,  when  withdrawn,  shows  httle 

shining  globules  of  metallic  lithium 

LiCl — ^U  +  C1 

Aluminium  is  commercially  produced  bv  electrolysing  a  »o*»<»«a 
of  alumina  (obtained  by  purifying  and  calcining  the  mineral  hydr- 
oxide,  bauxite)  in  cryolite,  AlF3,3NaF,  melted  at  1000   C. 
AI2O3— >2Al  +  30 

The  metal  sinks  to  the  carbon-lined  cathode  floor  and  ia  tapped 
off  twice  a  week,  the  oxygen  attacks  and  consumwi  the  ma-ivc 
coke-block  anodes  dipping  in  the  bath  from  above,  thua  mam- 
taining  the  necessary  heat ;  8000  ampdres  produces  about  H  cwl. 
per  week. 
II.  With  secondary  actions  on  the  liquids. 

'with  dilute  acid  of  any  sort,  say  sulphuric,  between  c-arbon  or 
platinum  electrodes,  two  volumes  of  hydrogen  «"- ?»;^"  "«  '"'"» 
the  cathode,  and  about  one  volume  of  oxygen  from  the  anode. 

HHISO4 
H2  off  at  cathode    |    +  H,0  =  H^SO^  f  O  at  anode. 


694  ELECTRICITY  [§  856 

The  oxygen  when  freed  from  its  ionic  charge  has  not  yet  collected 
into  its  customary  molecules  Og,  and  a  small  part  of  it  usually  gathers 
into  triatomic  molecules  O3  of  Ozone,  recognizable  by  its  odour. 

In  electrolysing  cold  potassium  acid  sulphate  part  of  the  anode 
oxygen  goes  to  precipitate  persulphate 

2KHSO4  +  0  =  HgO  +  2KSO4. 

The  British  Oxygen  Company  devote  serried  ranks  of  tanks  at 
N.  Wembley  to  the  production  of  pure  electrolytic  Hydrogen,  for 
use  in  the  the  hydrogenation  of  oils  to  solid  margarine,  to  suit 
British  taste  (?),  by  electrolysing  caustic  soda  between  iron  plates 
which  are  not  attacked. 


Na 
cathodes — H    OH 


HO 
=  KH^O  +  O). 


The  heat  generated  in  overcoming  the  resistance  of  the  electrolyte 
maintains  an  atmosphere  like  a  steam  laundry. 

Pure  electrolytic  hydrogen,  generated  in  the  test  solution,  is 
indispensable  in  making  Marsh's  test  for  arsenic,  in  medico -legal 
work,  for  even  the  purest  zinc  cannot  be  trusted  to  be  absolutely  free 
of  this  impurity. 

The  electrolysis  of  strong  brine  has  become  an  important  com- 
mercial process.  The  sodium  attacks  the  water  round  the  cathode 
to  form  caustic -soda  solution,  chlorine  accumulates  in  solution 
around  the  anode,  the  voltage  being  kept  too  low  to  liberate  gaseous 
oxygen. 

NalCl 
Hoff  +  NaHO^ — H2O+I 
[cathode  alkaU]  [neutral] 

For  the  manufacture  of  caustic  soda  the  cathode  is  a  pool  of 
mercury,  in  which  much  of  the  sodium  is  temporarily  retained  as 
an  amalgam,  the  metal  is  circulated  into  an  adjoining  tank  of 
water,  and  the  amalgam  slowly  decomposes  there  to  produce 
pure  caustic  solution. 

If  the  whole  liquid  is  gently  stirred  and  kept  cool,  chlorine  and 
caustic  interact  to  produce  sodium  hypochlorite,  bleaching  and 
disinfectant  {e.g.  '  Milton ')  ;  if  warm  the  interaction  produces 
sodium  chlorate,  for  explosives,  from  which  potassium  chlorate 
can  be  obtained  for  therapeutic  use.  About  7500  Units  produce 
a  ton. 

'  Pole-finding  paper  '  is  impregnated  with  sodium  sulphate  and 
phenolphthalein ;  when  moistened  and  laid  across  the  ends  of 
a  broken  circuit  it  turns  crimson  on  the  negative  wire,  owing  to 
the  alkali  set  free  at  the  cathode.     There  are  other  varieties. 

III.  With  secondary  actions  on  the  electrodes. 

In  electrolysing  copper  sulphate  between  copper  plates,  pure 
copper  is  deposited  on  the  cathode,  and  the  '  sulphions,'  instead 


§857]  ELECTRICITY   IN   LIQUIDS  695 

of  attacking  water  and  producing  acid  and  oxygen,  attack  the  anode, 
which  loses  weight  as  fast  as  the  cathode  gains  ;  or  rather  faMter, 
because  the  impurities  in  its  metal  also  fall  away 

CulSO^ 
deposited  on  cathode   |    +  Cu  dissolved  off  anode. 

This  electrodeposited  layer  of  copper  is  used  in  all  good  Electro- 
plating as  a  foundation  for  subsequent  layers  of  nickel,  chromium. 
silver  or  gold. 

All  copper  for  electrical  purposes  is  refined  elect  rolytically.  an 
the  impurities  of  the  anode  ingot,  whether  they  fall  away  or  di«»olvc 
in  the  liquid,  do  not  get  deposited  on  the  cathode. 

Silver  plating  is  done  in  a  bath  of  a  so-called  double  cyanide 
of  potassium  and  silver,  more  accurately  potassium  argent icyanide 
KAgCy2.  This  splits  into  cation  K  and  anion  AgiVj.  At  the 
cathode  the  potassium  atom  attacks  the  solution  thus, 

K  +  KAgCya  =  2KCy  -f  Ag 

and  by  this  action  the  silver  is  deposited  in  a  smooth  Uyer 
(whereas  directly  deposited  from  silver  nitrate  solution  it  ia  in 
separate  granular  crystals).  At  the  anwle  the  AgC-y,  attaokn 
the  ever-present  excess  of  potassium  cyanide  and  the  silver  anode 
plate  itself,  and  re-forms  the  argenti-cyanide  : — 

AgCy^  +  2KCy  +  Ag  =  2KAgCy,. 

Gilding  is  similarly  done  from  a  gold-cyanide  bath. 

A  trace  of  carbon  disulphide  in  the  bath  ensures  a  bright,  as 
opposed  to  a  frosted,  coating. 

A  less  poisonous,  but  effective,  electrolyte  for  silver-pUting, 
is  the  photographer's  '  spent  hvpo.' 

Articles  for  plating  are  first  boiled  in  alkali,  washed  and  pickled 
with  acid,  and  scratch-brushed  all  over.    They  are  then  '  gho«ted 
with  silver  by  a  minute's  immersion,  inspected  and  scratch -brushed 
all  over  again,  and  then  given  an  hour  and  a  half,  or  more,  in  the 
depositing  tanks. 

§  857.  The  study  of  this  subject  began  in  ISOO,  when  water  w«« 
first  electrically  decomposed  into  oxygen  and  hydrogen,  m  1^' 
Wollaston  deposited  silver  and  gold  on  baser  metals  and  laid  the 
foundation  of  the  art  of  electro-plating,  and  in  1807  Davy  de- 
composed  moist  caustic  soda  and  potaah  and  discovered  tlie  metala 

sodium  and  potassium.  i  t     *•*   *:«„  u^. 

Later  the  subject  was  taken  up  at  the  Royal  Institution  b> 
Faraday,  and  he  introduced  the  name  Electrolysis  and  the  vanojui 
terms  already  mentioned.  Having  first  satisfied  hmiHelf  n^  to  the 
identity  of  electricity  from  whatever  source  derive*!,  and  that 
in  an  electrical  circuit  the  current  was  the  same  all  the  way  round, 
he  in  1834,  enunciated  Quantitative  Laws  of  Electrolysb. 


696  ELECTRICITY  [§  867 

Law  I.  The  amount  of  chemical  action  taking  place  in  one  and 
the  same  electrolyte,  as  measured  by  the  mass  of  some  particular 
constituent  set  free,  is  proportional  to  the  quantity  of  electricity 
passed  through,  measured  electromagnetically,  as  in  §§  765,  766. 

This  is  to  be  proved  by  comparing  the  weights  of  copper,  for 
instance,  deposited  on  the  cathode  when  the  ammeter  reads 
1  amp.  for  60  min.  or  3  amp.  for  20  min.,  they  should  be  the 
same ;  or  by  comparing  the  volumes  of  hydrogen  or  oxygen 
given  off  from  dilute  acid  by  different  currents.  On  this  law  is 
based  the  use  of  the  Voltameters  described  below  for  measuring 
the  total  quantity  of  electricity  carried  through  by  any  current, 
however  variable. 

Notice  that  the  Voltage  does  not  come  into  account  at  all.  All 
electrolysis  is  conducted  at  low  voltage,  six  volts  or  so,  just  enough 
to  overcome  resistances  and  the  small  polarization  E.M.F.  It  is 
purely  a  question  of  Quantity  of  Electricity,  number  of  electrons. 

Law  II.  The  mass  of  an  element  set  free  by  the  passage  of  a  given 
quantity  of  electricity  is  proportional  to  its  chemical  Combining 
Weight  in  the  compound  being  electrolysed. 

To  investigate  this,  voltameters  (see  below)  containing  solutions 
of  dilute  acid  or  alkali,  of  copper  sulphate,  silver  nitrate,  mercurous 
nitrate,  etc.,  are  connected  in  series,  and  a  current  passed,  necessarily 
conveying  the  same  quantity  of  electricity  through  each.  Calculat- 
ing the  weights  of  the  volumes  of  hydrogen  and  oxygen  collected, 
and  weighing  the  various  cathode  deposits,  they  will  be  found  in 
the  ratio,  hydrogen  1,  oxygen  8,  copper  31-5,  silver  108,  mercury 
200,  etc.  These  are  the  same  proportional  weights  that  ordinary 
chemical  analysis  shows  capable  of  combining  with,  or  of  replacing, 
unit  weight  of  hydrogen. 

The  mass  of  a  substance  set  free  by  one  Coulomb  {=  1  ampere  X  sec.) 
of  electricity  is  called  the  Electrochemical  Equivalent  of  the  substance. 

Notice  at  once  the  different  meanings  of  Electrochemical  equivalent 
and  Chemical  combining  weight  (sometimes  called  chemical  equivalent 
weight).  But  notice  also  their  proportionality.  Notice  further, 
their  possible  variation,  in  the  case  of  some  substances,  between 
different  classes  of  compounds  ;  iron  would  be  deposited  with  greater 
economy  of  current  from  green  salts  than  from  yellow  :  you  know 
how  to  explain  this  on  the  Atomic  Theory,  but  be  careful  not  to 
confuse  the  Combining  Weight  disclosed  by  analysis  with  the 
Atomic  Weight  derived  by  the  subsequent  application  of  theory, 
although  probably  you  will  recollect  the  latter,  and  get  the  former 
by  dividing  by  the  valency. 

The  electrochemical  equivalent  of  silver  is  0-0011180  gm.,  of 
Hydrogen  0-00001044  gm.,  of  Copper  from  blue  copper  sulphate 
0-000328  gm.,  etc. 

Each  is  1/96,500  of  its  chemical  combining  weight  in  grams,  or 
96,500  coulombs  liberate  the  gram  equivalent  of  any  ion. 

This  Quantity  of  electricity  is  called  a  Faraday. 


868] 


ELECTRICITY   IN  LIQUIDS 


697 


§  858.  The  utilization  of  Faraday'n  Laws  of  Elcctrolysiii  for 
the  Measurement  of  Quantities  of'  Eleotrieity  prove*  extremely 
convenient  in  practice.  Tlie  electrolytic  celln  uHe<i  for  the  purfMjM 
are  called  Voltameters,  or  sometimcM  Couloineten*,  and  many 
varieties  of  them  have  been  devised. 

The  oldest  pattern  of  Gas  Voltameter  re«emblc«  Fig.  383 ;  there 
are  two  electro<les  of  platinum  foil  immcrHc<l  in  weak  Hulnhuhc 
acid,  the  leading  wires  are  covered  with  wateqiroof  imiulntion. 
Rising  above  each  is  a  glass  tul)e  graduated  in  cubic  centimotrra; 
the  hydrogen  rising  into  the  cathode  tul»e,  and  the  oxvgen  into  tho 
anode  tube,  displace  the  liquid  with  which  lK)th  are  fiflwl  at  fin»t. 

Another  pattern,  easy  to  refill,  is  shown  in  Fig.  3H4  ;  in  it  either 
hydrogen  or  oxygen  can  be  collected  according  to  direction  of 
current ;  hydrogen  preferablv,  because  the  volume  of  oxygm  is 
apt  to  be  unduly  diminisheii  by  its  greater  solubility  in  water. 
and  by  its  partial  ozonization. 


Fig.  383.         Fig.  384. 


Fig.  385. 


The  Mixed-gas  Voltameter,  Fig.  385,  is  a  small  jar  provided 
with  an  air-tight  bung  and  leading  tube  and  containing  caustic- 
soda  solution  in  which  dip  two  large  electrodes  of  sheet  nickel. 
cf .  §  856.  The  nickel  plates  are  not  attacked,  and  the  oxygen  is  free 
from  ozone.  The  mixed  gases  are  collected  in  a  graduated  tube  over 
water  ;   they  are  of  course  explosive. 

An  instance  of  the  Calculation  necessary  with  gaa  voltaroetert 
has  been  given  in  §278.  It  remains  only  to  point  out  that  since 
1  coulomb  liberates  the  electrochemical  equivalent  ()-iM«U(>44 
gm.  of  hydrogen,  the  weight  000541  gm.  there  calcuUte<l  indicate* 
the  passage  of  0  0054 1/0  0000 1044  =  519  coulombs.  (If  this  were 
collected  in  240  sec.  during  which  the  current  waa  kept  steady  by 
galvanometer,  that  would  indicate  519/240  =  2-16  amp.,  etc.) 

Roughly,  1  coulomb  produces  1/9  c.c.  of  hydrogen,  or  1/6  c.c.  of 

mixed  gas.  .  n  .     i    ^ 

The  Copper  Voltameter,  Fig.  386,  consists  of  a  small  t«nk  ol 
fairly  strong  blue  solution  of  copper  sulphate  slightly  •^^J"^ 
with  sulphuric  acid  ;  in  it  dip  a  couple  of  anode  platea  of  sh|Mi 
copper,  and  between  them  a  thin  removable  cathode  plate.  TTJJ 
latter  is  scoured,  rinsed,  dried,  and  weighed  at  the  start,  and 


698  ELECTRICITY  [§  858 

rinsed,  dried,  and  weighed  at  the  finish,  the  grammes  gain  in 
weight  divided  by  0-000328  :=::  coulombs  passed.  It  is  largely- 
used  for  commercial  testing  purposes. 

Figs.  385,  386  represent,  of  course,  the  arrangement  of  an  experi- 
ment on  Law  II,  for  comparing  the  electrochemical  equivalents 
of  hydrogen  and  copper,  and  so  determining  the  latter's  combining 
weight. 

The  Wright  House  Meter  electrolyses  mercurous-nitrate  solution, 
between  a  pool  of  mercury  and  a  metal  thimble ;  the  mercury 
globules  drop  from  the  thimble  into  a  graduated  tube,  whence  they 
periodically  siphon  over  into  a  second  wider  graduated  tube.  The 
whole  is  hermetically  sealed  and  is  reset  for  use  by  inverting  it  for 
a  moment,  when  the  mercury  runs  back  into  the  pool. 

A  way,  that  will  appeal  to  the  chemist,  of  measuring  small 
quantities,  is  to  electrolyse  potassium  iodide  between  platinum 
electrodes,  and  titrate  the  iodine  with  thiosulphate.  Since  10  litres 
of  decinormal  solution  contain  the  combining -weight  in  grams,- 
and  this  is  set  free  by  a  Faraday,  evidently  1  c.c.  of  decinormal 
solution  =  9-65  coulombs. 

In  the  Silver  Voltameter,  silver  is  deposited  in  little  granular 
crystals  on  a  platinum  bowl  which  holds  100  c.c.  of  10 — 20% 
solution  of  the  purest  silver  nitrate.  You  can  see  that  you  will 
not  be  called  upon  to  use  this.  Under  carefully  prescribed  conditions 
it  is  trustworthy  within  1  part  in  10,000  parts,  and  is  by  far  the 
most  accurate  we  possess. 

Indeed,  one  may  say  it  is  the  only  one,  for,  to  tell  the  truth, 
most  substances  honour  Faraday's  Laws  more  in  the  breach  than  in 
the  observance  :  §  856  has  suggested  already  that  there  are  many 
ways  of  dodging  them. 

In  the  Copper  Voltameter  itself — and  it  is  the  next  best — '  blue- 
stone  '  dissolves  in  water  to  a  shghtly  turbid  solution,  and  if  this 
is  electrolysed,  the  copper  ions  drag  down  with  them  colloidal 
basic  sulphate,  and  the  deposit  is  too  heavy.  So  one  adds  a  little 
sulphuric  acid  to  keep  the  solution  clear,  and  then  the  acid  attacks 
the  deposited  copper,  at  a  rate  depending  upon  concentration  and 
temperature  :  with  warm  acid  solution  and  very  small  current  there 
might  conceivably  be  no  deposit  at  all.  The  0-000328  given  is 
a  compromise  :  fortunately,  under  ordinary  laboratory  conditions, 
strong  cool  feebly-acid  solution,  current  somewhere  about  1  amp./ 
sq.  decimetre,  it  is  reliable  to  1  part  in  400  parts. 

The  aluminium  manufacturer  does  not  obtain  a  theoretical 
yield,  while  he  of  the  chlorate  gets  40%. 

This  does  not  mean  that  you  can  slight  these  Laws,  or  compare 
them  disparagingly  with  those  in  your  chemistry  books.  It  has  been 
shrewdly  said  that  most  chemical  actions  are  electrolyses  reversed. 
The  laws  of  the  Medes  and  Persians  altered  not — for  bakshish. 
If  you  take  an  opportunity  of  going  round  a  chemical  works,  you 
will  get  some  eye-openers,  and  will  come  away  with  an  increased 
respect  for  the  men  who  could  make  out  laws  at  all. 


1 


§861]  ELECTRICITY    IX   LIQUIDS  699 

§859.  Electro-osmosis  and  cataphoresis.  Not  only  do  ulu  in 
solution  form  charged  ions,  but  colloid  and  other  minute  particl«i 
in  suspension  become  electrically  charged,  and  can  accordingly 
be  driven  through  the  liquid  by  efectric  field. 

Thus,  in  the  purification  of  fine  china  clay,  the  negative-charged 
particles  travel  through  the  water — cataphore^iH — and  nettle  on  the 
anode  ;  and,  as  they  can  get  no  further,  the  water  from  among  them 
gets  forced  back  the  other  way — electro-o8mo«i» — ami  the  layer 
consolidates.  Using  50  volts  pressure,  to  keep  up  a  serviceable 
speed,  12  kw.-hrs.  has  deposited  7  tons  of  dry  clay,  an  *  electro- 
chemical equivalent '  of  S  gm.  per  coulomb. 

By  passing  a  current  through  silicate  of  soda  Holution.  the  mmU 
can  be  driven  out  through  diaphragms,  leaving  the  Hilicic  acid 
behind,  for  use  up  to  10%  concentration  for  therapeutic  purpose*, 
and,  when  stronger,  polymerizing  into  a  pure  Silica  (Jel  of  great 
adsorbent  value  in  the  treatment  of  wounds. 

Oil  drops  can  be  driven  out  of  watery  emulsions,  and  oil  can  be 
purified  from  moisture ;  tannin  can  be  driven  rapidly  into  hidea, 
from  weak  tanning  liquors. 

Anti-diphtheritic  serum  is  obtained  free  from  the  numerotia 
objectionable  substances  naturally  present  in  horse-senira,  and 
6 — 10  times  concentrated  ;  and,  from  glue,  gelatine  is  prepared 
so  pure  and  free  from  all  bacterial  food-material  that  its  solutiont 
keep  unchanged  for  months. 

§860.  Electrolytic  Polarization.  In  the  gaa  voltameter  we 
started  with  a  couple  of  platinum  plates  immersed  in  weak  acid, 
electrolysis  covered  one  with  oxygen  and  the  other  with  hydrogen, 
and  we  have  now  virtually  a  plate  of  oxygen  and  a  plate  of  hydrogen. 
Chemical  action  being  really  electrical,  very  different  E.M.F.'s  arise 
at  these  plates,  and  either  assist  or  impede  the  passage  of  current ; 
together  they  impede  it. 

Actually,  a  11 -volt  Daniell  cell  is  quite  unable  to  drive  a  current 
through  the  gas  voltmeter  ;  even  a  2-volt  accumulator  raises  scarcely 
a  sign  of  gas  on  the  electrodes,  with  two  in  series  decomposition  goes 
on  merrily.  . 

But  if,  after  trying  only  the  Daniell,  we  switch  it  out  of  circuit. 
by  a  two-way  switch,  and  bring  in  a  galvanometer,  this  shows  a 
strong  deflection  lasting  several  seconds.  The  plates  were  polarised, 
the  cell  had  developed  a  polarity,  and  the  current  was  sent  by  the 
back  E.M.F.  of  Electrolytic  PolarizaUon  as  long  as  supplies  of  ions 
lasted  round  the  plates. 

§861.  Polarization  E.M.F.  Solution  Pressure.  Whatever  plate 
is  dipping  into  a  solution,  there  will  arise  a  iK>lari7.ation  differrnce 
of  potential  between  it  and  the  solution.  This  is  expUmed  on  the 
Theorv  of  Solution  Pressure  in  this  way  :  There  arealre^dy  in 
solution  a  number  of  free  ions,  of  hydrogen  in  the  fofegoing 


700 


ELECTRICITY 


[§861 


of  zinc  in  zinc-sulphate  solution,  etc.,  and  these  of  course  exert  an 
Osmotic  Pressure,  and  endeavour  to  drive  some  of  their  number 
into  the  plate.  As  they  carry  +  charges,  that  would  raise  the  poten- 
tial of  the  plate  above  the  solution. 

But  the  plate  retaliates  by  sending  out  +  ions  of  its  own  sub- 
stance into  the  solution — evaporates  into  the  liquid,  so  to  speak — 
up  to  a  definite  '  Solution  Pressure,'  like  a  saturated  vapour  pressure, 
and  that  has  the  opposite  effect  on  the  P.D. 

These  Solution  Pressures  differ  enormously  for  different  substances, 

and  the  nett  result  is  that  some  metals  stand  above  the  potential 

of  the  solution — being  those  that  retaliate  feebly,  with  only  a  low 

solution  pressure,  e.g.  H  0-25  volt,  Cu  0-60,  Hg  0-99, 

I    I  while  others  arrive  at  equilibrium  below  the  solution, 

m  e.g.  Zn  at  -  0-51  volt,  Cd  -  0-19,  Fe  -  0-06. 

■ These  figures  are  obtained  by  aid  of  the  Dropping 

W  Electrode,  which  is  strongly  reminiscent  of  the  Sand 

I  Dropper  of  §  715.     A  fine  stream  of  mercury  from  a 

I     (.  burette.  Fig.  387,  breaks  up  into  drops  as  it  enters 

I I  a  mercurous  sulphate   solution  over- spreading  mer- 

Tj|_  \T  cury.     The  newly  formed  mercury  surface  of  the  jet 

expands  and  breaks  away  much  faster  than  the 
necessary  ions  can  be  supplied,  and  consequently 
it  remains  at  the  same  potential  as  the  solution 
(with  which  it  hasn't  had  time  to  quarrel,  so  to 
speak).  Hence  the  P.D.  between  mercury  in  burette, 
and  mercury  in  pool,  gives  really  the  P.D.  between 
the  latter  and  the  solution,  0-99  volt ;  whence  other 
P.D.'s  by  building  up  voltaic  cells  against  mercury. 

§  862.  The  measurement  of  the  Potential  Difference  between  an 
electrode  and  a  solution  has  come  into  prominence  as  the  means  of 
ascertaining,  with  great  exactness,  Hydrogen-ion  Concentration, 
which  determines  the  '  acidity  '  of  very  feebly  acid  solutions,  a  matter 
of  amazing  importance  in  many  vital  pro- 
cesses, and  one  of  which  you  will  hear 
a  great  deal  in  your  later  work. 

From  the  value  of  the  slight  conduc- 
tivity which  persists  in  the  purest  water 
ever  obtained,  by  distillation  and  freez- 
ing, 0-04  micromho,  and  the  mobilities  of 
the  H  and  HO  ions  given  in  §  855,  it  is 
deduced  that  at  about  25°  C.  1  litre  of 
pure  water  contains  one  ten-millionth, 
10~^,  gm.  of  free  Hydrogen  ions. 

Ten  times  as  many,  10"^,  mean  acidity  ; 
10"*  means  alkalinity,  because  always  H 
ions  X  HO  ions  will  =  10-^*.  Conveni- 
ently, the  logarithm  only  is  quoted,  with- 
out its  minus  sign,  and  is  called  ^H. 
Thus^H  of  Neutrality  is  7  ;  of  Acids,  below  7  ;  of  AlkaHes,  above  7. 


Fig.  387. 


Fig.  388. 


§  863]  ELECTRICITY   IN   LIQUIDS  701 

A  curious  voltaic  cell  is  made  up,  Fig.  388  :  on  the  right  is  a 
Hydrogen  Electrode,  a  platinum-blacked  platinum  pUt«  pant 
which  pure  hydrogen  has  been  flowed  for  20  min.,  on  the  left  in  a 
Calomel  Electrode,  wherein  mercury,  overspread  with  calomel. 
lies  at  the  bottom  of  a  tube  full  of  saturated  (or  else  decinormal) 
KCl  solution  saturated  with  calomel  (very  feebly  soluble).  The 
hydrogen  electrode  behaves  as  an  actual  plate  of  hydn>gen.  the 
calomel  electrode  has  been  found,  as  in  §861,  to  have  an  K.M.F. 
0-25  volt  in  saturated  KCl,  or  0-335  in  .V/lO-KCl,  against  the 
normal  hydrogen  electrode  pH  =  \,  which  has  an  E.M.F.  0-or>8 
volt. 

The  beaker  is  filled  with  the  liquid  under  test,  and  the  E.M.F.  of 
the  whole  cell  is  measured  by  the  Potentiometer,  then 

^H  =  (E.M.F.  measured  —  calomel  E.M.F.)/0058 

§  863.  Concentration  cells.  In  §  861  it  was  explained  that  metal« 
drive  out  +  ions  of  themselves,  into  the  liquid,  raising  it«  potential, 
until  a  balance  is  struck  between  the  '  solution  pressure  '  of  the 
metal  and  the  osmotic  pressure  of  its  ions  already  in  solution. 
This  balance  is  reached  sooner  if  these  latter  are  numerous,  i.e.  there 
results  a  less  potential  difference  between  a  metal  and  a  strong 
solution  of  one  of  its  own  salts,  than  with  a  weak  solution. 

A  Daniell  cell  put  by  and  forgotten  provides  an  excellent  example  : 
a  sheet  of  copper  stands  with  its  foot  in  copper-sulphate  cryHtalu, 
while  at  the  top  the  solution  has  grown  quite  weak.  The  good- 
conducting  copper  is  at  the  same  potential  throughout  (but  for 
some  minute  fraction  of  a  microvolt),  consequently  the  weak 
solution  is  at  a  higher  +  potential  than  the  strong,  and  this  means 
an  E.M.F.  driving  the  copper  ions  out  of  the  weak  into  the  stronger 
solution,  and  there  dumping  them  on  the  copper  plate,  raising  its 
potential  throughout,  and  enabling  it  to  drive  more  ions  out  into 
the  weak  solution  :   a  slow  current  is  always  flowing  up  the  metal. 

Thus  the  plate  is  corroded  away  at  the  top  and  grows  thicker 
at  the  bottom.  The  porous  pot,  left  with  its  zinc  inside,  actii 
likewise  as  a  conductor  at  uniform  potential,  and  crystals  of  copper 
slowly  grow  outside  its  lower  edge. 

A  stick  of  tin,  in  acidulated  stannous  chloride  solution,  strong  and 
weak,  shows  the  same  action. 

You  see,  the  metal  contrives  to  fall ;  and  if  you  think  about  it. 
Gravity  is  really  the  motive  force  in  the  cell. 

A  tiny  beaker  of  mercury  stands  on  the  bottom  of  a  large  jar, 
and  another  is  hung  near  the  top.  Insulated  wirea  dip  into  each, 
and  are  brought  out  to  a  sensitive  galvanometer. 

Weak  sulphuric  acid,  which  has  been  shaken  with  the  feebly 
soluble  mercurous  sulphate,  so  as  to  saturate  it,  is  pourwl  in  to  fill 
the  jar,  drowning  both  beakers.  

A  current  runs  perpetually  through  the  wire  from  lower  mercury 
to  upper  :  why  ? 


702 


ELECTRICITY 


[§863 


Because  by  going  out  into  solution  from  the  upper  beaker, 
positively  charged  mercury  ions  will  ultimately  be  able  to  fall 
into  the  lower  beaker  ;  in  the  end,  the  upper  will  be  emptied  into 
the  lower,  the  falling  weight  drives  the  cell. 

But  how  does  the  upper  mercury  know  it  is  on  top  ? 

Gravity  pulls  all  the  time  on  everything,  everything  is  falling. 
As  long  as  particles  are  small  enough,  molecular  movement  prevents 
them  all  precipitating  into  a  solid  ;  as  Brownian  movement  it  keeps 
fine  mud  suspended,  but  the  mud  grows  denser  downwards,  as  does 
the  atmosphere,  and  so  do  the  heavy  molecules  of  every  solution — 
not  much,  no  need  to  shake  an  ordinary  bottle — and  so  do  the 
mercury  ions  :  they  are  sparser  round  the  upper  beaker ;  it  is 
Concentration  Cell. 


1 


§  864.  The  Capillary  Electrometer  puts  polarization  to  practical 
use.  In  Fig.  389  a  slightly  sloping  capillary  tube  joins  two  little 
reservoirs,  the  lower  end  contains  mercury,  which  also  rises  up  the 
capillary,  but  not  to  the  full  level,  for  it 
is  held  down,  as  in  §  345,  by  the  surface 
tension  in  the  meniscus  separating  it  from 
the  weak  sulphuric  acid  in  the  rest  of  the 
tube  and  second  reservoir.  At  the  bottom 
of  this  latter  is  a  broad  pool  of  mercury ; 
wires  are  connected  to  both  lots  of  mercury. 
When  a  fraction  of  a  volt  is  applied 
between  these  wires,  the  pool  being  positive, 
the  meniscus  surface  of  course  polarizes  and  stops  any  current. 
But  the  polarization  increases  the  surface  tension,  and  drives  the 
mercury  farther  down  the  capillary  tube,  past  a  scale  which  can 
be  graduated  by  preliminary  trials  with  known  fractions  of  a  volt. 
The  instrument  is  a  sensitive  electrometer  or  voltmeter,  for 
anything  below  0-9  volt,  it  takes  no  current,  it  is  easy  to  make, 
and  manage,  and  it  finds  much  favour  in  physiological  work.  At3 
higher  voltages  hydrogen  bubbles  form. 


Fig.  389. 


§  865.  A  curiously  exaggerated  sort  of  polarization  occurs  witl 
aluminium  electrodes,  and  is  utilized  for  '  rectifying  '  alternating 
current,  i.e.  stopping  out  the  back  flow  and  transmitting  only  th( 
direct  rushes  of  current,  so  that,  e.g.,  accumulators  may  be  charge( 
from  A.C.  mains.  The  Electrolytic  Rectifier  is  simply  a  Lead  plate 
and  an  Aluminium  rod  in  a  jar  of  solution  of  borax,  or  of  ammonium 
phosphate  ;  when  the  aluminium  is  cathode  the  1  volt  back  E.M.F. 
of  hydrogen  upon  it  is  easily  overcome,  but  when  reversed  current 
makes  it  anode,  it  is  instantly  overspread  by  a  non-conducting 
oxide  film,  of  exceeding  thinness,  but  quite  capable  of  preventing 
current  being  driven  back  by  ordinary  domestic  voltages. 

The  same  exceedingly  thin  insulating  film  serves  in  the 
Electrolytic  Condenser.  Aluminium  plates  are  exposed  as  anodes  for 
some  hours,  in  ammonium  borate  solution,  to  100  volts  ;  they  then 


§  866]  ELECTRICITY   IN  LIQUIDS  703 

serve  as  condenser  electrodes  against  the  solution  up  to  90  voltn 
800  cycles  A.C.,  a  couple  of  o-in.-square  plates  having  the  enormouil 
capacity  of  6  microfarads,  incomparably  the  cheapest  and  moiit 
compact  form  of  condenser. 

§866.  If  we  could  accumulate  much  larger  quantities  of  oxygen 
and  hydrogen  on  the  plates  of  our  electrolytic  cell  in  §  HIM)*  the 
polarization  E.M.F.  would  drive  a  current  for  us  for  quite  a  useful 
length  of  time.  Something  towards  this  may  be  done  by  coating 
the  electrodes  with  platinum-black,  which  has  a  great  'jKiwer  of 
occluding  gases,  but  nothing  of  practical  yalue.  To  store  adequate 
amounts  we  must  get  them  into  some  easily  decompo«*able  chemical 
combinations  ;  and  the  many  oxides  of  lead  have  been  in  sen'icc 
since  1860. 

Let  us  take  a  '  Main  and  1 1th  '  street-car  at  Niagara  to  the  T.S.L. 
Plant,  and  watch  the  mushroom  growth  of  6-yolt  automobile- 
engine-starter  batteries  for  half  the  States;  Secondary  Batteries, 
Storage  Batteries,  Accumulators,  call  them  what  you  will. 

The  automatic  casting-machine,  one  of  four  which  together  would 
supply  the  continent,  is  ejecting  thin  flat  skeleton  grids  which  look 
like  plans  of  an  American  city  ;  the  '  blocks,*  holes  about  }  x  ^  in., 
the  streets,  thin  feather-edged  strips  of  hard  lead  :  in  foum  they 
leave  the  mould,  to  be  cut  up  speedily  and  trimmed  to  shape. 

In  the  next  shop  a  burly  black,  protected,  like  everybody  else, 
by  an  anti-lead-dust  mask,  is  tipping  a  tawny  gravel  of  litharge  and 
sulphuric  acid  out  of  a  concrete-mixer;  and  with  this,  men  using 
trowels  and  rolls  and  hand  rammers  pack  the  grids,  now  travelling 
on  under  compressing  rolls,  which  clinch  down  the  feather-edget 
of  the  strips,  and  so  '  pocket '  the  '  paste,'  and  then  through  a  steam 
drying  box.  Double-crossed-belts  carry  the  already  hard  plates 
past  trimming  knives,  and  deliver  them  for  inspection  and  assembly. 

Left  and  right  they  are  packed,  separateti  by  thin  compressed 
veneers  of  Oregon  cypress,  the  intense  sweet  aroma  of  which 
spreads  far  from  the  vats  in  which  the  little  s(}uares  are  being 
boiled  in  caustic  soda,  to  rid  them  of  obstructing  resin,  and  make 
them  thoroughly  porous  ;  prior  to  being  '  candled,*  like  red  glass, 
for  possible  faults,  and  trimmed  to  exact  size  by  chopping  knives. 

The  assembled  blocks  stand  on  the  travelling  belt,  lead  connecting 
strips  and  stubs  are  laid  on,  the  hissing  blue  tongue  of  an  oxy- 
acetylene  blowpipe  licks  them  back  and  forth,  touches  a  nul  of 
lead,  which,  instantly  turning  to  gum,  sticks  all  secure,  and  the 
trickiest  job  of  all  has  been  done  while  you  read  this  sentence. 

Now  they  are  dropped  by  threes  into  the  familiar  black  boxes. 
These  are  compounded  of  bitumen  and  ground-up  old  tyre  canvas, 
kneaded  into  a  black  dough,  kept  in  an  oven,  and  weightnl  out  in 
lumps  which  are  dumped  into  beautifully  made  and  noIishe<l  steel 
moulds  in  hydraulic  presses.  Ten  minutes'  steam-cooking  in  these, 
and  the  press  is  lowered,  the  mould  knocked  open,  and  the  glossy 
black  three-celled  case  has  a  few  minutes  to  cool,  on  the  conveyor, 


704  ELECTRICITY  [§  866 

before  the  tester  fits  it  with  loose  electrodes  inside  and  out,  switches 
on  25,000  volts — and  an  instant  puff  of  smoke  sends  just  one  now 
and  again  back  to  the  melting-oven. 

The  lids  are  tapped  on,  outside  connectors  are  dropped  into  place, 
projecting  stubs  ripped  off,  collars  put  on  terminals,  and  another 
deftly-handled  blowpipe  finishes  off  all  the  metal  work  you  see  on 
the  top  of  the  battery,  in  a  matter  of  seconds. 

Without  a  spilt  drop,  the  lids  are  run  in  twice  with  bitumen 
coming  down  a  flexible  copper  pipe,  kept  suitably  hot  by  1000 
amperes  in  its  wall.  Air  blasts  cool  them,  triple  automatic  nozzles 
deliver  acid  from  the  pump,  s.g.  1-2  ;  those  cells  in  this  case  whose 
sponsors  intend  to  extract  extra  money  by  smart  salesmanship 
get  a  few  seconds'  beauty  treatment,  and  then  all  join  the  serried 
battalions  on  the  charging-room  floor. 

Here  they  imbibe  their  long  first  charge,  at  a  cost  of  nearly  half- 
a-cent  apiece,  and  next  day  are  pushed  along  into  the  freight-car, 
to  be  stacked  layer  on  layer,  taped  tight  to  withstand  the  shocks 
of  the  shunting-yards,  and  despatched  east  and  west  to  save  the 
crank  arms  of  all  good  citizens  of  the  U.S. 

These  are  cells  in  which  structural  strength  is  subordinated  to 
enormous  discharge  rate,  up  to  120  amperes  or  more  for  brief  spells 
of  '  starting,'  their  plates  made  thin  that  the  ions  may  have  ready 
access.  Many  will  come  back  as  junk  in  a  couple  of  years,  to  be 
pitched  without  ceremony  into  the  melting-pot,  where  what  doesn't 
melt,  or  fume  up  chimney,  is  raked  out  to  feed  the  fire. 

Massive  compared  with  these  are  the  other  batteries  made  here, 
destined  to  carry  the  name  of  our  genial  companion  throughout 
the  length  and  breadth  of  America,  in  the  W.  L.  Bliss  lighting- 
system  of  its  railroad  cars.  Here  the  negative  plates  are  packed 
with  a  black  powder  made  by  rolling  half-pound  nubbles  of  lead 
in  a  barrel  in  hot  air,  a  process  not  recognized  by  the  chemistry 
books,  but  resulting  in  the  accretion  of  7%  of  oxygen,  very  nearly 
PbO  ;  while  the  positive  plates  are  of  solid  lead,  which,  by  a  sort 
of  knuckling  action  between  curved  steel  dies,  has  been  gradually 
squeezed  into  thick  slabs,  lined  on  both  sides  by  grooves  much  deeper 
than  wide,  so  that  the  actual  surface  is  many  times  increased. 
These  are  subsequently  '  formed,'  in  the  original  Plante  mamier, 
by  repeated  charging  and  discharge,  until  the  surface  is  deeply 
rusted  into  a  thick  deposit  of  active  peroxide. 

§  867.  The  electrolyte  is  Sulphuric  Acid  of  best  conductivity 
strength,  about  1  in  4,  or  sp.  gr.  1-20,  and  when  charged  the  positive 
plate  is  covered  with  a  dark-grey  electrolytic  hyper-oxide  which 
analyses  out  to  (3Pb02)0,  while  the  negative  plates  are  grey  spongy 
metallic  lead. 

During  discharge 


(3Pb02)0    H2ISO4    Pb 


§  S68J  ELECTRICITY   IN   LIQUIDS  705 

the  positive  plate  reverts  to  bright  brown  PbO,  ;  the  acid  decreaaet 
m  specific  gravity,  on  account  of  the  using  up  of  H-SO^  and  produc- 
tion  of  H2O,  usually  by  about  002  e.g.,  so  that  little  hydrometera 
are  useful  in  testing  the  electrical  condition  of  your  batt«TV 
(provided  you  first  find  out  the  s.g.  of  the  acid  when  fully  charK«l)'; 
and  the  spongy  lead  is  converted  without  change  of  apjwaranct- 
into  a  colloidal  sulphate,  which  only  consolidates  into  ini»olubU' 
white  lead  sulphate  as  the  result  of  long  months  of  neglect  or 
misuse — when  the  '  sulphated  '  plate  must  be  nursed  back  to  capacity 
by  long-continued  slow  charging. 

During  charge  >. 

1+  + 

(3PbOo)    OlHg    SO^Pb 

the  current  is  forced  back  by  attaching  the  -f  wire  from  the  d>iuimo 
to  the  +  terminal,  the  acid  regains  its  strength,  the  negative  plate 
is  invisibly  cleared,  the  brown  positive  plate  bhickens,  and  the  end 
of  charge  is  announced  by  copious  '  gassing,'  which  is  simply  the 
production  of  oxygen  and  hydrogen  at  electrodes  impervious  to 
any  further  action,  and  does  no  harm,  except  to  use  up  H,(),  ami 
dry  out  your  plates  unnecessarily  ;  so  that  you  must  '  top  up  * 
with  distilled  water  more  often — and  do  this  only  before  charging, 
which  will  mix  it  well. 

There  is  an  older  theory,  which  insists  that  the  PbO,  itself 
decomposes — as  it  undoubtedly  does  if  you  let  the  cell  flounder 
unsteadily  along  after  its  voltage  has  collapse<l  by  10% — and 
probably  there  is  truth  in  both,  for  in  point  of  fact  few  cnemical 
reactions  can  abide  the  straitness  of  the  books,  but  1  have  avoi<ied 
an  explanation  which  makes  the  charged  positive  plate  brown, 
whereas  every  boy  with  a  toy  accumulator  knows  that  colour 
perfectly  well  as  a  sign  of  emptiness. 

§868.  Lead  Accumulators,  like  other  fat  heavy'  people,  are  of 
delicate  constitution.  They  are  desjx^rately  susceptible  to  the 
effects  of  metallic  poisons  ;  traces  of  iron  rust,  or  of  venligris  from 
corroded  brass  terminals,  cause  sulphation  of  one  or  other  plate 
at  every  charging,  and  progressive  loss  of  capacity  ;  the  acitl  nniiit 
be  free  from  arsenic  ;  tap- water  is  inadmissible  for  *  topping  up, 
and  even  rain-water  contains  too  much  ammonia  in  towns ;  while 
salt  water  liberates  chlorine,  which  has  slain  the  crew  of  more  than 
one  submarine.  The  froth  developed  in  celluloid-caseil  accumulatom 
is  merely  messy,  except  that  it  gets  at  the  terminals,  which  rwjuir© 
protective  greasing. 

Neither  charge  nor  discharge  may  exceed  the  rate*  preacnbcd 
by  the  makers  of  the  battery.  The  minimum  penalty  is  loiw  of 
capacity  :  for  the  active  subsUnce  of  the  jKJsitive  plate  undergoes 
considerable  change  in  bulk,  and  if  this  occurs  too  rapidly  the  pUte 
buckles  (hence  there  is  always  one  less  -f  pi«te  than       ,  no  that 

AA 


706  ELECTRICITY  [§  868 

both  sides  get  equal  use)  and  crumbles  to  pieces,  and  bits  fall  out 
and  cause  short  circuits,  and  '  washing-out '  merely  fetches  out  more 
of  them  ;  new  positives  are  the  only  remedy. 

'  Block,'  '  ironclad  '  (really  ebonite),  etc.,  constructions  are  more 
or  less  successful  attempts  to  cope  with  this  disintegration  difl&culty  ; 
but  their  positive  lugs  crack  through.  Negative  brass  terminals 
corrode  solid  with  unnoticed  zinc  sulphate. 

'  Unspillable  '  accumulators  have  their  acid  either  stuffed  with 
glass  -wool,  or  '  gelled  '  with  silica,  from  silicate  of  soda  admixture,  w 

There  is  no  way  but  the  hydrometer  of  gauging  the  charge  re- « 
maining  in  your  cell,  and  then  you  must  have  checked  the  density 
of  the  acid  when  full,  and  cold.  The  voltmeter  is  no  guide,  for  it 
will  show  full  2  volts  per  cell  almost  until  the  end  ;  this  steadiness 
is  a  most  excellent  property  of  lead  accumulators.  Strictly  entre 
nous,  attach  half-a-yard  of  wire  to  one  terminal,  and  flick  the  end 
as  quick  as  you  can,  once,  across  the  other  terminal ;  a  fiery  snap 
back  at  you  means  all's  well,  reject  a  weak  sparker. 

Be  quick,  or  you  will  burn  your  fingers,  and  serve  you  right,  for 
spoiling  the  cell ;  for  short-circuiting,  or  anything  approaching  it, 
however  accidental,  is  speedy  death  to  most  accumulators  ;  their 
internal  resistance  is  almost  vanishingly  small  (Ex.  2,  §  779),  and 
gives  them  no  protection. 

In  charging,  2-5  volts  per  cell  is  necessary ;  to  charge  from  the 
mains,  a  Resistance  must  be  employed  to  keep  down  the  current, 
lamps  for  small  rates  (amps.  =  watts  marked  -^  voltage),  a  small 
domestic  heater  for  larger  cells.  From  A.C.  mains  a  Rectifier  of 
some  description  must  be  used  as  well.  In  either  case  make  sure 
that  the  current  is  going  in  to  the  red  +  terminal ;  by  compass 
and  Ampere's  rule,  or  by  pole-paper.  Copious  gassing,  and  a  voltage 
well  above  2  with  charging  current  off,  indicate  full  charge. 

Lead  accumulators  want  charging  up  every  month  or  so,  in  use  or 
not,  or  they  sulphate  and  lose  capacity. 

A  wireless  or  car  battery  that  is  perpetually  hungry  wants  scrap-  • 
ping. 

The  efficiency  of  a  battery,  whether  merely  as  ampere-hours  of 
discharge  ^  ditto  of  charge,  or  the  more  scientific  watts  output/ 
watts  input,  is  a  serious  question  for  the  submarine  commander 
or  the  central -station  engineer  ;  it  may  well  be  90%  or  80%  respec-^ 
tively,  but  not  on  heavy  discharge. 

§869.  In  spite  of  its  fragility  and  corrosiveness,  the  lead-acid 
accumulator  has  outlived  many  competitive  Alkaline  Accumulators, 
which  have  been  troubled  by  fanciful  variations  of  voltage  during 
charge  and  discharge.  A  successful  Nickel-Cadmium-alkali  cell, 
in  which  this  difficulty  has  been  overcome,  is,  however,  now  being 
produced  by  Batteries  Ltd.,  Hunt  End  Works,  Redditch,  as  th© 
'  Nife  '  Battery. 

This  cell  gives  a  steady  1-3  volts  only,  instead  of  2,  so  that  more 
cells  are  needed  in  a  battery,  but  this  may  altogether  be  only  half 


§  870]  ELECTRICITY  IN   LIQUIDS  707 

the  weight  of  a  lead  one,  and  the  cell8  come  up  Miniling  under  the 
worst  treatment  of  every  description. 

The  cell  containers  are  of  rust -proof  steel.  welde<l  up  completHy, 
and  are  supported  on  ebonite  insulators  in  the  hanlwood  cratni. 
The  electrolyte  is  KOH  solution  of  sp.  gr.  1I«,  which  doen  not 
vary  in  action,  and  does  not  corrode  any  part  of  the  battery.  The 
plates  are  thin  flat '  biscuits  '  in  which  the  active  material  in  packe<l 
in  cellules  between  thin  perforated  steel  plates  ;  none  drops  out,  and 
the  plates,  which  are  separated  by  thin  ebonite  rods,  do  not  buckle. 

The  positive  plate  is  packed  with  nickel  hydroxide,  and  the 
negative  with  cadmium  hydroxide,  and  the  action  is,  briefly  : — 

2Ni(OH)3  +  Cd  ^ >  2Ni(0H),  -f  Cd(OH),. 

Gas  is  evolved  only  at  the  end  of  charge,  when  the  usual  mixture 
comes  off,  and  of  course  involves  occasional  topping  up  with  din- 
tilled  water. 

No  self -discharge  whatever  takes  place,  no  matter  how  long  the 
battery  is  left,  nor  in  what  condition  ;  the  cell  works  well  clown 
to  —  15°  C,  and  freezing  does  no  permanent  harm.  The  batten- 
can  be  charged  in  an  hour,  and  heavy  rates  of  discharge  do  not 
injure  it,  a  60-ampere-hour  12-volt  battery  can  give  quarter-minute 
bursts  up  to  3  h.p.  The  original  Diesel -engined  railroad  car  ha« 
been  started,  to  date,  half  a  milHon  times,  by  its  original  batten*. 

The  battery  is  a  great  relief  after  the  lead  accumulator,  amf  an 
altogether  admirable  proposition  for  Cautery  work,  etc. 

§  870.  Primary  Batteries  produce  currents  as  soon  as  they  are 
put  together,  without  any  previous  '  charging  with  electricity.* 

The  earliest  was  the  invention  of  V'olta  (whence  primary  Iwtteriea 
are  often  called  Voltaic  Cells) ;  it  consisted  of  a  plate  of  zinc  and 
a  plate  of  copper  dipping  in  salt  water  or  weak  acid.  The  zinc 
dissolves,  tiny  bubbles  of  hydrogen  overspread  the  copper,  and  a 
charm  compass  shows  the  passage  of  a  current  from  copper  to  uno 
along  a  wire  joining  them  outside  the  liquid. 

Alessandro  Volta  became  Professor  of  Natural  Philooophy  at 
Pavia  in  1774,  and  maintained,  in  opposition  to  Luigi  Cialvani, 
Professor  of  Anatomy  at  Bologna,  1702,  that  moisture  only  waa 
necessary  to  excite  an  electrical  action  between  metals,  ami  not  the 
vitality  of  frog's  legs — you  will  repeat  an  experiment  rather  like 
Galvani's  in  your  physiology  course. 

Volta  built  a  Pile  of  many  paired  discs-in-contact  of  diaatmilar 
metals,  separated  by  moistened  cloth,  and  obtained  the  <>f^ 
electric  attractions  and  repulsions  from  its  entls  (he  is  alao  credited 
with  the  invention  of  the  Electrophorus).  These  discoveriee  mu»t 
have  been  surprising,  seeing  that  moisture  had  been  the  great  foe 
of  electrification. 

How  the  current  arises  you  see  from  §«61  :  on  account  of  the 
driving  out  of  ions,  described  there,  the  cop|x»r  assumes  a  jjotenlial 
0-6  volt  above  that  of  the  liquid,  and  the  zinc  0-5  below ;  coiwequenUy 


708 


ELECTRICITY 


[§870 


1-1  volts  is  available  to  drive  current  from  copper  to  zinc,  along  a 
wire  or  any  other  metallic  contact.  The  copper  therefore  provides 
the  +  pole  of  the  cell,  whereas  the  zinc  is  the  more  electropositive 
metal.  Signs  are  always  a  nuisance,  and  you  can  get  badly  tangled 
up  in  voltaic  cells ;  much  the  best  place  to  think  it  out  in  is  the 
diving-tank. 

The  numerous  zinc  ions  meet  and  combine  with  the  negative 
'  sulphions  '  or  '  chlor-ions  '  of  the  acid,  leaving  the  liquid  positively 
charged  with  superabundant  H  ions,  which  give  up  their  charges 
to  the  copper  electrode,  and  form  gas  on  its  surface. 

But  this  accumulation  of  hydrogen  '  polarizes  '  the  electrode, 
and  the  current  dwindles  to  practically  nothing.  The  hydrogen 
must  be  got  rid  of  somehow ;  merely  scrubbing  the  copper  with  a 
wire  brush  has  some  effect,  but  voltaic  cells  were  not  a  success  until 
chemical  means  were  employed  to  remove  the  hydrogen. 

You  see  that  this  use  of  the  word  '  polarize  '  is  diametrically 
wrong,  for  the  polarity  of  the  cell  is  destroyed ;  it  should  be 
'  depolarize,'  and  the  '  depolarizers  '  to  be  described  directly  should 
be  '  repolarizers.'  However,  it  has  the  sanction  of  long  misuse ; 
don't  let  it  worry  you,  this  is  England,  and  nobody  cares  how  it  is 
used,  or  spelt. 

§  871.  Daniel!  surrounded  the  copper  with  a  blue  solution  of  its 
own  sulphate,  kept  from  mixing  with  the  weak  sulphuric  acid 
round  the  zinc  by  means  of  a  '  porous  pot.'  The  H  ions  diffusing 
through  the  pot  possibly  carry  most  of  the  current,  but  ultimately 
appropriate  the  sulphions,  and  leave  their  charges  to  the  copper 
ions,  which  deposit  them,  and  themselves,  on  the  cathode.  Here, 
then,  is  a  zinc-copper  cell  that  never  gets  choked  with  hydrogen. 


Zn 


SO4 


H. 


■>S04 


Cu  — >-  Copper,  + 


The  Daniell  Cell  has  taken  a  variety  of  forms,  the  laboratory  It 
pattern,  !Fig.  390,  left,  commonly  consists  of  a  jam-pot,  a  copper 
calorimeter  past  mending,  the  porous  pot,  and  a  spare  bell-battery 
zinc.     In  a  '  gravity  '  pattern.  Fig.  390,  right,  the  copper  plate  lay 

at  the  bottom  of  a  deep  dish  under 
a  layer  of  blue  crystals  (which  of 
course  got  used  up) ;  weak  sulphuric 
acid  was  filled  in,  and  the  zinc  plate 
supported  horizontally  near  the  top  ; 
the  slowness  of  liquid  diffusion  pre- 
vented the  copper  solution  reaching 
the  zinc,  so  long  as  the  cell  was  kept 
in  action,  as  it  was  in  the  early  days 
of  telegraphy,  when  messages  were 
merely  breaks  in  the  current. 
There  is  no  need  to  cough  over  the  hydrogen  rising  from  the 


In  liiHaSO^orZnSOi^ 

I  POROUS  POT 

|-|-CuLTvCaSO^+ 


FiG.  390. 


§  871]  ELECTRICITY   IN  LIQUIDS  709 

*  local  action  '  of  the  acid  on  the  crude  zinc,  for  the  cell  works  per- 
fectly well  on  neutral  zinc-sulphate  solution 


>Jcu 


Zn  — >  SoJZn  — >  SoJcu  — y  Copper  -h 


an  action  that  stultifies  the  popular  aAscrtion  that  the  *  zinc  naiuraUy 
dissolves  in  the  acid  and  so  drives  the  cell,*  and  nccc»MitAt<«  the 
fuller  explanation  on  the  lines  given  in  §  H«l . 

Evidently  this  cell  is  reversihle,  a  current  forced  in  at  the  copper 
end  gradually  removes  the  deposited  copi)er  and  redepodtn  the 
dissolved  zinc,  but  this  comes  down  in  incoherent  crystaui,  and  no 
attempt  has  been  made  to  utilize  the  cell  as  an  accumuUtor. 

If  potassium  cyanide  be  used  in  place  of  the  blue  copper  Hulphate, 
the  cell  spontaneously  works  backwards,  with  K.M.t.  half  a  volt. 
The  copper  dissolves  to  colourless  cupricyanide,  in  which  it  i» 
anionic  ;  i.e.  there  never  are  any  Cu^+  ions  in  the  liquid,  the  copper 
finds  it  an  *  ionic  vacuum,*  and  goes  on  *  evaporating  *  f  chamt 
into  it  faster  than  even  the  zinc  can,  and  so  succeedji  in  getting 
below  the  potential  of  the  zinc. 

This  action  is  worth  trying,  and  there  is  no  need  to  riiik  poiiionin^. 
for  you  can  at  the  same  time  satisfy  yourself  that  a  Voltaic  Oil  is 
merely  two  metals  and  some  damp.  Attach  a  strip  of  zinc  to  one 
wire  from  an  ordinary  sensitive  g^vanomet«r,  put  one  drop  of 
water  and  one  of  CUSO4,  in  contact  with  it,  on  paper,  touch  the 
zinc  and  the  other  copper  wire  in  them,  and  there  is  your  IHuiiell 
cell.  Now  put  a  drop  of  ZnSO^  and  one  of  KCN,  with  one  of 
some  sodium  salt,  to  keep  the  peace,  between  them  (eliie  stno 
cyanide  precipitates) ;  try  again,  and  the  needle  swingn  the  other 
way.     This  size  cell  works  on  a  potentiometer  admirably. 

Like  its  ally,  the  Tangent  Galvanometer,  the  Daniell  (ell  repays 
study  in  the  junior  laboratory,  though  nowadays  it  never  gom 
outside  it,  for  its  internal  resistance  is  seldom  l^ns  than  an  ohm 
(the  solutions  are  poorly  conductive,  see  Table  §  777)  and  iU 
E.M.F.  is   a  volt ;   thus  its  steady  currents  have  little  |iowrr. 

Left  standing,  the  blue  solution  soon  diffuses  in.  and,  reduced 
by  the  zinc,  covers  it  completely  with  brown  copper  mud.  when  it 
is  useless.  But  if  you  rub  this  off  under  the  tap,  and  refill  the 
pot  with  water  and  2  or  3  c.c.  of  sulphuric  acid,  you  at  once  have 
a  serviceable  cell  of  E.M.F.  from  1-07  to  MO  volt,  according  to 
concentration  of  solutions. 

With  saturated  sulphate  solutions  at  65*  F.,  ami  pure  mrlaU, 
the  E.M.F.  is  1095  volt,  with  a  very  slight  dependence  on  tem- 
perature, on  account  of  the  greater  solubility  of  the  «ul|>hatni  m 
warmer  water. 

It  was  for  long  the  Standard  of  E.M.F..  but  thi«  wandenng 
propensity  of  the  CuSO^  had  to  be  exorcised  somehow. 


710  ELECTRICITY  [§  872 

§  872.  In  the  Standard  Cadmium  Cell  mercury  replaces  copper, 
and  cadmium  is  used  instead  of  zinc  ;  pairs  of  closely  related  metals, 
but  possessing  the  desired  dissimilarities. 

The  mercurous  sulphate  forms  a  plaster-like  plug  above  the  mer- 
cury in  Fig.  391,  it  is  nearly  insoluble,  but  if  any  does  diffuse  over 
and  reach  the  cadmium  it  can  do  no  harm,  for  the  cadmium  is  already 
mixed  with  mercury  into  a  1/8  amalgam.  Cadmium  sulphate  is 
no  more  soluble  hot  than  cold,  the  solution  filling  the  cell  remains 
of  invariable  strength,  and  the  E.M.F.  is 
1-0188  volts  at  20°  C,  diminishing  only 
1/20,000  per  °C.  warmer. 
_  _  The  cell  made  up  and  sealed  in  an  H 

-_  solri.  tube  can  be  sent  by  post  without  derange- 
ment. Its  internal  resistance  is  about 
650  ohms,  consequently  it  is  useless  except 
with  the  potentiometer.  If  permitted  to 
send  more  than  a  momentary  milliamp. 
it  polarizes,  but  recovers  in  a  minute  or 
two. 

§  873.  The  batteries  that  really  fought  the  good  fight  in  the  days 
before  the  dynamo,  buried  in  their  brown  fumes,  valiantly  running 
arc-lamps  and  everything  else,  were  the  Grove  and  the  Bunsen, 
with  their  platinum  or  carbon  in  strong  nitric  acid  for  '  depolarizer  ' ; 
but  both  they,  and  the  less  obtrusive  though  more  easily  tired 
Bichromate,  have  gone  down  before  the  Accumulator. 

Their  survivor  is  the  Leclanch6,  with  its  infinite  capacity  to  stand 
and  wait,  always  ready  to  yield  moderate  currents,  requiring  only 
a  drink  of  fresh  water  now  and  again  to  make  up  for  evaporation, 
and  once  a  year  or  so  a  pinch  of  the  not  very  corrosive  sal-ammoniac  ; 
excellent  for  bells,  telegraphs  and  telephones,  in  the  absence  of  any 
central  supply. 

The  containing  jar,  Fig.  392,  left,  is  usually  square,  to  stand  close  ; 
and  of  glass,  so  that  the  liquid  level  can  be  seen  ;  the  zinc  is  a  plain 
rod,  and  the  solution  is  a  saturated  one  of  ammonium  chloride 
(sal-ammoniac),  which  has  no  action  whatever  on  zinc  until  the 
circuit  is  closed.  Then  it  attacks  the  zinc  to  form  zinc  chloride, 
which  crystallizes  as  the  double  chloride  of  zinc  and  ammonium, 
sets  free  ammonia,  which  remains  in  solution  and  can  be  smell 
if  the  liquid  is  warmed,  and  produces  the  hydrogen  ions.  Thest 
are  oxidized  by  the  solid  depolarizer,  black  oxide  of  manganese^ 
which,  in  granules,  mixed  with  carbon  to  make  the  mass  conductive, 
is  either  packed  round  the  carbon  positive  plate  in  a  porous  poti 
(Fig.  392,  left),  or  is  strapped  on  to  it  in  baked  blocks,  or  is  incorpor- 
ated in  the  hollow  positive  cylinder  of  baked  carbon  itself. 

Zn  +  4NH4CI  =  ZnCl2.2NH4Cl  +  2NH3  +  2H  reduces  MnOg 

The  black  MnOg  gets  partially  reduced  to  a  mixture  of  lower 
oxides.     It  is  a  rather  slow  oxidizing  agent,  because  it  is  solid,  and 


§875] 


ELECTRICITY   IX  LIQUIDS 


71 


slLT" 


Fio.  Z92. 


the  H  ions  have  to  hunt  round  and  find  it ;  and  this  unfitJi  the 
Leclanche  for  sending  strong  currents. 

The  electromotive  force  is  1-45  volts,  but  it  decreaiiM  with  mtm 
to  less  than  a  volt.     The  interna!  resistance  of  the  pint  size  with 
porous  pot  is  about  an  ohm,  and  this  size  will  very  steadily  maintain 
the  0-2  amp.  for  a  bell,  but  fails  when 
larger  currents  are  demanded. 

Most  so-called  Dry  Cells  are  not -so- 
wet  Leclanche's  :  their  outer  case  of  zinc, 
Fig.  392,  right,  contains  a  cream  of  sal- 
ammoniac,  in  cloth  or  pulp,  and  the  mass 
of  coke-dust  and  MnOo  is  also  consoli- 
dated, round  the  central  carbon  plate  or 
rod,  by  about  l/8th  its  weight  of  sal- 
ammoniac  and  zinc  chloride,  mixed  in. 

Inventors  continue  to  introduce  Cells 
using  CuO,  FeClg,  and  other  depolarizers, 

and  some  are  good  ;  but  they  are  a  little  more  trouble,  or  rather 
more  expensive,  or  something ;  and  altogether  the  demand  for 
Primary  Batteries  of  quality  seems  limited. 

§  874.  Battery  Electromotive  Force  can  sometimes  be  calcuUt«d 
on  these  lines  : — 

1  chemical  combining  weight  in  grammes  (32|)  of  zinc  disnolveit 
to  form  zinc -sulphate  solution  with  the  evolution  of  54,23()  talorieji. 

The  removal  of  1  combining  weight  in  grammes  of  oxygen  from 
nitric  acid  to  form  nitrous  acid  is  found  to  require  91 '»()  calories. 

In  voltaic  cells  the  energy  appears  as  electrical  energj*  instead 
of  heat,  96,500  coulombs  cause  the  deposition  or  solution  of 
1  combining  weight  in  grammes,  §857.  Hence  the  output  of  this 
quantity  of  electricity  from  a  Bunsen  cell  is  accom{>anie<i  by 
(54,230  -  9150)  X  4-2  =  45,080  X  4-2  =  189,000  joules  of  enersy, 
or  1-96  joules  per  coulomb.  And  since  joules  —  coulombs  x  volU 
(§  811),  therefore  1-96  is  the  voltage  of  the  Bunsen  cell. 

As  a  matter  of  fact,  it  is  only  1-82,  and  the  discrepancy  i«  due* 
to  some  of  the  energy  still  going  into  heat,  the  cell  gets  hot  in  action  : 
the  whole  question  is  highly  thermodynamical. 

§  875.  Battery  Resistance.  The  Internal  Resistance,  6,  of  • 
Voltaic  Cell,  is  an  aggregate  of  all  the  various  hindrances  to  its 
action  with  which  it  is  infested  ;  as  well  as  mere  frictional  rej«istancc 
to  the  motion  of  ions  in  the  electrolyte  there  is,  e.g.,  the  difticuliy 
of  the  hydrogen  ions  finding  more  or  less  hidden  soli<l  depolanzer 
quickly  enough  :  this  difficulty  increases  with  the  crowd,  if.  with 
the  current,  so  that  you  need'not  expect  to  find  6  independent  of 
the  current,  as  ohmic  resistance  should  be. 

In  §  783,  if  you  take  care  that  wires  and  galvanometer  are  of 
trifling  resistance— only  a  yard  or  two  of  No.  22  copper— A  is  ail 
the  resistance  there,  and  consequently  the  additional  K  that  hahw 


712 


ELECTRICITY 


[§875 


the  current  =  b,  or  that  which  reduces  it  to  l/nth  is  {n  —  l)b  ; 
Fig.  393,  I,  sufficiently  reproduces  Fig.  348. 

In  II  the  Voltage  V  is  first  measured  '  on  open  circuit,'  i.e.  when 
the  cell  is  not  at  work  :  if  you  want  extreme  accuracy,  and  feel  that 
merely  actuating  the  usual  high-resistance  Voltmeter  is  too  much  of 
a  drain  on  the  cell,  use  the  Potentiometer. 

Then  a  known  R  is  connected  across  the  cell  terminals  as  a  working 
circuit,  and  the  voltage  falls,  to  v. 

It  is  as  if,  in  the  old  coal-burning  days,  all  the  potential  power  of 
a  Channel  steamer  were  displayed  by  all  her  stokers  being  on  deck, 
for  a  breather,  while  in  harbour,  there  being  no  demand  for  steam — 


Fig.  393. 


an  occasional  blast  of  the  whistle  would  fairly  represent  the  volt- 
meter's feed. 

Slowing  out  of  harbour,  two  or  three  must  go  below ;  faster, 
means  that  more  must  descend  to  those  depths  whereunto  we  may 
not  penetrate  ;  the  greater  the  demand  for  steam  (current)  the  less 
stokers  (voltage)  left  in  sight — in  the  metal  circuit,  where  only  can 
the  voltmeter  obtain  information  (unless  it  were  armed  with  the 
elaborate  fishing  tackle  of  §  862). 

Plainly,  the  lost  (V  —  v)  is  now  below,  driving  through  the  cell 
itself  the  same  current  which  v  drives  through  the  metal  R, 
.*.  assuming  Ohm's  Law  applicable 

— r —  =  =^  =  the  current. 
0  rC 

In  III,  the  working  circuit  is  made  up  with  any  convenient  bit 
of  resistance  wire  and  an  ammeter,  which  reads  A ;  again  Y  —  v 
is  below  driving  A  through  6,  .*.  V  —  v  =  A6. 

These  are  very  favourite  practical  exam  questions,  simple  enough,j 
but  tricky  and  elusive  :   get  to  understand  them  by  doing  them- 
but  not  on  an  accumulator,  its  resistance  is  too  small  for  safety,  s( 
Question  24. 

There  are  other  ways  :  after  all,  the  truest,  that  which  gives  the 
best  approximation  to  the  actual  Ohmic  resistance,  is  to  put  the 


§  877]  ELECTRICITY   IN  LIQUIDS  7|j 

cell  as  the  unknown  into  the  gap  of  an  ordinary  Metre  Bridge, 
§  785,  and  to  feed  the  bridge  with  A.C.  from  a  toy  medical  coil 
run  by  a  Leclanch^,  and  listen  in,  for  silence,  with  headphones 
in  place  of  galvanometer.     Try  this  also,  using  only  low-res»Uiioe 

coils. 

§  876.  How  to  arrange  a  number  of  voltaic  cells  can  beet  be  seen 
by  considering  a  *  battery  wireless.' 

Where  it  is  a  question  of  driving  a  heating  current  through  ahort 
thick  wires  of  but  small  resistance,  quite  a  small  voltage  sufiicee. 
Hence  all  the  active  materials  are  packed  into  one  cell,  the  plates 
are  large  and  close  together,  and  the  liquid  is  highly  conductive, 
§  777,  so  the  cell  wastes  little  energy  pushing  current  through  itwif. 

Indeed,  if  the  cell  is  provided  with  multiple  plates,  as  all  high- 
capacity  cells  are,  these  can  be  '  paired  off,'  you  have  a  batter>'  of 
so  many  cells  '  in  parallel,'  each  providing  only  its  share  of' the 
chemical  action,  using  up  only  its  share  of  the  active  materials,  of 
which  a  big  cell  contains  enough  for  hours  running,  and  the  voltage 
is  just  that  of  one  cell. 

In  contrast  is  the  H.T.  battery,  also  a  heavy  supply  of  active 
materials,  but  now  in  many  little  packets.  The  outsiaehindrances 
to  be  overcome  are  enormous,  driving  force  and  yet  more  driring 
force  is  called  for,  a  high  voltage  must  be  built  up  by  letting  celU 
climb  on  one  another's  shoulders,  connecting  them  all  *ln  series,* 
carbon  to  zinc  throughout ;  all  the  E.M.F.'s  add  up. 

So  do  their  internal  resistances,  but  the  sum  total  still  remaiiw 
small  compared  with  the  external  circuit ;  so  again  the  battery 
does  not  waste  an  excessive  proportion  of  energy  on  itself. 

The  consumption  of  chemicals  is  large  in  the  aggregate,  for  the 
whole  voltaic  action  is  repeated  in  every  cell  :  fortunately  they  are 
littles,  for  a  small  current,  of  some  few  milliamperes,  is  all  that  is 
needed.  Yet,  as  you  know,  it  is  possible  to  be  too  sting>-  over  the 
capacity  of  the  cells  :  the  H.T.  battery  is  the  modem  equivalent 
of  the  Voltaic  Pile,  but  that  had  too  much  resistance  and  too  little 
chemicals. 

An  intermediate  case  is  provided  by  6- volt  valve«  :  here  you  hs%'0 
to  '  series '  three  of  the  '  parallel-plated '  2-volt  accumulators, 
getting  the  same  power  with  only  1/3  the  current  that  must  have  been 
distributed  had  you  stuck  to  2  volts.  A  similar  thing  crops  up 
in  Pocket  Torches,  of  all  descriptions,  little  and  big. 

Observation  and  common-sense  solve  these  problems :  old- 
fashioned  academic  queries  about  arranging  whole  hosts  of  cells, 
like  regiments  of  toy  soldiers,  are  completely  out  of  date. 

§877.  Electrolytic  corrosion.  Unsuspected  voltaic  batteries 
abound. 

If,  in  dissolving  zinc  in  HCl  to  make  soldering  flunl.  the  action 
is  slow,  drop  a  bit  of  copper  wire  on  the  zinc  :  you  have  made  s 
short-circuited  voltaic  cell,  and  the  effervescence  is  soon  vigorous— 


714  ELECTRICITY  [§  877 

from  both.  From  the  copper,  according  to  §  870,  from  the  zinc 
itself  on  account  of  '  local  action  '  between  adjacent  parts  of  varied 
alloy. 

*  Galvanizing  '  iron,  i.e.  coating  it  with  zinc,  often  nowadays 
by  electro-plating,  protects  it  against  atmospheric  corrosion  even 
when  patches  are  worn  bare,  for  the  zinc  dissolves  in  moisture,  and 
the  iron  surface  remains  coated  with  hydrogen  ions,  which  inhibit 
oxidation. 

The  cables  of  the  '  Grid  '  have  a  steel  core,  for  strength,  tightly 
enclosed  by  the  aluminium  strands  which  carry  the  current.  There 
is  no  fear  of  the  steel  corroding,  the  aluminium  meets  the  attacks 
of  acid  or  alkali,  pure  water  is  harmless. 

Steel-cored  copper  cable  would  be  hopeless ;  like  copperized 
iron  wire  or  tinned  steel-plate,  where  as  soon  as  moisture  can  find 
minute  holes  in  the  coating  it  corrodes  the  iron,  which  is  electro- 
positive now  to  tin,  and  the  holes  grow  and  spread.  Burnished 
steel  needles  and  knives  rust  in  spots  in  the  same  way,  the  '  flowed  ' 
burnished  material,  §  145,  is  more  resistant  than  the  crystalline 
mass  beneath  ;  like  tin-plate,  they  are  rust -resisting  at  first,  but  are 
routed  as  soon  as  the  first  line  of  defence  is  penetrated. 

Hard  water  is  safely  supplied  through  lead  pipes  ;  but  in  a  cistern 
where  a  brass  cylinder  was  bolted  on  to  thick  leaden  lugs,  these 
disappeared  in  ten  years  :  they  should  have  been  insulated  by 
fibre  washers. 

Per  contra,  you  have  seen  old  wrought -iron  railings,  set  into  the 
stone  sill  by  molten  lead,  thinned  down  or  eaten  right  to  a  point, 
at  the  base,  by  London  rain-water,  and  the  now  electronegative 
lead. 

The  instant  copperizing  of  bright  iron  when  dipped  in  copper  ^ 
solutions,  the  growth  of  '  lead  trees  ' — many  other  voltaic  actions 
will  occur  to  you. 

To  clean  blackened  silver-ware,  pile  it  on  an  aluminium  plate 
and  drown  in  warm  washing-soda  ;  the  electropositive  plate  decays, 
and  the  hydrogen  ions  attack  the  sulphide  ;  wipe,  rinse,  and  rub 
dry. 

A  propos  of  the  Conservation  of  Energy,  if  you  exert  yourself 
to  wind  up  a  clock  spring,  and  then  throw  it  in  acid  and  let  it  all  \ 
dissolve  away,  what  becomes  of  the  energy  you  put  into  it  ?  '  Oh, 
it  dissolves  with  the  production  of  greater  heat  of  solution.'  But 
the  difference  is  too  small  for  direct  measurement,  far  inside  the  limit 
of  experimental  error  ? 

Clean  6  in.  of  old  clockspring  with  emery  cloth,  break  it  in  halves, 
and  solder  one  end  of  each  to  a  sensitive  galvanometer  wire.  Hold 
them  side  by  side,  one  in  each  hand,  by  finger  and  thumb  on  their 
ends,  with  their  bulges  touching  a  filter  paper  wet  with  weak  acid. 
Let  the  galvanometer  settle  down,  and  then  bring  finger  and  thumb 
closer,  so  as  to  bend  them  more,  alternately,  and  watch  the  spot. 
Decide  what  it  means  by  touching  down  the  copper  wire  instead 
of  one  of  them,  giving  an  iron-copper  cell,  in  which  current  always 
comes  out  from  copper. 


§  878]  ELECTRICITY   IN   LIQUIDS  715 

The  earliest  of  the  fast  turbine  ships  had  much  trouble  with  their 
propellers,  the  blades  rotted  in  great  holes  at  the  root,  on  the  after 
side,  and  snapped  of! ;  Admiralty  and  Cunanl  alike  got  tired  of 
docking  them.  At  last  the  reason  appeared :  l^oth  the  reaction  of 
the  water  on  the  blade,  and  the  centrifugal  force  due  to  itn  rapid 
rotation,  were  bending  it  back  (i.e.  tip  forward),  the  greatest  iitrain 
was  at  the  root,  and  it  dissolved  there,  the  ions  nishing  into  Holution 
with  the  extra  energy  of  the  strain.  A  more  reHintant  bronze 
was  devised,  and  a  stouter  neck  in  which  the  strain  never  roue  tto 
high,  and  the  trouble  ceased.  Incidentally,  the  steel  tail-iihalt 
carrying  the  propeller  is  bronze-sheathetl  right  through  »tem- 
tube  and  gland,  into  the  dry  shaft  tunnel,  or  the  whole  propeller 
might  presently  go  a-missing. 

§  878.  Mercury,  soiled  and  contaminated  in  '  mercur\'.bivakH  '  or 
elsewhere,  can  be  cleaned,  for  all  purposes,  with  minimal  labour, 
loss,  and  mess,  as  follows  : — 

Fit  a  stout  conical  filter-flask  with  an  air-tight  bung  and  inlet 
tube  to  bottom,  and  connect  the  side-tube  to  a  filter-pump,  through 
a  pipe  which  runs  a  few  feet  up  and  down  the  wall,  so  that  no 
water  can  draw  back  into  the  flask  in  case  of  temporary'  failure  of 
the  pump.  Pour  in  a  quarter-inch  depth  of  strong  sulphuric  ackl. 
and  mercury  up  to  a  couple  of  inches,  and  bubble  air  through  for 
a  day  or  two.  Drain  away  all  the  acid  you  can,  and  then  empty 
the  flask  over  a  pound  of  oil-shop  *  whitening,'  roughly  crushwi.  in 
a  china  basin.  The  residual  acid  attacks  the  carbonate,  to  produce 
a  thirsty  calcium  sulphate  ;  crepitation  presently  ceases,  and  you 
pour  out  the  purified  metal,  through  any  dust-trap,  warm,  dry,  and 
enduringly  bright. 


EXAM  QUESTIONS,   CHAPTER  LII 

The  first  five  §§  introduce  and  doscribo  the  lona  vou  an*  Roing  lo  employ 
as  carriers ;  a  glance  at  their  work,  and  then  follow  their  oxtrvn»ely  importani 
trade  union  rules,  illustrated  by  such  instancee  aa  you  ineot  «;»»»»"»  »«•  "^ 
The  rest  of  the  chapter  is  occupied  with  the  conaideration  of  what  tn*N>o« 
do  •  on  their  own  initiative,'  and  includoe  the  older  way  of  pnxlunnir  elodrto 
current,  useful  as  ever  for  small  quantititw.  C'hnptor  aii.l  Qtw^turM  mm 
practical  in  character,  and  you  must  examine  laboratory  ap|Hkmtuii,  .pUI 
open  an  old  dry  cell,  etc.  :  further  theory  ia  givon  m  troattaea  oo  Uectfo. 
Chemistry. 

1.  What  are  Ions,  and  what  part  do  they  pUy  in  the  irwiaport  of  ©tectnriiy 
through  liquids,  and  through  gasea  ?     (  X  4) 

2  A  battery  is  joined  to  an  electrolytic  call  cont^ning  *^f^*^^^f!^ 
sea-wat«r,  petrol,  milk,  sugared  toa.  and  dilute  acid,  in  auccentoo.     umenm 

what  happens  in  each  case.     (  X  2) 


716  ELECTRICITY 

3.  A  4-volt  pocket  battery  is  connected  to  a  stainless  knife  blade,  and  to  tb 
teaspoon  of  salt  water  into  which  it  dips.  Describe  what  happens  :  wha1 
would  be  the  effect  of  warming  the  liquid  ? 

4.  Describe  how  current  is  carried  through  copper  sulphate  solution.  Wh, 
does  any  E.M.F.,  even  the  smallest,  result  in  an  alteration  of  the  weight  o: 
the  electrodes  ?     How  would  you  impel  iodine  into  the  human  body  ?     (  X  2' 

5.  State  the   laws   of  electrolysis,   and  describe   experiments  illustrat 
them. 

If  the  electrochemical  equivalent  of  hydrogen  is  0-0001045  in  c.g.s.  uniti 
what  current  will  decompose  1  gm.  of  water  in  an  hour  ?     (  X  2) 

6.  Distinguish  between  the  chemical  and  the  electrochemical  equivalen 
of  a  substance. 

3  amp.  flowing  through  a  solution  of  its  sulphate  (at.  wt.  63-6)  for  half  an 
hour,  deposits  1-78  gm.  of  copper.     Calculate  ec.  eq.  of  hydrogen.     (  x  3) 

7.  What  are  Faraday's  laws  of  electrolysis  ? 

A  metal  plate  10  cm.  square  and  1  cm.  thick  is  to  be  electroplated  with 
silver.  If  a  current  of  1-5  amp.  is  used  how  long  will  it  take  to  make  a  deposit 
0-005  cm.  thick  ?     Sp.  gr.  of  silver  10-5.     (  x  2) 

8.  Three  voltameters,  containing  dilute  caustic  soda,  copper  sulphate  solu- 
tion, and  silver  nitrate  solution,  are  run  in  series ;  90  c.c.  of  mixed  gas  are 
evolved  from  the  first ;  calculate  the  weights  of  copper  and  silver  deposited 
(1)  ignoring  corrections,  (2)  correcting  for  temperature  17°,  saturation  with 
moisture,  and  barometer  77  cm. 

9.  A  current  passes  through  a  copper  voltameter  and  a  tangent  galvano- 
meter with  10  turns  of  wire,  radius  8  cm.,  and  deflection  45° ;  how  much 
copper  is  deposited  in  half  an  hour  ?    H  =  0-18. 

10.  What  are  the  relative  advantages  and  disadvantages  of  galvanometers 
and  voltameters  ?     (  X  2) 

11.  The  current  which  causes  a  steady  deflection  of  45°  in  a  tangent  galvano- 
meter having  20  turns  of  wire  of  16  cm.  radius  is  observed  to  deposit  0-27  gm. 
of  copper  in  an  hour.     Calculate  H. 

12.  What  evidence  is  there  that  in  electrolysis  the  current  does  not  ionize 
the  solution,  but  merely  utilizes  ions  already  present  ?  Calculate  the  current 
which  in  5  minutes  liberated  100  c.c.  of  saturated  hydrogen  at  15°  and  74  mm. 
barometric  pressiu-e.     (  X  2) 

13.  How  many  coulombs  deposit  5  gm.  of  nickel  from  NiS04,  and  what 
bulk  of  mercury  would  they  liberate  from  mercurous  nitrate  solution  ?    (  X  2) 

14.  What  current  is  theoretically  necessary  to  produce  1  cwt.  of  aluminium 
per  24  hr.  ? 

15.  Two  pads  soaked  in  potassiiun  iodide  are  fastened,  one  on  each  side 
of  the  forearm,  7  cm.  thick,  and  a  gradient  of  1  volt  per  cm.  is  produced  through 
the  arm.  If  the  pads  are  each  9  sq.  cm.,  and  2  milliamps.  flows  per  sq.  cm., 
what  is  the  resistance  between  them  ?  If  the  mobility  of  the  iodine  ion  is 
0-0007  cm.  per  sec.  per  volt  per  cm.,  how  far  will  the  ion  penetrate  in  half  an 
hour?     (X  2) 

16.  Why  is  1-5  volts  required  to  electrolyse  acidulated  water  ? 
A  cm-rent  is  passed  for  1  hr.  through  dilute  sulphuric  acid,  and  500  c.c.  o: 

saturated  hydrogen  is  collected  at  76  cm.  barometer  and  10°  C.     If  an  ani' 
meter  in  circuit  registers  1-3  amps.,  what  is  its  error  ? 

17.  Six  volts  being  applied  to  a  water  voltameter  of  3  ohms  resistance  and 
1-5  volts  back  E.M.F.,  calculate  the  weight  of  water  decomposed,  and  th©j 
calories  liberated,  per  hour. 

18.  A  pail  of  washing  soda  solution,  into  which  dips  an  iron  rod,  is  a  commorf 
resistance  for  regulating  long -continued  heavy  currents.  What  chemic  ** 
and  physical  actions  would  you  expect  to  observe  in  its  operation,  and  what] 
renewals  would  you  have  to  make  from  time  to  time  ? 

19.  Of  two  copper  voltameters,  one  has  twice  the  resistance  of  the  other, 
Compare  the  rates  of  copper  deposition,  and  of  heat  production,  in  them, 
when  they  are  run  (a)  in  series,  (6)  in  parallel.     Neglect  the  battery  resistance. 


ELECTRICITY   IN   LIQUIDS  717 

20.  Describe  an  arrangement  of  apparatus  to  demonstrate  polaritatkm 
in  an  electrolytic  cell :    explain  this  action,  and  mention  aome  ^tti*tt  for  it 

(X  4) 

21.  Describe  the  effect  of  connecting  one  dry  cell  to  (a)  two  plattnum 
wires  dipping  in  dilute  acid,  (6)  two  copper  wires  dipping  into  copfwr  sulpliale 
solution. 

22.  Describe  the  construction  of  a  secondary  cell  or  accumulator,  aod  give 
an  outline  of  the  chemical  actions  which  occur  in  it  during  charinng  and 
discharging. 

What  is  meant  in  practice  by  the  '  capacity  '  of  sucli  a  cell  and  oo  what 
factors  does  it  depend  ?     (  X  3) 

23.  Account  for  the  variation  in  density  of  the  acid  during  charge  and  die* 
charge  of  a  lead  accumulator.  What  are  the  advantages  and  disa<i\'ant«g«B 
of  accumulators  as  compared  with  dry  cells  7     (  X  2) 

24.  A  2-volt  secondary  cell  has  its  two  platee,  12  cm.  square,  aenarated  by 
a  liquid  space  of  8  mm.  If  the  conductivitv  of  the  acid  u  0-7,  csJcukle  the 
current  on  short  circuit,  the  resistance  of  pUtes  and  wire  being  0-03  ohm. 

25.  A  battery  of  accumulators  has  an  E.M.F.  of  IIO  volta  and  an  internal 
resistance  of  0-10  ohm.  How  many  50-watt  lamps  mav  be  lighted  by  the 
battery  if  each  lamp  requires  a  voltage  of  100  anii  if  the  roaiatance  of  the 
leads  to  the  battery  is  0-30  ohm  ?     (  X  2) 

26.  A  dozen  lead  accumulators,  each  with  an  effective  back  E.M.F.  of  2-5 
volts,  are  to  be  charged  from  100-volt  mains;  what  resistance  must  be  pui 
in  circuit  to  keep  the  charging  current  down  to  3  amps.  T 

27.  Two  lead  platos  immersed  in  dilute  sulphuric  acid  are  cotmected  to 
the  poles  of  three  Daniell  cells  in  series,  and  after  tM>mo  time  are  dierotuMded 
from  these  and  joined  together  by  a  wire.  Describe  and  explain  all  |inin— 
taking  place. 

28.  Describe  a  battery  entirely  suitable  for  operating  a  cautery. 

29.  Describe  the  construction  of  some  form  of  coll  suitable  for  a  standard 
of  electromotive  force,  and  explain  the  chemical  reactions  when  a  current 
is  passing. 

30.  Describe  any  two  tyjies  of  voltaic  cell,  pointing  out  for  what  usee  they 
are  suitable  and  how  they  are  made  '  constant.'  ( X  3) 

31.  A  student  has  connected  a  voltmeter  and  an  ammeter  in  aaries  and  to 
a  voltaic  cell,  and  has  re&d  1-4  volts,  and  about  002  amp.  Being  told  to  aher 
his  connections,  he  now  connects  each  directly  to  the  cell,  and  obtains  simol* 
taneous  readings  1-4  amp.  and  about  002  volt.  What  may  he  deduce  from 
these  observations  ?     (  X  3) 

32.  A  Leclanch6  cell  of  1-5  volts  and  resistance  3  ohms  is  found  to  sand 
0-4  amp.  through  a  galvanometer.  What  information  can  you  gather  about 
the  circuit  ? 

33.  What  is  meant  by  joining  cells  '  in  series  *  and  '  in  parallel  *  ?  In  what 
circumst€tnces  is  one  or  other  to  bo  preferred  ? 

Two  equal  cells,  each  of  E.M.F.  1-5  volts,  joined  in  series,  ssnd  0-25  amp. 
through  10  ohms.  If  the  cells  are  now  joined  in  parallel  what  current  wiU 
they  give  through  the  10  ohms  ?     (  X  2) 

34.  Distinguish  between  the  E.M.F.  of  a  cell  and  the  P-^.  bjtJJ^f*  j»J 
terminals.  What  condition  holds  good  if  two  similar  cells  ^^  J^^'  J^ 
volts  give  the  same  current  through  a  resistance  of  10  oluns  whether  they 
are  connected  in  series  or  in  parallel? 

Calculate  the  P.D.  between  the  cell  terminals  in  either  case. 

35.  Explain  the  terms  electromotive  force  and  internal  reeielanee  aa  appl»«<d 
to  a  voltaic  cell.  .  .  .        .  ^  >.  .  _» ^ 

Given  three  cells  each  of  E.M.F.  1  volt  and  mtemal  resistance  0^4  "hm, 
show  how  to  find  the  E.M.F.  and  internal  nwistance  of  the  various  iMluwwi 
that  may  be  constructed  by  using  all  of  the  cells. 


718 


ELECTRICITY 


36.  State  Ohm's  Law,  specifying  its  limitations. 
A  battery  of  E.M.F.  6  volts  has  an  internal  resistance  of  2  ohms.     When 

the  poles  are  connected  by  a  wire  A  the  P.D.  between  them  is  5  volts,  and 
when  by  a  wire  B,  4  volts.  Calculate  the  resistances  of  A  and  B,  and  the 
currents  flowing  in  them. 

37.  State  Ohm's  Law  for  a  circuit  containing  battery  and  resistance. 
What  are  the  voltage  and  internal  resistance  of  a  cell  which  shows  1-4  volts 
with  500  ohms  in  circuit,  and  1-2  volts  with  50  ohms  ? 

38.  Explain  fully  how  you  would  determine  the  internal  resistance  and 
the  E.M.F.  of  a  Daniell  ceil,  with  the  help  of  an  ammeter,  an  accumulator, 
and  a  length  of  wire  of  uniform  thickness  and  known  resistance.     (  X  2) 

39.  How  would  you  use  the  Potentiometer  to  measure  (a)  the  internal 
resistance  of  a  voltaic  cell,  (&)  a  voltage  of  several  hundred  volts  ? 

40.  Calculate  the  internal  resistances  of  cells  in  which  (a)  1-08  volts  on 
open  circuit  fell  to  0-9  when  cell  was  sending  current  through  5  ohms,  (6)  1-4 
falls  to  1-1  on  10  ohms,  (c)  8  falls  to  6  on  12  ohms.     (  X  4) 

41.  It  is  required  to  charge  an  accumulator  with  a  definite  quantity  of 
electricity  by  the  use  of  Daniell  cells  each  of  E.M.F.  1-1  volt  and  internal 
resistance  1-5  ohm.  There  is  a  back  E.M.F.  of  2-5  volts  in  the  accumulator. 
Compare  the  times  necessary  using  (a)  three,  (6)  four  cells. 

42.  A  battery  of  E.M.F.  1-4  volts  and  resistance  4  ohms  is  connected  through 
two  resistances  of  10  ohms  and  20  ohms  in  parallel.  What  current  will  flow 
through  each  resistance  and  what  will  be  the  P.D.  between  the  terminals, 
of  the  battery.     (  X  2) 


PRACTICAL   QUESTIONS 


Measure  the  electrochemical  equivalent  of  copper ;   or,  of  hydrogen. 

Obtain  the  '  constant '  of  a  galvanometer  by  using  a  copper  voltameter. 

[Allow  for  any  zero  error  on  an  ammeter. 

Arrange  to  reverse  the  current  through  a  tangent  galvanometer,  but  not 
through  the  voltameter.  Don't  forget  to  wash  voltameter  plates  before 
drying  to  weigh  :  if,  by  mismanagement,  you  have  taken  copper  off,  instead 
of  putting  on,  mention  the  mistake,  which  is  not  serious,  and  calculate  out  as 
usual.] 


CHAPTER   LUI 
THE  TRANSPORT  OF  ELECTRICITY  THROUGH  GA8E8 


§881.  Free  electrons.  Let  the  negative  end  of  the  wire  along 
which  the  electrons  have  drifted,  §778,  be  enclosed  in  a  vacuum 
of  modern  completeness,  say  a  thousand-millionth  of  a  millimetir. 
§  107,  in  which  there  can  be  little  to  hamper  their  movementji.  They 
refuse  to  leave  this  metal  cathode.  Put  on  a  heavy  voltage,  «ay 
80  kv.  (kilovolts,  80,000  volts),  and  they  make  noisy  JemonHtrationii 
in  the  air  outside,  sparks  of  which  we  shall  tell  later ;  but,  like 
Nature  in  bygone  days,  they  '  abhor  the  vacuum.' 

Heat  your  Cathode,  until  it  glows  yellow- or  white-hot,  and  electrons 
distil  out,  without  any  further  trouble,  in  quantities  rapidly  in- 
creasing with  rise  of  temperature  ;  from  2000°  to  2500°  A.  the  curve 
is  almost  indistinguishable  from  the  vapour- pressure  curve  of  Fig. 
82 — from  0°  to  110°.  In  fact,  the  electrons  behave  like  a  volatile 
substance  dissolved  in  the  solid,  just  as  they  diffused  osmotically 
in  §§  778,  800. 

By  alloying  the  tungsten  with  a  little  thorium,  the  gaa-mantle 
metal,  a  lower  temperature  suffices,  while  filaments  coated  with 
lime,  strontia,  or  baryta,  as  in  the  dull-emitter  valves  of  radio 
receiving  sets,  run  from  1100°  to  1300°  A.;  but  for  long  hard 
heavy  work  the  plain  tungsten  filament  glows  at  2400 — 2500*  A., 
at  which  latter  temperature  it  emits  an  ampere  per  sq.  cm. 

Somewhere  in  the  vacuum  bulb  must  be  a  cold  Anode  plate  to 
take  the  (negative,  of  course)  current  away  ;  then,  by  applying  a 
negative  potential  to  the  cathode,  a  current  passes  (in  size  ordinarily 
from  micro-  to  milli-amperes).  If  the  cathode  is  verj*  hot,  there 
will  be  abundance  of  electrons  ready  to  carry  current,  so  that  a  low 
voltage  will  suffice  to  drift  the  multitude  along  ;  but  with  a  cooler 
cathode  higher  and  higher  voltages  are  needed  to  drive  the  few 
electrons  fast  enough  ;  the  power  required  by  the  tube  in  WatU 
being,  as  always,  volts  X  amperes. 

§  882.  There  is  nothing  to  see  in  the  tube,  but  as  the  voltaffo 
increases,  the  glass  walls  begin  to  fluoresce,  with  a  light  usuajly 
greenish.  Crookes,  using  the  highest  vacua  obtainable  in  1878, 
studied  this  '  radiant  matter,'  and  found  that 

{a)  it  travelled  in  '  Cathode  Rays '  straight  away  from  the  cathode. 

(6)  made  the  glass  fluoresce  (and  phosphoresce  aften»ards)  where 
it  hit,  and  made  many  minerals  fluoresce  brilliantly  in  various  colmirn 
(so  that  bouquets  of  glass  flowers  were  painted  with  them  and  put 

in  the  tubes) ;  •  i.  *  # 

(c)  the  discharge  from  a  concave  cathode  had  a  ver>'  not  locun. 

(d)  the  '  rays  '  were  deflected  by  a  magnet,  across  tts  field, 
'  '    *  719 


720  ELECTRICITY  [§  883 

§  883.  In  1896  Joseph  John  Thomson  began,  at  the  Cavendish 
Laboratory  at  Cambridge,  the  series  of  researches  on  these  dis- 
charges which  shattered  the  chemical  Atom  and  opened  up  an  entirely 
new  chapter  in  the  history  of  science,  that  of  intra-atomic  physics  ; 
researches  which  not  only  put  that  Laboratory  in  the  forefront 
of  the  attack  on  the  mysteries  of  matter,  but  put  Research  itself 
on  a  higher  footing  ;  researches  that  have  been  pursued  ever  since, 
with  ever-increasing  intensity,  by  a  world  which  has  long  known, 
honoured,  and  loved  him  as  'J.  J.' — Master  of  Trinity,  close  wrapt 
in  researches  still. 

This  was  before  the  discovery  of  hot  cathodes,  the  development 
of  which  began  in  that  Laboratory  in  1903  ;   but  in  such  vacua  as 

were    then    called    high,    the  cold 

cathode  of  the  long  tube   of  Fig. 

394,   under  the   steady  urge  of   a 

Wimshurst  machine,  sent  its  stream 

through  two  small  holes  in  metal 

plates,  to  strike,  as  a  bright  green 

Fig.  394.  fluorescent   spot,  on  the  willemite 

(zinc  silicate)  screen  at  the  far  end. 

The    '  tangent  galvanometer   coils '   h  h  produced  a  Magnetic 

Field,  in  your  line  of  sight,  across  which  the  stream  had  to  pass, 

and  plates  /  /  inside  the  tube  could  be  electrified  to  produce  an 

Electric  Field  F  straight  across  between  them,  at  right  angles  to 

the  other. 

The  mechanical  force  exerted  per  cm.  length,  electromagnetically, 
on  a  current -carrying  conductor  =  magnetic  field  H  X  current 
(absolute  units),  §  749.  Assuming  the  Cathode  Stream — let  us 
drop  the  term  '  rays  ' — to  be  composed  of  flying  particles,  which 
J.  J.  called,  first,  'corpuscles,'  and  later,  electrons,  carrying  a  negative 
charge  e,  the  '  current '  that  each  one  represents  =  its  charge 
e  X  its  speed  v — for  think  of  the  '  current '  of  water  that  a  bucket 
of  water  passed  from  hand  to  hand,  towards  a  fire,  is  equivalent  to. 
Hence  the  force  deflecting  each  electron  as  it  crosses  the  field 
H  in  Fig.  394  is  Hev,  pressing  it  downwards  in  the  plane  of  the 
diagram,  and  making  it  take  a  curved  path,  of  radius  r,  which  calls 
into  play  a  centrifugal  force  mv^jr  equal  and  opposite  to  the 
electromagnetic  force 

Hev  =  rnv'^jr 

hence,  dividing  throughout  by  Hmv 

-r>   , .    charge  of  electron        e         v 

Ratio  ^ —  =  _  =^ 

mass  of  electron        m       Hr 

in  which  H  is  calculable  much  as  in  §  767,  and  r  is  not  difficult  to 
compute  from  the  displacement  of  the  spot  up  or  down  on  the  ^ 
screen. 

An  electrostatic  field  F  is  now  set  up  between  the  plates  ff,  and  ; 
adjusted  until  it  lifts  the  electrons,  each,  of  course,  with  the  force 


§  883]  ELECTRICITY   IN  GASES  7ti 

Fe,  §  722,  just  as  much  as  the  magnetic  field  depreiwes  them   and 

the  spot  returns  to  zero. 

Then  Hev  =  Fe 
/.  »  =  F/H 

This  gave  for  v  enormous  values,  more  than  lOO.OUO  kilometre 
per  second,  dependent  on  the  driving  voltage  (actually  r  km.  '«ec  » 
eOOVvolts),  and  putting  these  into  the  expression  above,  gare 

e/m  =  176,000,000  coulombs  pergm. 

A  check  on  this  was  obtained  by  another  tulx?,  in  which  was  a 
diminutive  copper  cup  instead  of  a  screen.  The  cathode  stream 
was  deflected  into  this  by  a  magnet,  and  gave  up  all  the  charges 
e  of  its  electrons,  §711,  raising  an  attached  Olo-mfd.  condenav 
about  5  volts  per  second.  At  the  same  time  it  gave  up  all  their 
mechanical  energy,  Jmv^  ergs,  as  heat  to  the  cup,  which  had  a 
thermo- junction,  §  799,  embedded  in  it. 

This  Jmt;2  =  energy  given  to  charge  e  by  potential  fall  V  =  eV. 

Now,  in  Electrolysis,  e/m  for  At.  Wt.  1  is  96,'>00,  since  it  would 
take  that  many  coulombs  to  liberate  one  gram,  §  857. 

The  Question  arose.  Was  m  the  same  in  both  cases,  or  was  e  ? 

Were  these  particles  hydrogen  ions,  flying  from  the  cathode  at 
a  speed  many  thousand  times  that  of  their  natural  molecular  motion. 
and  each  carrying  a  charge  176,(X)O,0OO  96,r)00  times  as  great  as  in 
electrolysis,  or  were  the  charges  identical  and  the  particles  IH40 
times  smaller  than  hydrogen  atoms  ? 

Avogadro's  number,  that  of  the  molecules  in  the  gram-molecule 
mass  of  any  element,  was  known  as  61  x  lO**,  conaequeiitly 
the  actual  charge  on  the  hydrogen  ion  in  electrolysis  is  96,iMI0/ 
61  X  1023  =  15-8  X  10-20  coulombs  :  what  was  the  charge  on  the 
cathode  particle  ? 

Means  became  available,  as  we  shall  see  later,  of  obtaining  what 
were  undoubtedly  similar  particles  in  the  open  air,  and  not  in  hurried 
movement.  Millikan  introduced  a  fine  spray  of  oil  drops  into  a 
space  between  upper  and  lower  metal  plates,  and  watche*l  the 
movement  of  individual  droplets  with  a  horizontal  micrometer 
microscope.  From  the  rate  of  fall  of  a  drop  through  air  it  ia  easy 
to  determine  its  weight  7ng  (cf .  §  334).  Electrifying  the  platea  -f 
and  —  had  no  effect  at  all  on  uncharged  drops,  but  it  was  obaer%*ed 
that  suddenly  a  drop  would  start  off  towards  the  -f  plate  ;  evidently 
a  wandering  electron,  with  its  charge  e,  had  met  and  stuck  to  it. 
The  electric  field  F  between  the  plates  was  then  adjustwl  until, 
the  +  plate  being  on  top,  the  drop  hung  motionless,  Ve  upwards 
exactly  equal  to  mg  downwards  (until  another  elwtron  was  suddenly 
captured  and  lent  a  hand  with  the  lifting) ;  and  «  proved  to  fc« 
15-9  X  10-20  coulombs,  identical  with  that  in  electrolysis. 


722  Electricity  [§  sss 

The  electronic  e/m  being  176,000,000,  this  gives  m  as  1/1839  that 
of  the  H  atom  (atomic  weight  1-0078)  or  15-9/17-6  x  lO^o  x  10^ 
=  0-905  X  10-27  gm.,  the  electronic  mass  (at  no  great  speed). 

'  In  the  far,  far,  frozen  North,'  ran  a  monkish  legend  of 
the  Middle  Ages,  '  guarding  the  grim  passes  that  lead  to 
the  happy  land  of  the  Hyperboreans,  stands  a  Rock,  100  miles 
wide,  and  100  miles  thick,  and  100  miles  high.'  There  are  boulders 
on  the  Norway  coast  a  mile  each  way,  but  this  is  a  million  of  them. 
'  And  every  hundred  years  comes  a  little  bird,  and  perches  on  a 
corner  of  it,  and  scrapes  his  bill,  and  so  the  rock  is  worn  away,  to 
nothing,  and  yet  shall  not  one  second  of  eternity  be  passed.' 

Leaving  the  pious  chronicler  to  his  exciting  vigil,  let  us  indulge 
in  a  little  calculation  in  which  the  infinities  shall  have  no  say. 
Suppose  the  bird  scrapes  away  a  milligram  each  visit,  so  that  since 
the  beginning  of  historic  time  he  has  wasted  one  apothecaries'  grain 
of  granite  :  the  rock  will  last  lO^^  years,  the  number  of  electrons 
totalling  a  mass  of  1  gm.,  the  number  actually  contained  in  1840 
times  that  mass  of  anything,  a  4-lb.  iron  weight,  for  instance. 

And  you  and  I  have  no  personal  experience  of  even  100  years  ! 

§  884.  The  three  outstanding  applications,  at  the  present  day, 
of  this  direct  stream  of  electrons  from  the  cathode,  are  the  Rectifying 
and  amplifying  tube  almost  universal  in  '  Radio,'  the  Cathode - 
Ray  Oscillograph  ;  and  the  X-ray  tube  of  the  next  chapter. 

The  great  feature  of  the  first  of  these  is  the  Grid  which  acts  as  a 
gateway  across  the  path  of  the  electrons,  practically  stopping  them 
when  enough  have  fallen  upon  it  to  charge  it  up  to  the  same 
negative  potential  as  the  hot  cathode  filament.  This  is  a  very  small ; 
charge,  and  by  admitting  comparable  +  charges  from  the  incoming 
signals  the  gate  is  virtually  opened,  little  or  much,  and  the  electrons 
fly  through,  to  fall  on  the  anode  plate,  in  swarms  the  numbers  of 
which  depend  on  the  electrical  build  and  make-up  of  the  tube. 
That  gives  the  amplification,  and  the  rectifying  valve-action  is  due, 
of  course,  to  the  plate  being  cold  and  unable  to  emit  electrons.  All 
else  about  these  valves  you  will  read  in  your  Wireless  books. 

The  modern  Cathode-Ray  Oscillograph  differs  from  the  original 
Fig.  394  in  no  essential  which  would  make  a  fresh  figure  worth 
while.  It  has  a  better  vacuum — that  goes  without  saying  nowadays, 
but  doesn't  show  in  a  picture,  except  that  it  involves  a  hot  cathode 
— and  it  has  a  second  pair  of  parallel  plates  fixed  at  right  angles 
to  the  first,  so  that  the  electric  field  between  them  points  straight 
at  you  in  Fig.  394.  The  cathode  stream  is  focussed  to  a  beautifully 
sharp  and  brilliant  tracing  spot  on  the  fluorescent  screen. 

Suppose  it  is  required  to  trace  the  wave- form  of  an  ordinary 
alternating  current.  A  '  time-base  '  circuit  is  built  up  which  charges 
a  condenser  attached  to  the  right  and  left  attracting  plates  steadily 
throughout  each  period,  and  then  discharges  it  quickly.  This 
pushes  the  spot  steadily  across  the  screen  from  left  to  right,  a  straight 


§  885] 


ELECTRICITY  IN  GASES 


7iS 


base  line.  Meanwhile  the  A.C.  passes  through  the  coiU,  ami  iU 
magnetic  field  gives  the  spot  up-and-down  motion,  and  you  wee  th© 
whole  wave  standing  on  the  screen. 

For  higher  frequencies,  the  magnetic  field  procedure  becomM 
hopelessly  slow,  and  the  top  and  bottom  pair  of  plates  is  connected 
up,  and  draws  charges  from  the  circuit  at  some  convenient  place. 
You  whisper  into  a  microphone,  for  instance,  a  valve  amnUfim  th© 
microphone  current  and  supplies  it  to  the  p!at(>^,  and  the  line  of 
light  quivers  into  a  shining  detailed  record  of  the  thousands  of  hissing 
sound  ripples  per  second. 

Or  connected  up  to  a  Television  circuit,  the  spot  *  scans  '  the  screen 
all  over,  several  times  a  second,  stopping  in  its  to-and-fro  fiight 
43,000  times  in  the  second,  for  the  right  length  of  time  to  exciir, 
at  each  place,  just  the  exact  amount  of  fluorescent  light  for  the 
whole  to  build  up  into  the  picture. 

The  millionth  of  a  second  represents  leisure  to  th©  Cathode- Ray 
Oscillograph. 

§  885.  The  admission  of  a  very  small  trace  of  gas  into  the  vacuum 
tube  exempts  one  from  the  necessity  of  heating  the  cathode,  pn»\  idcd 
a  high  kilo- voltage  be  used.  The  cathode  .stream  of  negative  ions 
is  now  plainly  got  from  the  gas  ;  are  there  any  Positive  Ions  ? 


Fig.  305. 

There  are,  though  by  no  means  so  easy  to  find,  awl  th€»y  stream 
back,  through  a  perforated  cathode,  away  from  the  repelling  anocie. 
They  are  studied  by  magnetic  and  electric  fields,  as  iK'fore,  but 
now  it  is  found  necessary  to  work  in  the  polar  gap  of  a  great 
electromagnet,  in  a  field  6f  3000  gauss  or  more,  instead  of  that  of 
a  few  turns  of  wire.  The  reason  is  that  they  are  more  maamve, 
a  much  greater  force  must  be  employed  to  deflect  their  momentum. 
in  fact  elm  drops  below  100,000  ;  they  are  evidently  hydrogtn  atams^ 

The  apparatus  employed  is  sketched  in  tig.  2\^,  the  cathode 
is  a  narrow  metal  tube,  and  the  electric  field  between  it  and  the  ancwie. 
in  a  recess  on  the  right,  extracts  positive  ions  from  the  gas  m  the 
bulb,  and  accelerates  them  so  that  they  .shoot  thnmgh  the  narrow 
bore  and  strike  the  screen  or  photographic  plate  at  the  end  of  the 
conical  tube  on  the  left.  NS  is  the  electromagnet  and  the  .le. «re 
was  adopted  of  having  the  electric  field  parallel  to  the  magnet ^, 
little  slices  of  iron  being  sawn  off  the  i>ole-pieces.  insuUted  with 
mica,  and  used  as  the  electric  attracting  plates  4    ^  . 

This  throws  the  electric  deflection  out  at  right  anglea  to  the  mag. 


724  ELECTRICITY  [§  885 

netic,  and  the  result  was  to  develop  a  family  of  parabolas,  showing 
where  the  projectiles  hit  the  plate,  as  suggested  on  the  left. 

Any  one  species  of  atom  might  carry  e,  2e,  or  even  3e,  which  gives 
perfectly  well  separated  and  identifiable  curves,  and  atoms  might 
combine  into  molecules  such  as  Og  and  O3,  which  would  have  more  | 
momentum  per  charge,  and  therefore  fly  straighter  and  make  dis- 
tinct smaller  parabolas.  And  if  anything  like  H3  managed  to  hang 
together  for  the  fifty-millionth  of  a  second  necessary  for  its  slow — 
8000  km. /sec.  or  so — flight  to  the  plate,  it  made  its  distinct  faint 
parabola.  In  fact,  it  was  easy  to  scale  off  the  plate  the  comparative 
masses  of  the  atoms  concerned  :  when  Neon,  atomic  weight  20-2, 
made  no  such  mark,  but  a  heavy  one  at  20  and  a  lighter  one  at  22, 
J.  J.  had  discovered  Isotopes  :  for  which  see  §  924. 

§  886.  Let  further  traces  of  gas  be  admitted  to  the  tube,  and 
the  current  increases,  for  there  are  more  carriers.  And  more ; 
and  the  streams  of  +  and  —  carriers  travelling  opposite  ways 
begin  to  jostle  each  other,  and  the  result  you  see  in  the  gleaming 
red  and  blue  tubes  on  the  shop-fronts,  the  blue-green  and  yellow 
of  main-road  lighting,  the  aurora,  the  electric  spark,  lightning. 

It  is  plain  from  the  great  length  of  these  tubes  and  flashes,  and 
from  the  fact  that,  if  the  pressure  in  the  discharge  tube  is  much 
more  than  a  millionth  of  an  atmosphere,  the  cathode -stream 
cannot  reach  the  wall  at  all,  but  ends  in  a  haze  of  light  in  the  gas, 
that  there  must  be  many  more  carriers  employed  in  these  luminous 
discharges  than  ever  started  from  their  ends ;  there  must  be  some 
means  of  '  handing  on  the  torch.' 

This  is  found  in  Ionization  by  Collision  ;  it  takes  place  at  the 
cold  cathode  between  gas  and  metal,  it  takes  place  over  and  over 
again  along  the  tube,  perhaps  many  times  per  cm. 

If  you  will  stand  near  a  red  neon  tube,  and  sweep  your  eyes 
rapidly  across  it,  you  will  see,  of  course,  that  the  discharge,  which 
is  A.C.,  is  very  intermittent,  but  in  some  tubes  you  will  see  also 
that  each  discharge  is  '  striated,'  the  line  of  light  is  dotted,  the 
successive  flames  perhaps  a  finger-width  apart.  You  don't  see  them 
in  the  whole  discharge,  because  they  flicker  to  fresh  positions  every 
flash.  They  are  like  standing  ripples  in  the  discharge.  Fig.  135 ; 
two  lots  of  ions  have  colHded,  and  brought  each  other  to  harmless 
slowness,  the  electric  fleld  along  the  tube  accelerates  them  up  to 
speed  again — across  a  dark  space  where  there  is  a  sharp  drop  of 
voltage,  and  presently  they  crash  in  another  blaze  of  light :  American 
football  in  fact. 

§  887.  First  we  must  digress  to  show  that  Ions  are  produced  by 
Collision.  C.  T.  R.  Wilson,  native  of  a-  land  of  mist,  performs  the 
experiment  of  Fig.  86,  to  get  washed  wet  air  which  will  no  longer 
form  droplets  of  mist  when  chilled  by  expansion,  in  a  glass  pill-box, 
of  which  the  bottom  can  be  suddenly  pulled  down,  so  as  to  expand 
its  volume  by  half,  Fig.  396. 


888] 


ELECTRICITY   IN  GASES 


725 

.  nln^^"*?  are  electrified  ions  present,  they  act  as  nuclei,  §  312.  and 
a  drop  of  water  condenses  on  each  from  the  cold  supwUturtted 
vapour,  and  in  a  strong  light  a  snap-shot  can  be  taken.  ^nd^nK^ 
to  show  individual  drops,  if  not  too  densely  packed  ^""^'K^ 

We  shall  see  later  on  that  radium  gives  of!  two  kinds  of  electrified 
particles,  a  and  p,  which  are  extra-fast  Positive  Ions,  and  Klectrona  • 
therefore  one  puts  a  minute  speck  of  it  near  the  wet  chamber* 
and  hres  the  expansion,  and  the  camera.  Straight  rocket  trackji 
appear,  and  their  photographs,  when  enlarged,  look  very  like  what 


Fio.  396. 

you  would  get  by  exposing  a  thread  of  wool  to  a  fine  water-spray  : 
there  are  the  courses  of  a  particles,  the  massive  positively  charged 
atoms  (of  helium),  which,  crashing  through,  have  each  produced 
over  200,000  ions.  In  the  very  woolly  track  on  the  right  in  Kig. 
396  the  ions  had  had  half  a  second  to  diffuse  lH?fore  the  ex|mnHi<»n 
was  fired  and  caught  them.  The  faster  and  less  destnictive  Hight 
of  the  electrons,  curving  as  they  pass  over  a  S  pole  beneath  the 
chamber,  is  sketched  in  the  drop-studded  tracks  in  ^  and  ^  magnifi(<<l 
As  the  flying  charge  passed  by  an  atom  of  the  gas,  it  dragge<l  an 
electron  out  of  it,  and  this  electron  was  capture*!  oy  another  atom  ; 
thus  a  pair  of  charged  ions  has  been  produced. 

§  888.  Ions  such  as  these  travel  off  opposite  ways  in  an  electric 
field,  and  that  is  how  a  gas  conducts.     When  they  collide,  they 


726  ELECTRICITY  [§  888 

*  recombine,'  whether  under  the  pull  of  the  field  ;  or  in  the  natural 
molecular  dance,  after  a  life  of  a  second  or  more. 

Their  vigorous  recombination,  in  bulk,  causes  the  emission  of 
Light — not  only  the  light  in  these  tubes,  where  the  pressure  of  the 
gas  is  from  0-001  to  0-01  atmo.,  and  the  tube  is  scarcely  warm  to 
the  touch,  but  the  blue  Light  of  a  Bunsen  Flame,  or  the  well-known 
colours  imparted  to  it  by  sodium,  copper,  etc.  That  the  flame  is 
hot  is  effect  rather  than  cause,  for  its  luminosity  accompanies  the 
chemical  combination  going  on,  which  is  ionic  combination  :  merely 
heating  gases  ever  so  hot  does  not  make  them  luminous. 

[Incidentally,  the  feeble  '  blue '  of  a  bunsen  is  the  nearest  known 
approximation  to  daylight.] 

§  889.  The  Aurora  Polaris  shows  this  discharge  on  the  grand  scale. 
Unfortunately,  it  is  seldom  noticed  near  London ;  at  a  recent 
students'  conference  on  the  subject  at  South  Kensington  nobody 
would  claim  to  have  ever  seen  it. 

That  was  probably  wrong,  in  view  of  §  572,  but  faint  flickering 
crimson  streamers  spreading  over  half  the  northern  sky  have  a 
poor  chance  near  street -lighting  and  town  haze,  while  a  greenish 
false  dawn  in  the  north,  though  giving  light  enough  to  make  the 
night  angler  doubt  the  church  clock,  might  be  put  down  to  clear 
air  and  twilight. 

It  is  visible  about  5  times  a  year  in  the  Channel ;  and  35  in  the 
north  of  Scotland,  even  reaching  overhead ;  from  Norway  it  may 
be  seen  to  the  west ;  and  it  is  frequent  enough  in  the  St.  Lawrence 
and  in  Canada  generally,  all  within  2000  miles  of  the  Magnetic  Pole. 
It  often  takes  on  the  appearance  of  a  green  arch,  from  which  arise 
long  straight  streamers  in  erratic  motion — the  headlights  of  cars 
coming  up  over  a  distant  hill,  then  suddenly  turning  aside  as  they 
reach  the  busy  cross-road  along  the  ridge. 

It  lies  mostly  between  55  and  80  miles  high,  and  up  to  more  than 
120 ;  so  high  that  when  actually  over  N.  Scotland  it  would  be  in 
sight  of  London. 

The  chief  of  its  four  equidistant  spectrum  lines  is  the  green  0-56 
micron,  due  to  oxygen. 

The  Sun  not  only  sends  us  much  ultra-violet,  which,  as  we  shall 
see  later,  arrives  in  8  minutes,  ionizes  the  attenuated  upper  atmos- 
phere, and  produces  a  *  conducting  layer  ' ;  but  also  acts  as  a  hot 
cathode,  sending  us  ions  in  just  over  a  day.  There  is  '  space  charge  ' 
enough  all  round  him  to  keep  most  of  them  back,  but  when  he  turns 
towards  us  a  face  suffering  from  one  of  those  fiery  eruptions  we 
recognize  from  their  spottiness,  a  rush  of  electrons  may  break  through, 
and  aurorae  and  magnetic  storms,  §  698,  both  of  them  evidences 
of  currents  circulating  overhead,  are  to  be  expected.  Undoubtedly 
too,  positively  charged  particles,  probably  '  positrons,'  are  contained 
in  the  solar  stream. 

Pointing  an  idle  telescope  to  the  sun,  one  blazing  day  in  May, 
I  looked  straight  into  a  big  black  bull's-eye  of  a  spot ;   within  24 


§890]  ELECTRICITY   IN   GASES  727 

hours  the  Abinger  magnetograph  wandered  clean  oflf  ite  acale ;  an- 
fortunately  the  hght  skies  of  that  time  of  year  hid  any  auroral 
display. 

The  electron  stream  of  §  8H4  would  curl  round  the  linos  of  a 
magnetic  field,  no  stronger  than  the  Earth's,  in  complete  cirrle« 
small  enough  to  have  no  visible  diameter  at  50  mileit ;  the  electrons 
do  not  arrive  at  right  angles  to  the  dipping  lines  of  field,  but  with 
a  component  of  their  motion  parallel  to  them,  consequently  they  go 
*  cork-screwing  '  along  the  lines  in  imperceptibly  close  sp'irals,*  the 
auroral  streamers,  all  aimed  northerly,  their  strict  parallelism  to 
the  lines  of  force  becoming  convergence  through  the  effect  of  per- 
spective. 

§  890.  Vapour-filled  discharge  tubes  not  only  enliven  our  streeU. 
but  are  now  making  it  practicable  for  public  authorities  to  light 
their  highways  adequately  for  modern  fast  traffic  ;  so  they  deserve 
a  paragraph. 

Hawksbee  in  1700  rotated  an  evacuated  glass  globe  against  his 
dry  hands,  in  a  wheelwright's  lathe,  and  it  became  full  of  glimmering 
light ;  the  Abbe  Nollet  showed  in  1744  a  yard-long  tube  of  Htcady 
glow ;  Faraday  began  the  systematic  study  of  the  discharge  in 
1838  ;  Greissler  of  Jena  produced,  from  1851  on,  an  infinity  of  pretty 
bits  of  fancy  glass-blowing,  which  were  run  by  the  newly- invented 
sparking  coil,  §  826,  and  these  became  the  simple  little  straight 
narrow  tubes  which  you  have  seen  in  use,  filled  with  H,,  CO,,  etc., 
to  a  pressure  of  about  001  atmo.,  for  the  production  of  the  bright- 
line  spectra  of  these  gases,  §  555.  Attempts  to  use  these  for  publio 
lighting,  about  1900,  were  frustrated  by  the  '  clean-up '  effect, 
by  which  the  gas  presently  disappears  from  the  tube,  condensing 
in  its  walls  (cf.  §  912) ;  or  in  the  case  of  sodium  vapour,  by  the 
blackening  of  the  glass  by  reduced  silicon. 

Neon  is  pure  luck  ;  not  only  is  it  the  brightest  glower  of  all  per- 
manent gases,  but  its  *  clean-up  '  is  a  matter  of  months,  hence 
those  brilliant  red  tubes  carrying  about  60  milliamps.,  at  a  voltage 
to  suit  their  length,  and  giving  about  2  c.p.  per  watt.  Its  colour 
is  against  it  for  general  lighting,  but  as  an  aerial  fog-beacon  it  is 
unequalled,  it  wastes  no  energy  on  the  production  of  short  wave* 
of  inferior  penetrating  power,  §§  568,  569,  and  is  a  score  times  more 
efficient,  per  watt,  than  a  tungsten  wire  lamp  screened  by  ruby  glaas. 

An  anti-sodium  glass  having  been  evolved  (often  lined  with 
fused  borax),  lamps  containing  Sodium  can  be  started  on  neon  and 
warmed  up  in  10  min.  to  vaporize  enough  of  it  to  give  iU  well- 
known  monochromatic  golden  glow  at  3  c.p.  per  watt.  Though 
this  is  still  onlv  one-eighth  of  the  light  output  ideally  obtainable 
from  the  tube,' in  which  there  are  many  actions  going  on  betidee 
the  useful  one  here  described,  it  is  well  ahead  of  the  1  cpjpcf  wmii 
of  street-lighting  tungsten-wire  lamps.  Of  course  it  redooee  all 
colours  to  monochrome. 

Mercury-vapour  tubes,  also  containing  a  little  noon  starter,  warm 


728  ELECTRICITY  [§  890 

up  in  5  min.  to  a  2-7-c.p./watt  efficiency,  and  1500-hr.  life,  at  a 
fair  fraction  of  an  atmosphere  pressure.  Their  pale  blue  light,  see 
Fig.  223,  compound  of  yellow,  strong  green,  and  violet,  contains 
no  red,  and  is  therefore  ghastly ;  but  by  adding  a  little  cadmium, 
its  red  line  shows  up  to  2-5%  and  restores  warmth  to  faces  and 
fabrics,  though  it  reduces  luminous  output  by  1/3,  and  there  is 
15%  red  in  daylight.  The  rival  companies  are  busy  with  further 
additions,  and  the  '  whiteness  '  of  the  output  is  being  immensely 
improved. 

A  trace  of  turpentine,  or  some  such  organic  impurity,  imparts 
that  intriguing  wriggle  to  the  mercury  discharge,  and  can  produce 
striation  and  other  effects. 

The  glass  of  green  tubes  is  faintly  yellow  with  uranium,  which 
not  only  stops  the  violet  line  from  getting  out,  but  also  converts 
it,  and  a  good  deal  of  invisible  ultra-violet,  into  a  general  green 
fluorescent  glow  assisting  the  green  and  yellow  emission. 

§891.  The  mercury  discharge  emits  an  immense  amount  of 
Ultra-VioIet ;  see  Fig.  418,  which  gives  the  chief  bright  lines  of  its 
whole  spectrum;  this  is  refused  egress  by  the  glass,  but  fused- 
silica  glass  passes  it  freely,  and  is  employed  in  lamps  for  Ultra- 
VioIet  Treatment,  for  sterilizing  water,  etc.  Advantage  is  taken  of 
the  heat-resisting  capacity  of  the  silica,  §  172,  to  run  the  lamps 
hard,  their  electrodes  being  cooled  by  metal  fins.  They  need 
watching  with  an  ultra-violet  quantimeter,  §  984,  because  some 
unseen  deposit,  without  changing  the  visible  light,  may  reduce 
the  ultra-violet  output  to  l/6th  :  they  are  restored  by  cutting  open 
and  washing  out  with  nitric  acid. 

Most  of  these  Vapour  Lamps  start  more  easily,  and  run  harder 
on  lower  voltage,  with  a  hot  cathode,  which  is  a  little  stick  of  baryta 
wrapped  in  a  heater-spiral  of  tungsten.  Cold-cathode  types  can 
be  started  by  a  momentary  flash -over  from  a  special  high -voltage 
device ;  small  mercury  lamps  are  often  started  by  tilting  until 
the  liquid  short-circuits  the  electrodes,  passing  a  large  current, 
which  then  starts  the  discharge  as  the  familiar  self-inductance 
flash,  at  '  break.' 

The  heat  of  this  leaves  a  glowing  spot  on  the  mercury  surface, 
and  that  continues  to  function  as  the  hot  cathode. 

Consequently  a  Mercury  Vapour  lamp,  with  this  hot  spot 
wandering  over  a  pool  of  mercury,  and  from  two  to  six  cold  iron 
anodes,  which  electrons  cannot  leave  at  the  running  voltage,  will 
act  as  a  Rectifier  for  1-  or  3-phase  A.C.,  and  very  large  installations, 
2000  kw.  and  upwards,  have  been  put  in  at  docks,  etc.,  where  hauling 
and  lifting  are  much  better  done  by  D.C.  The  pale  lambent  blue 
flame  looks  a  feeble  thing  to  hang  ten-ton  crane-loads  on,  but  I 
recollect  setting  one  to  work  in  1912, to  produce  electrolytic  hydrogen, 
and  it  speedily  hanged  an  indiscreet  individual  who  was  more 
familiar  with  the  arsenical  content  of  fly-papers  than  with  the 
minutiae  of  Marsh's  Test. 


§  892]  ELECTRICITY   IN   GASES 


729 
are 


§  892  This  has  brought  us  to  The  Are— in  fact  these  lamps  *.., 
commonly  called  mercury-arcs— but  now  let  us  work  at  Atmos 
pheric  Pressure. 

The  simplest  practical  way  of  arriving  at  an  Arc  in  to  break  an 
ordmary  lighting  circuit  slowly  :  a  flame  starts  between  the  receding 
contact-pieces,  raises  a  little  cathode-spot  on  one  of  them  to  the 
necessary  high  temperature,  and  the  discharge  continues  destruc- 
tively.    That  is  why  all  switches  have  a  quick  spring-break. 

Lightning  striking  an  overhead  transmission  line  is  led  to  earth 
through  '  Lightning  Arrester  '  spark-gaps  at  intervals  :  it  is  quit© 
a  problem  to  prevent  the  much  lower  pressure  current  from  following 
in  the  trail  the  lightning  has  blazed  ;  it  is  best  broken  into  a  succeii- 
sion  of  gaps  between  cold  knurled  masses  of  brass,  on  which  it  is 
difficult  to  maintain  hot  spots. 

Any  rapid  succession  of  strong  sparks  is  apt  to  coalesce  into  a 
quiet  writhing  arc,  in  which  oxygen  and  nitrogen  bum,  to  NO,, 
with  absorption  of  energy,  but  production  of  abundant  ions,  which 
carry  the  current.  Arcs  40  ft.  long  are  burning  in  Norway,  pro- 
ducing '  artificial  nitrate  '  for  fertilizer. 

Such  machines  as  *  mercury- breaks  '  for  sparking-coils,  where 
an  arc  must  not  be  established,  are  run  in  coal-gas,  which  contains 
no  constituents  capable  of  combination. 

The  Carbon  Arc,  by  far  the  most  intense  artificial  illuminant, 
eight  times  as  good  as  the  next  best,  §  612,  indispensable  in  .search- 
lights, cinema-projectors,  etc.,  is  formed  between  two  ro<b  of 
'  carbon  ' — a  mixture  of  lampblack,  petroleum-coke,  and  pitch, 
calcined  at  1200°  in  carbon-dust.  These  are  left  together  when  the 
current  is  switched  on,  and  incandescent  spots  quickly  develop, 
from  mere  resistance,  at  the  rough  point  of  contact :  now  separated, 
a  discharge  continues  steadily  at  60  volts  or  upwards,  the  arc  being 
2  or  3  mm.  long,  but  much  longer  at  higher  voltages.  The  electronic 
stream  from  the  hot  cathode  (negative  carbon)  blasts  a  '  crater  * 
in  the  positive,  and  the  positive  ions  from  this  keep  the  cathode  hot. 

The  Arc  itself  gives  little  light,  its  spectrum  is  composed  of  the 
bright  lines  of  carbon  and  any  impurities  present.  Any  substance 
put  into  the  crater  volatilizes  and  adds  its  lines,  from  infra-red  to 
the  most  distant  ultra-violet  observable  in  air ;  the  arc  is  our  best 
approximation  to  the  atmosphere  of  the  sun. 

Calcium  fluoride  incorporated  in  the  soft  core  of  the  -f  carbon 
maintains  a  highly  luminous  golden-red  arc,  cerium  fluoride  a  brilliant 
white  one,  useful  also  for  Ultra-Violet  Treatment. 

The  Positive  Crater  is  at  3820°  A.,  which  is  presumably  the 
subliming  temperature  of  carbon  ;  it  is  the  highest  temperature 
we  can  reach  artificially,  and  is  only  exceeded  in  the  vacuum 
furnace  of  §  612,  which  borrows  its  flashes  from  the  sun.  (Jreater 
voltage  merely  lengthens  the  arc,  greater  current  increases  the  area 
of  the  crater,  which  is  1-3  sq.  mm.  per  am|)^re,  pro|)ortionately 
increasing  the  output  of  light,  but  not  the  intrinsic  brightness  jkt 
sq.  mm. 


730  ELECTRICITY  [§  892 

The  crater,  obstructed  as  little  as  possible  by  the  negative  carbon, 
therefore  faces  the  projector  mirror,  §  612,  for  from  it  comes  at  least 
3/4  of  the  total  light,  probably  6  c.p.  per  watt.  Cinema  arcs 
employ  from  80  amperes,  searchlights  twice  as  much. 

It  has  been  maintained  that  the  transport  of  the  electricity  is 
mostly  electrolytic,  the  tetravalent  C,  with  chemical  equivalent 
only  3,  passing  from  crater  to  cathode.  In  vacuo,  perhaps  it  might 
be,  but  the  arc  is  unrecognizable  there  ;  in  air  both  carbons  actually 
burn  away,  the  positive  about  twice  as  fast,  so  it  is  made  1-4  times 
thicker  diameter,  and  both  must  be  fed  in  slowly,  by  hand  or  by 
automatic  mechanism,  at  equal  speed. 

Too  quick  a  feed  encourages  the  arc  to  wander  to  exposed  areas, 
which  burn  freely,  providing  an  excess  of  ions,  the  arc  hisses  and  the 
current  increases  ;  instability  such  as  this,  to  which  it  is  prone, 
has  to  be*  counteracted  by  a  '  ballast  resistance  '  in  series. 

Arcs  with  one  or  both  electrodes  of  Magnetite,  or  of  Tungsten, 
give  a  much  greater  proportion  of  ultra-violet  than  the  carbon  arc — 
for  it  is  the  gaseous  arc  itself  that  does  most  of  this,  not  the  merely 
hot  continuous-spectrum  electrode — and  are  useful  in  localized 
Ultra- Violet  Treatment. 

In  Fig.  324  the  arc  is  shown  deflected  by  a  magnetic  field.  Its 
own  current  produces  a  magnetic  field,  and  with  that,  and  the 
earth's,  the  Arc  is  always  more  or  less  curved,  and  so  got  its  name. 

Electric  Furnaces,  such  as  those  used  for  carborundum  and 
graphite  production,  are,  in  one  stage  of  their  operation,  seething 
with  arcs  between  their  conducting  contents,  carrying  6000  amp. 
and  upwards,  see  §  816. 

The  Pointolite  Lamp  is  a  vacuum  arc  in  which  the  electronic 
blast  from  a  hot  strip  cathode  maintains  a  little  pill  of  tungsten 
near  3000°  C.  It  is  a  luxury  for  fine  optical  work,  but  has  not  a 
quarter  the  brilliance  of  the  carbon  arc. 

§  893.  Passage  of  electricity  through  air  at  atmospheric  pressure 
and  temperature.  Now  we  must  leave  these  fireworks,  though 
with  the  promise  of  even  brighter  later  on,  and  consider  the  passage 
of  a  current  of  electricity  through  air,  or  other  gas,  at  the  ordinary 
pressure  and  temperature. 

The  current  will  probably  be  a  very  small  one,  far  too  small 
for  any  galvanometer,  and  one  just  charges  up  an  insulated  pair 
of  parallel  plates  of  any  convenient  small  size  and  distance  apart, 
and  connected  to  a  gold-leaf  or  other  sensitive  electrometer,  §  735  ; 
and  then  watches  the  rate  at  which  the  leaf  collapses  when  the  gas 
is  between  the  plates,  and  they  are  discharging  through  it. 

When  you  have  perfected  the  insulation,  you  find  that  there  is 
no  leakage  through  air  at  all,  it  is  the  complete  insulator,  dry  or 
wet — for  the  mischief  that  damp  does  to  frictional  electrical  ex- 
periments lies  in  its  spreading  as  a  film  on  the  surface  of  the  insulators. 

More  critical  observation  shows  that  air  does  conduct  just  a 
very  little,   about   a   hundred-millionth   of   an   electrostatic   unit 


I 


§893]  ELECTRICITY   IN   GASES  731 

per  c.c.  per  second.     This  is  traceable  to  it«  ionization   by  the 
radio-activity  of  the  earth,  and  by  *  Cosmic  Rays  '  from  outer  space. 

Water  can  scarcely  conduct  until  substances  are  put  into  it, 
which,  under  its  influence,  §  853,  split  into  -f  and  -  ions,  which 
carry  the  current. 

Air  cannot  conduct  at  all,  until  some  external  influence  splitjt  iu 
molecules  up  into  +  and  —  ions. 

These  do  not  differ  sharply  in  chemical  character  as  do  solution 
ions. 

There  are  various  ways  of  ionizing  a  gas  :— 

{\)  An  intense  electric  field.  We  have  use<i  this  alrea<ly.  at  the 
cold  cathode,  from  which  the  electrons  could  not  othen*n»e  escape. 
In  all  probability,  its  action  is  that  stray  ions,  already  present  in 
the  gas,  are  accelerated  by  the  strong  field,  so  that  they  crash  out 
ions  from  neutral  molecules,  as  in  §  885,  especially  from  gas  in  actual 
contact  with  the  metal. 

An  old-fashioned  easy  way  of  getting  an  intense  field  is  to  use 
an  Electrified  Point. 

Look  at  the  pointed  end  of  the  oval  conductor  in  Fig.  313 :  the 
lines  are  crowded  together,  sho\*nng  an  intense  electric  field. 
Roughly,  one  can  think  of  the  end  as  the  sphere  of  §  727  ;  the 
potential  this  produced  was  inversely  as  the  radius  from  its  centre. 
Let  a  sharp  needle-point  with  a  hemispherical  end  perhaps  0-(K)l  cm. 
radius  be  attached  to  a  conductor  at,  say,  10  e.s.  units  of  potential. 
At  radius  0-01  cm.  around  the  point  the  potential  is  roughly  one- 
tenth  of  this,  a  drop  of  9  units  in  0-009  cm.,  at  the  average  rate  of 
1000  e.s.  units  (or  300,000  volts)  per  cm.,  far  more  than  the  air  can 
sustain.  Fix  a  needle  on  the  prime  conductor  of  an  electrical 
machine  and  turn  the  handle,  electricity  makes  a  quiet  or  slightly 
hissing  escape  ;  there  is  no  spark,  but  in  the  dark  a  tiny  bluish 
glow  is  seen  at  the  point. 

Now  (a)  any  insulated  conductor,  e.g.  an  elect rascope  cap,  held 
near  the  point,  gets  a  charge  :  remember  the  use  of  sharp-pointed 
combs  to  collect  charge  from  the  plates  in  electrical  machines, 
§  715.     Evidently  the  air  is  conveying  electricity. 

(6)  A  silk  thread,  or  a  candle  flame,  held  near,  is  blown  aside  by 
a  Wind  from  the  discharging  point. 

Thus  there  is  actual  Convection  of  charged  jMirticles,  like  the 
motion  of  the  ions  in  electrolysis,  but  much  faster  in  the  more 
mobile  fluid. 

(2)  The  splashing  and  spraying  of  itxiter,  cf.  §  897. 

(3)  Chemical  action,  especially  combustion. 

Gilbert  found  that  flame,  and  the  fumes  just  above  it,  conducted 
away  electrification,  §  702. 

TRY  holding  a  flame  under  the  spark-gap  of  a  Wimshurst,  or  a 
spark-coil. 


732  ELECTRICITY  [§  893 

Meteorologists  use  a  little  flame,  on  the  end  of  a  fishing-rod,  to 
ascertain  the  potential  of  the  air  around  it,  ions  passing  freely  in 
and  out  to  an  electrometer.  The  hot  exhaust  of  an  aeroplane 
keeps  it  at  the  same  potential  as  the  surrounding  air. 

Chemical  combination  being  combination  of  ions,  this  action  of 
flames  is  only  to  be  expected. 

H.  A.  Wilson  sprayed  various  salts  into  a  flame,  and  found  they 
electrolysed  with  exactly  the  same  charge,  1  faraday  per  gm.-moL, 
as  in  solution. 

He  measured  the  actual  speeds  of  the  ions  in  a  hot  flame,  and 
found  the  +  ion,  metal,  travelled  at  about  60  cm. /sec.  for  1  volt 
per  cm.  driving  force  (contrast  §  855),  but  the  —  ion  travelled  1000 
cm. /sec.  This  is  much  too  fast  for  OH,  and  the  conclusion  is  that 
at  first  it  is  a  free  electron,  only  getting  caught  in  an  ion  of  atomic 
size,  or  larger,  after  some  length  of  flight. 

This  throws  light  on  the  size  of  the  ions  in  gases  generally ; 
they  are  atomic,  or  larger — even  up  to  water  drops,  §  887 — and  have 
one  or  more  electrons  too  many  or  too  few. 

Ions  are  easily  filtered  out  of  air  by  cotton  wool. 

(4)  Incandescent  solids  ionize  air,  the  free  electrons  of  §  881  being 
speedily  caught. 

(5)  X-rays  and  (6)  radioactive  substances  put  a  quick  stop  to 
frictional  electrical  experiments  ;  see  Chapters  LIV  and  LV. 

(7)  The  photo-electric  effect  is  described  in  Chapter  LVI. 

§894.  The  current  through  an  ionized  gas.  When  X-rays,  for 
instance,  are  passing  athwart  the  air  space  between  two  small 
metal  plates  which  are  connected  to  a  charged  condenser  and 
attached  electrometer,  a  discharging  current  immediately  flows 
across.     As  the  voltage  between  the  plates  is  raised  (by  having  used 

more  and  more  cells  to  charge  the  con- 
denser) the  current  increases,  but  not 
proportionally.  In  fact,  after  a  certain 
limiting  potential  difference  has  been 
reached,  never  more  than  1000  volts 
VOLTAGE  PER  CM.  jzo.ooo       pcr  cm.   of  air  gap,  the  current  does 


I 


Fig.  397.  ^^^  increase   at   all,    and   is   called    a 

saturation  current.  Fig.  397. 

And  now  if  the  air  gap  is  lengthened  and  the  voltage  per  centi- 
metre length  kept  the  same,  the  saturation  current  increases,  almost 
proportionally  to  the  quantity  of  air  between  the  plates  and  exposed 
to  the  ionizing  influence. 

A  satisfactory  explanation  is  that  the  +  and  —  ions,  as  soon  as 
formed,  begin  to  move  towards  the  —  and  +  plates,  with  speeds- 
about  proportional  to  the  forces  acting  on  them,  i.e.  to  the  field, 
the  volts  per  cm. 

In  weak  fields  the  motion  is  slow,  and  the  majority  get  time  to 


J 


§  895]  ELECTRICITY   IN  GASES  733 

recombine  spontaneously  into  neutral  molecules,  hence  only  a  few 
give  up  theu-  charges  to  the  plates,  and  the  current  U  small. 

Strong  fields  drag  the  ions  out  so  fast  that  few  get  the  chance 
of  recombming  ;  since  the  total  production  of  ions  depmda  on  the 
external  ionizing  influence,  a  still  stronger  field  will  gather  no  more 
ions  ;  the  current  is  *  saturated.* 

The  wider  the  space  the  more  ions,  hence  the  greater  the 
maximum  current  obtainable. 

In  another  experiment  a  very  strong  field  is  put  on  at  a  definite 
interval  after  the  ionizing  rays  have  been  cat  off,  and  the  total 
discharge  obtained  is  measured  on  an  electrometer,  lu  diminu- 
tion with  increase  of  time-interval  enables  the  rate  at  which  the  ions 
have  been  naturally  recombining  into  neutral  moleculee  to  be 
calculated. 

Presently  the  curve  turns  up,  and  the  current  increases  with  a 
rush  :  the  intense  field  is  accelerating  the  ions  up  to  the  speed  at 
which  they  ionize  other  atoms  by  the  crash  of  collision.  §888. 
and  these  are  accelerated  in  turn,  so  that  their  numbers  increase 
in  '  snowball '  fashion,  and  from  plate  or  point  (of  (1)  §  893)  fliea  a 
Spark. 

§  895.  Thus  the  actual  Electric  Spark  sometimes  noticeably 
hangs  fire,  while  ions  are  being  multiplied  up  in  the  gap  ftxnn  the 
500  pairs  per  c.c.  normally  present  in  the  atmosphere.  The  strong 
field  causes  a  hissing,  and  in  the  dark  you  see  a  Brush  quivering 
out  from  the  Positive  electrode,  or  a  tiny  purplish  Glow  on  a  Negative 
point.  These  are  outgrowths  of  the  silent  Corona  Discharge,  which 
limits  the  voltage  of  high-tension  electric  mains,  or  produces  Oiooe. 

Anyone  who  has  seen  good  rubber  tubing  fall  off  an  ozoniier  after 
very  few  seconds,  will  scarcely  need  to  be  told  that,  in  any  concentra- 
tion, ozone  can  be  not  merely  a  nasty  smell  but  a  corrosive  nuisance, 
and  a  depressant  poison  to  operators  in  the  X-ray  room,  where 
corona  must  be  avoided  by  doing  away  with  points  and  thin  wires, 
§  893,  (1),  and  ventilation  must  be  good. 

If  you  examine  very  small  Sparks  under  the  Microscope  you  will 
see  just  the  same  structure  that  appears  in  a  wide  glass  tul>e  as  the 
air  pressure  is  reduced  to  a  few  mm.  of  mercury,  and  the  thin 
straggling  spark,  which  began  at  a  few  cm.  pressure,  widens  out  to 
fill  the  tube. 

The  cathode  is  enveloped  in  a  Purple  Glow,  not  touching  it,  but 
separated  by  a  narrow  Dark  Space.  It  is  in  this  Spw^  that  the 
electrons  fly  free ;  it  fills  the  high -vacuum  tube.  Then  oooiea  a 
longish  Scarcely  Luminous  Part,  which  you  see  aa  a  ver>*  thin  bit 
at  the  negative  end  of  all  thin  sparks,  such  aa  thoee  from  an 
electrophorus ;  and  then  the  rest,  up  to  any  length,  is  luminous 
pink  *  Positive  Column.' 

No  names  are  given  here,  because  the  close  study  of  this  is  now 
of  only  historic  interest,  and  your  examiners  are  fed  up  with  descrip- 
tions and  figures  of  it  copied  out  of  books. 


6 

12 

8 

22 

16 

40  (12-cm.  knobs) 

35 

90 

55 

135 

85 

175 

734  ELECTRICITY  [§  895 

The  minimum  sparking  voltage  in  air  is  350,  which  represents 
the  field  necessary  to  accelerate  ions  up  to  '  ionization  by  collision,' 
and  is  valid  for  low  pressures  only.  At  atmospheric  pressure  it 
takes  about  1500  volts  to  produce  the  smallest  sparks. 

Sparking  Potentials  in  Kilovolts  (peak)  in  the  open  air. 

spark  gap.  Between  needle  points.  Between  spheres. 

2  mm.  ?  5-5  (2-cm.  knobs) 

5     „ 

1  cm. 

2  „ 
5     „ 

10     „ 
20     „ 

The  figures  for  needle-points  are  unreliable,  on  account  of  the  varying 
amounts  of  ionization  they  promote,  §  893  (1). 

The  Sparking  P.D.,  at  varying  lengths  and  pressures,  is  simply 
proportional  to  the  mass  of  air  in  the  gap. 

Thus  a  magneto  on  test  must  show  a  spark  several  times  longer 
than  the  plug-point  gap,  for  it  has  to  jump  this  in  compressed 
air. 

The  meandering  course  of  a  Long  Spark  is  dependent  upon  the 
supply  of  ions — the  spark  must  travel  where  it  can,  see  §  898  and 
effect  of  flame,  §  893  (3). 

A  stream  of  Strong  Sparks  in  air  often  coalesces  into  the  quiet 
writhing  Arc  of  §  892,  woolly  with  flame  burning  to  NOg  ;  which, 
and  not  Ozone,  is  the  smell  of  sparks  in  air. 

The  Energy  of  the  Spark,  apart  from  this  (which  of  course  does 
not  occur  in  coal-gas  or  other  non-self -combustible  gases),  is  mostly 
spent  in  heating  the  small  quantity  of  air  along  its  track,  and  the 
abrupt  expansion  of  this  starts  the  sharp  sound-wave  we  hear  as 
a  crack.  From  the  long  tortuous  lightning  flash  comes  a  long 
irregular  crackle  ;  some  '  reaches  '  of  it  will  happen  to  lie  their  whole 
length  across  the  line  joining  them  to  the  ear,  so  all  the  wave 
deriving  from  that  length  arrives  at  once,  and  makes  a  louder  snap  ; 
refracted  and  reflected  among  clouds  the  whole  softens  into  the 
roll  of  Thunder. 

The  Quantity  passing  in  such  sparks  as  one  commonly  gets  from 
an  electrophorus  is  probably  a  few  tenths  of  a  microcoulomb. 
Filled  into  a  ley  den  jar  of  tinfoil  surface  about  fitting  your  hand, 
until  you  could  draw  a  quarter-inch  spark  with  the  other  knuckle, 
you  might  get  4  microcoulombs,  carrying  about  half-a-million 
ergs,  say  half  a  kilogram-centimetre,  of  kick  ;  and  you  would  quite 
likely  be  content  with  that  modest  dose. 

The  effect  of  Quantity  in  a  Spark  is  best  seen  by  working  a 
Wimshurst  with,  and  without,  attached  leyden  jars,    try  it. 

Connected  across  the  gap  of  a  sparking  coil,  §  826,  a  jar  shortens 
and  fattens  the  spark,  for  its  capacity  is  too  great  to  be  filled  to  very 
high  potential  by  the  small  quantity  available,  so  one  has  to  reduce 


I 


§  896]  ELECTRICITY   IN   GASES  735 

the  gap,  until  it  is  suddenly  filled  with  the  same  energy  as  was 
previously  drawn  out  thin. 

A  20-cm,  spark  carrying  a  milli-coulomb,  representing  2000  of 
the  little  jar  shocks  suggested  above,  is  usually  treated  with  distant 
respect :  '  the  Grid  '  will  provide  you  with  100  per  second,  and 
without  the  quantity  limit ;  stepped  up  to  a  milHon  volts,  and  backed 
by  1000  h.p.,  the  writhing  storm  of  7-ft.  sparks  is  an  impressive 
spectacle. 

Until  one  thinks  of  real  Lightning. 

§  896.  Atmospheric  electricity.    On  a  fine  day,  in  the  open,  the 

ground  is  negatively  charged,  with  about  1  milli-coulomb  per  sq. 
kilometre,  and  there  is  a  vertical  potential  gradient  of  about  lOO 
volts  per  metre.  This  means  that  round  your  head  is  a  region 
nearly  200  volts  in  potential  above  that  of  the  soil :  it  does  not  imply 
that  there  is  any  charge  there,  §  728. 

Indoors,  under  trees,  or  among  buildings,  virtually  *  inside 
a  closed  conductor,'  there  is  no  potential  gradient :  above  them  it 
is  increased  for  a  distance,  to  catch  up  with  the  general  run  in  the 
open. 

Whether  it  has  any  physiological  effect  on  us,  or  on  plants,  has 
never  been  decided.  An  electro-culture  of  early  vegetables  under 
glass  has  evolved,  from  experiments  on  it,  but  this  is  probably  an 
entirely  different  effect,  connected  with  active  nitrogen  being  driven 
into  the  soil. 

It  varies  during  the  24  hr.,  its  minimum  at  4  a.m.  being  roughly 
half  its  maximum  at  7  p.m. 

It  is  about  twice  as  great  in  winter  as  in  summer  ;  and  is  increased 
in  fog,  when  charged  ions  get  attached  to  loads  of  rubbish  and  have 
to  tow  them  about,  so  that  an  atmospheric  current  Hows  leas 
readily. 

There  are  minute  traces  of  radioactive  materials,  §  944,  in  the 
soil  everywhere,  and  in  the  air  above  it,  and  these  produce  about 
4-3  pairs  of  ions  per  c.c.  per  second.  They  are  missing  over  the  sea, 
but  ever3rwhere  there  are  ions  due  to  cosmic  radiation,  §  947 ; 
which  produces  another  2  pairs  per  c.c.  per  second,  and  many  more 
at  high  altitudes.  The  result  is  that  there  are  somewhere  alx)ut 
1000  ions  per  c.c.  normally  present  in  the  air,  but  usually  more 
positive  than  negative.  Under  a  potential  gradient  of  100  volts 
per  metre  these  move  at  about  1  cm.  per  second. 

The  potential  gradient  disappears  at  10  km.  height,  where  the 
potential  has  reached  about  a  miUion  volts  ;  this  means  that  1k»1ow 
this  level  are  enough  positive  ions  for  all  lines  of  force  from  the  ground 
to  end  on,  and  they  are  steadily  drawn  downwards,  producing 
the  air-earth  fine  weather  current  of  2  micro-amp.  per  sq.  km.,  or 
about  1000  amp.  for  the  whole  surface  of  the  earth. 

This  current  would  discharge  the  whole  negative  charge  of  the 
surface  in  500  sec.  :  where  is  the  compensating  process  which 
prevents  this  ? 


736  ELECTRICITY  *  [§  897 

§  897.  Thunderstorms.  We  have  seen  in  §  317  that  a  thunder- 
cloud is  produced  by  the  violent  uprush  of  warm  moist  air,  which 
begins  to  condense  copiously  at  about  a  mile  high,  and,  warmed 
by  the  evolved  latent  heat  of  condensation,  soars  up  from  2  to  5 
miles  higher  yet,  cf .  Fig.  97,  which  was  a  friendly  pup. 

There  is  plenty  of  water  in  a  thundercloud,*  and  wind  rushing 
through  it.  Splashing  of  water  ionizes  the  air,  as  near  waterfalls, 
§  893  (2).  In  1842  Lord  Armstrong  blew  wet  steam,  from  a  boiler 
on  glass  legs,  through  tortuous  nozzles,  which  trapped  most  of  the 
wet  and  let  the  vapour  escape,  and  the  boiler  became  strongly 
negatively  electrified. 

Theory  has  got  no  further  ;  as  a  general  rule  the  wet  lower  part 
of  the  thundercloud  is  negative  and  the  top  is  positive,  but  sometimes 
they  are  the  other  way  up. 

The  blowing  of  the  top  two  or  three  miles  above  the  bottom  is 
the  lifting  of  the  top  plate  of  the  electrophorus,  and  of  course  in- 
creases the  P.D.  between  the  charges,  the  necessary  energy  coming 
from  the  latent  heat  of  the  condensed  vapour  :  the  stormcloud  is  a 
great  steam  engine  and  influence  machine  combined. 

The  potential  gradient  near  the  ground  changes  from  100  to 
over  10,000  volts  per  metre,  now  usually  negative,  but  changing 
suddenly.  People  get  thunderstorm  headaches.  It  increases 
until  tree-tops  and  lightning-conductors  and  all  sharp  points,  such 
as  the  tips  of  the  grass,  are  spraying  off  a  coronal  discharge,  pro- 
ducing a  dense  blanket  of  ions,  and  ozone,  which  the  rain  brings 
down.  Exceptionally  we  have  seen  a  lambent  flame,  as  of  burning 
alcohol,  running  on  the  path,  flickering  from  its  edging,  leaping  up 
from  every  splash  of  the  rain,  a  positive  brush  discharge,  St.  Elmo's 
fire  come  down  from  the  masthead,  in  response  to  a  field  of  possibly 
a  quarter-million  volts  per  metre. 

This  abundant  supply  of  ions,  from  point  discharge,  and  from 
splashing  rain,  carries  up  a  Neutralizing  Current  to  the  cloud,  which 
may  even  reach  an  ampere,  or  2  or  3. 

§  898.  Lightning.  Failing  this,  or  expediting  matters,  comes 
a  Lightning  Flash,  bringing  20  coulombs  down  a  mile  and  a  quarter, 
under  the  drive  of  a  thousand  million  volts. 

The  charge  is  regenerated  in  from  10  to  20  seconds  :  three  a  minute; 
maintains  an  ampere,  which  at  that  pressure  of  course  means  a^ 
million  kilowatts,  double  the  working  capacity  of  the  biggest 
supply  station  in  this  country,  as  the  output  of  a  moderate  thunder-^ 
storm  in  lightning  flashes  to  earth  alone,  leaving  quite  out  of  account 
the  non-spectacular  neutralizing  current  just  mentioned,  and  also 
what  is  going  on  at  the  top  of  the  cloud. 

For  meanwhile  the  uprush  is  quite  likely  throwing  hailstones 
up  in  the  cloud,  sometimes  over  and  over  again,  §  317  (it  is  com- 
puted that  a  good- sized  hailstone  has  had  a  run  of  nine  miles),  and 
from  the  top  it  sweeps  its  remaining  moisture  frozen  into  the  fine 
snow- dust  of  the  '  anvil '  plume. 


^  S98]  ELECTRICITY   IN  GASES  757 

It  is  a  far  cry  from  the  oppositely  charged  top  of  the  storm  to 
he  conducting  Heaviside  layer,  §  984,  80  miles  high  or  thewabouU 
l)ut  the  upper  air,  thanks  largely  to  cosmic  radiation,  ig  toorai  of 
times  as  conductive  as  below,  and  1000  million  volt«  ia  penoMive. 
and  a  good  deal  of  quiet  discharge  goes  on  there,  probabiroften  thi 
source  of  the  far-distant  Summer  Lightning. 

Discharge  from  end  to  end  inside  the  cloud  itaelf  must  be 
difficult,  for  the  ions  are  all  water-laden  and  heavy  to  move  It  ia 
not  water  floating  in  the  air  that  makes  electrostatic  experimenU 
go  amiss,  but  a  condensed  water  film  on  the  apparatus,  frequently 
hygroscopic.  ^ 

Similarly,  lightning  seems  less  likely  to  strike  through  thedenaert 
rain  :  this  conveys  its  own  charge  down  with  it,  but  not  a  great  deal. 

One  never  sees  the  tributary  system  of  a  lightning  flash  which 
might  be  expected ;  if  it  exists  it  remains  hidden  in  the  cloud ; 
but  lightning  occasionally  branches  downwards,  and  most  of  these 
branches  end  in  the  air.  It  is  established  that  this  is  a  downcoming 
positive  discharge,  and  it  reminds  one  of  the  short -stalked  spreading 
brushes  that  distinguish  the  positive  end  of  a  Wimshurst  working 
in  the  dark.  The  negative  end  bears  a  small  point  of  steady  glow  ; 
and  the  negative  lightning  flash  is  unbranched. 

The  growth  of  a  long  spark. 

The  wandering  crinkly  path  of  all  sparks  of  any  length  prompta 
one  to  inquire  why  they  are  not  content  with  the  shortest  way, 
and  lightning  offers  the  best  prospect  of  an  answer. 

To  start  a  spark  at  all,  from  a  cold  conductor  in  air  at  atmosphere 
pressure,  experimentally  requires  a  potential  gradient  of  about 
30,000  volts  per  cm.  This  would  exist  at  the  surface  of  a  sphere 
of  250  m.  radius  charged  with  20  coulombs.  A  current  therefore 
breaks  out  in  one  direction,  just  like  a  crack  running  through  glaat, 
and  advances,  charging  the  air  as  it  goes,  and  so  altering  the  original 
distribution  of  the  field  of  force,  until  it  is  pulled  up  at  a  point 
where  the  amount  already  arrived  is  insufficient  to  produce  the 
necessary  30,000  volts/cm.  There  it  may  have  to  halt  for  rein- 
forcements, before  continuing  the  advance,  again  not  neoeesahlv 
in  the  geometrical  direction  of  supposed  maximum  strain, but  throuip 
the  most  '  friendly  country  '  where  pairs  of  ions  happen  already 
to  have  formed  in  most  abundance,  §  895,  in  response  to  the  strong 
field,  ete. 

So  the  spark  does  actually  work  its  way  along,  by  a  niooeHkm 
of  rushes,  where  best  it  can  ;  even  sometimes  having  to  break  into 
branches  which  dry  out,  like  a  desert  river. 

By  oscillograph,  §  884,  it  has  been  found  that  a  single  lightning 
flash  may  involve  as  many  as  five  partial  discharges,  and  photo- 
graphs  in  a  swinging  camera  have  sho^^-n  succoiBive  attampts 
reaching  further  and  further  along  the  track.  Theee  brief  partial 
discharges  may  be  as  long  as  1/3  sec.  apart,  the  impression  ooeoftan 
gets  of  a  flash  having  two  or  three  shots  at  it  is  perfectly  corract. 

BB 


738  ELECTRICITY  [§  898 

Once  blazed,  the  ionized  trail  is  an  easy  path,  for  a  second  or  two, 
and  a  swinging  camera  has  shown  three  identical  flashes  in  succession 
from  a  vigorous  cloud,  and  a  fourth  breaking  off  along  a  new  track 
half-way.  But  they  are  not  oscillatory  discharges,  §  833,  because 
the  resistance  is  too  great ;  if  a  return  half -wave  shows  at  all  it  is 
of  trifling  amplitude. 

Sir  C.  V.  Boys,  using  spinning  camera  lenses,  photographs  a  pilot 
flash,  of  limited  length,  working  its  way  down  from  tbe  cloud,  and 
a  great  return  discharge  striking  up  from  the  earth  along  its  trail,  at 
speeds  of  advance  about  30,000  km. /sec,  contrast  §§840,  901. 

Sheet  lightning  is  usually  lit-up  cloud  :  '  globe  lightning,'  a  slow- 
moving  '  ball  of  fire,'  is  well  authenticated,  and  is  usually  asserted 
to  be  a  still-burning  spark  of  oxygen  and  nitrogen  dropped  off  the 
main  flash,  which  causes  plenty  of  this  combination. 

Some  also  give  the  discharge  through  moist  air  the  credit  for  much 
of  the  atmospheric  ammonia,  and  as  rain  water  contains  from  1/4  to 
1  mg.  of  nitrogen  per  litre,  you  can  easily  calculate  that  the  soil  gains, 
from  atmospheric  electricity,  from  1  to  5  lbs.  of  fertilizing  combined 
nitrogen  per  acre  per  annum,  in  this  country,  and  probably  much 
more  in  the  tropics  where  thunderstorms  flourish.  That  means 
from  10  to  40  lbs.  of  nitrate  of  soda,  an  amount  which  any  farmer 
regards  with  respect. 

My  canny  friend  C.  T.  R.  Wilson  did  not  obtain  his  estimates 
of  the  appalling  power  of  lightning  by  standing  underneath  it.  The 
charge  on  the  cloud,  averaging  a  mile  high,  sends  exactly  the  lines 
of  force  to  the  flat  conducting  surface  of  the  earth,  the  line  drawn 
across  the  middle  of  Fig.  302,  as  it  would  to  a  mirror-image  of  itself 
a  mile  underground,  and  this  vertical  system  has  an  Electrical 
'  Moment '  =  (charge  X  distance  apart).  It  presents  its  '  broadside  ' 
to  anywhere,  and  accordingly  causes  a  vertical  electrostatic  field 
(charge  X  2  miles)  -f-  d^  from  storm,  cf .  §  690.  This  induces  a 
charge  up  into  a  suspended  aerial  of  calculated  capacity,  such  as 
a  galvanized-iron  roofing  sheet ;  and  this  discharges  through  one 
of  his  electrometers  when  the  lightning  flash  discharges  the  cloud 
temporarily,  destroying  the  field.  Interested  villagers  for  miles 
around  supply  time  and  place  of  storms  to  the  patient  watcher  in 
Cambridge  ;  and  you  see  his  calculation  is  rather  simple — and  pretty. 

§  899.  The  electrical  balance  sheet  of  the  atmosphere. 

Atmospherics  are  notoriously  due  to  electrical  discharges,  either 
alow  or  aloft :  they  sign  themselves  on  the  oscillograph  usually 
as  uni- directional  jolts,  or  with  a  very  small  reverse  half -wave 
as  just  mentioned;  their  duration  may  be  3/100,000  sec,  which 
is  long  enough  for  short-wave  wireless  to  be  blissfully  free  of  them. 
Fishing  round,  one  can,  of  course,  frequently  find  up  a  storm  :  and 
by  concerted  action  between  directional  stations  wide  apart, 
atmospherics  are  found  to  originate  mostly  in  the  main  thunder- 
storm belts  margining  the  tropics,  and  it  is  computed  that  altogether 
about  1700  storms  are  going  on  in  the  world  at  this  moment. 


§  900]  ELECTRICITY   IN   GASES  739 

Crediting  them  with  an  ampere  apiece,  recollecting  they  are  mostly 
positive  on  top,  but  allowing  for  a  proportion  being  upside-down  and 
cancelling  out  in  pairs,  you  see  you  can  soon  find  up  the  mining 
1000  amp.,  and  so  balance  the  budget  of  Atmospheric  Electricity 
quite  as  well  as  most  finance  ministries  can  balance  theirs  at  the 
present  time. 

§  900 .  Lightning  Conductors.  The  suddenness  of  a  simrk  diiicharge 
makes  it  comparable  to  a  half-wave  of  an  A.C.  of  high  frequency— 
quite  likely,  from  the  duration  of  atmasphericH,  2n,000  for 
lightning — and  it  obeys  the  rules  of  §  834.  Ohmic  resistance  is  a 
minor  .hindrance.  Inductance  is  the  great  obstruction;  the  Im- 
pedance of  the  circuit  is  almost  entirely  Reactance. 

Lightning  Conductors  mainly  occupy  themselves  in  snreading, 
from  their  sharp  points,  a  cloud  of  ions  which  tempers  down  the 
intense  potential  gradient  under  a  thunder-cloud.  Plainly,  for  that 
purpose,  they  should  be  continuous  and  well  earthed,  and  a  colony 
of  them  mostly  fends  off  lightning  :  if  one  does  get  struck,  straight- 
ness,  for  minimum  inductance,  will  be  its  saving  grace. 

Thickness  of  metal  is  wanted  to  avoid  getting  broken  by 
accidents,  or  to  counter  corrosion  ;  any  wire  can  carr>'  20  coulomba, 
but  a  break  caUs  forth  a  blaze  which  burns  it  wider.  Of  courae 
there  should  be  a  solid  spike  to  pierce  that  blazing  mass  of  air ; 
a  dozen  yards  of  our  aerial  disappeared  one  day,  but  the  inshore 
ends  merely  kicked  and  carried  on.  #      «•  i 

A  Uehtning  conductor  is  reputed  to  protect  an  area  of  rf^ms  equal 
to  its  height ;  so  among,  but  not  under,  high  trees,  should  be  a 
safe  place  Inside  a  '  closed  '  earthed  conductor  is  of  course  the 
best  one  can  do,  so  magazines  etc.  are  ..^11  ^""^*^**\«\ .»"„*  T^^ 
conductor  system,  and  iron  sheds  run  little  ^''[l^'^^ .^I'^'l' "^'^^^ 
engines  or  steel-masted  ships.  One  doubts  if  he  r»»;^^^>'«»J 
a  car  are  in  any  way  protective-the  case  is  ^^ffe-^^t  fmm  what 
occurs  in  dry  weather,  when  an  open  stream  of  l^^J^* J^^^, 
poured  into  or  out  of  a  tank  lorry,  and  it  becomes  ver>-  nece«ar) 

to  earth  it  by  a  trailing  chain  ^.ntjatirAllv    liiihtnina 

Caught  in  the  open,  don't  argue  ^^at,  statistical^ .  l^htn^^^ 

casualties  are  few ;   and  that,  considering  ^^«j;^~»«^ j^^'  Jl 
direct  damage  done  by  it  is  ridiculously  small .  lie  Mat  ana  gee 

^tightning  striking  a  tree  probably  travels  ;|«-"  ^^^^^^^^^ 
cambium,  destroying  it  and  Fod^cing  steam  and  blo^^^^^^^^^^^ 

of  bark  or  tearing  them  ^^  .^l^^^T^^^^'^'^^^^lnt^^^ 
trusted,  however,  to  keep  to  the  t^"t'.^^;,X7;"i  j^,  Ui^^^^ 
the  surface  of  a  large  ^yl^der  escapes  l^m^^  ^^,;^, 

netic  rings  of  force  which,  filling  the  «>. Under  ^;^^^^  ^^  ^ 

it  if  it  had  travelled  along  the  axis    i.e.  ^he  Y'^^'^';*^!*;;  earth 
less  inductance.     Consequently,  l^h^^^^^ 

down  over  the  wet  foliage,  and  it  is  J^f^  ^;*^"  ^  y^  j^^^a  on  vou 
underneath  :  the  '  tree-like  markmgs    that  might  De  lou 


740  ELECTRICITY  [§  900 

would  have  less  reference  to  the  arboreal  species  than  to  the  con- 
ductance and  course  of  the  subjacent  vessels. 

§901.  Electric  Shock.  Would  that  voltage  kill  you?  is  a 
question  almost  always  asked  as  a  noisy  torrent  of  foot -long  sparks 
crackles  overhead.  The  fact  is,  that  voltage  has  very  little  to  do 
with  you,  being  mainly  occupied  in  clearing  a  way  through  the  air, 
a  very  few  milUgrams  of  which  undoubtedly  get  a  severe  shaking 
up.  After  all,  if  Walter  Raleigh  had  struck  a  match  to  light  his 
pipe,  the  Elizabethan  Court  would  have  had  no  eyes  left  for  William 
Gilbert's  sparks. 

Men  have  been  killed  by  100  volts,  and  men  have  got  up  and  walked 
home  after  a  flick  from  the  tail  of  1000  million  volts.  It  is  the  amount 
of  electrolysis  initiated  in  the  nerves — ^which  carry  their  messages 
as  an  electrochemical  impulse  at  100  m.  sec. — that  causes  trouble, 
i.e.  it  is  the  quantity  of  electricity  getting  in  under  the  skin,  the  great 
protector  against  everything.  The  resistance  of  this,  20,000  ohms 
between  dry  hands,  §  787,  can  be  brought  down  to  a  tenth  by  per- 
spiration, or  very  low  by  large  contact  pads,  wet  with  saline  or  almost 
any  moisture  :  once  inside,  nerves,  and  vessels  full  of  blood,  are 
good  conductors. 

One  gets  used  to  casual  100-volt  contacts,  but  few  people  can 
tolerate  upwards  of  150  volts  D.C.  without  complaining  of  a 
burning  sensation ;  A.C.  is  perceptible  at  lower  voltages  and  tickles 
more.  Higher  domestic  voltages  can  be  painful,  even  alarming 
if  your  hands  won't  let  go  when  the  spasmodic  contraction  of 
arm  and  body  muscles  would  naturally  throw  you  off.  It  is  a 
question  of  what  quantity  is  likely  to  be  driven  in ;  one  unhappy 
friend  managed  to  get  electrocuted  in  his  bathroom  by  current 
from  feet  to  hand  (along  the  spine)  on  a  faulty  switch,  whereas  in 
another  the  family  merely  passed  the  word  not  to  turn  off  water 
and  light  at  once. 

A  maintained  current  of  0-02  ampere  is  unbearable  and  dangerous 
and  0-1  amp.  is  deadly. 

Quite  thin  dry  fabric  is  proof  against  the  600  volts  of  the  '  third 
rail.' 

Sparks,  of  course,  add  the  pain  of  burns,  and  the  high-frequency 
discharge,  which  else  causes  no  shock,  §  835,  is  no  exception  in  this 
respect.  Again,  a  jab  from  a  sparking  coil  with  a  milliamp.  output 
is  a  different  matter  from  a  blast  from  a  modern  X-ray  apparatus 
which  lights  up  the  whole  room.  Yet  Shock,  as  always,  is  largely 
what  the  victim  makes  of  it,  and  the  lady  assistant  who  caught 
the  last  carried  on,  after  tea  and  attention  to  the  burns. 

Naturally,  there  are  limits;  robust  constitution  and  hardiness 
of  nerve  don't  save  Dartmoor  ponies  from  being  killed  by  lightning, 
but  whenever  the  patient  can  be  kept  warm,  and  artificial  respiration 
persisted  in,  that  treatment  should  be  tried  with  a  good  deal  of  hope. 


ELECTRICITY   IN  GASES  741 


EXAM   QUESTIONS.  CHAPTER   LI  II 

The  first  eight  §§  of  this  chapter  are  indisponsahlo  in  the  uniiervUnding 
of  the  chapters  that  follow.  The  whole  of  the  mathornaticH  is  in  f  »H3,  you 
will  simply  recognize  old  friends  with  new  faces:  iu  ropro<luction  haa  not 
hitherto  been  asked  for.  The  rest  of  the  chapter  deals  with  the  manifold 
activities  of  the  electron,  and  resolves  itself  into  short  definite  diMM-riptions 
of  several  things  with  which  you  almost  certainly  have  alrea<iy  somo  «<>qimint. 
ance.     §  896  to  end  is  off  the  main  track  and  will  detain  you  only  at  ploasuro. 

The  Questions  are  few,  but  they  are  uncommonly  soarchinf;.  and  in  thaw 
regions  of  modern  physics  they  may  be  expected  to  increase  in  frequency. 

L  On  what  ejqperiments  would  you  rely  to  prove  the  fundamental  identity 
between  electricity  produced  by  friction  and  that  obtaincnl  from  a  volt*io 
cell? 

2.  How  are  cathode  rays  produced  ?  Describe  experiments  to  ahow  thai 
their  beam  behaves  like  a  current  of  electricity.  What  concluaiooa  aa  to 
their  nature  can  be  drawn  ?     (  X  3) 

3.  How  are  cathode  rays  produced  and  utilized?  How  'm  it  concluded 
that  they  are  negative  particles  moving  very  fast  ?     (  X  2) 

4.  Give  a  general  explanation  of  how  electrical  current  trax'enea  a  gaa, 
exhausted  to  various  low  pressures.     (  X  2) 

5.  Describe  an  electric  arc,  and  give  a  diagram  of  its  circuit.  IHacuaa  ita 
'  striking.'  What  properties  are  desirable  in  protective  goggles  for  uae  with 
it?     (x2) 

6.  To  what  is  the  electrical  conductivity  of  a  gas  due,  what  tests  show  it, 
how  is  it  measured,  and  how  increased  ?     (  X  4) 

7.  What  is  a  '  Saturation  Current '  ?  Is  there  any  other  sort  of  current 
through  a  gas  ?  Describe  any  one  type  of  instrument  for  measuring  currents 
through  gases. 


CHAPTER   LIV 


X-RADIATION 


§911.  This  Cathode  Stream  of  Chapter  LIII,  can  it  get  out  of] 
the  tube  ? 

Lenard  investigated  this  in  1898  ;  he  sealed  on  another  vacuum 
chamber  and  opened  a  little  window  between  the  two,  covered 
with  very  thin  aluminium  foil.  A  very  attenuated  stream  did 
shoot  through,  and  proved,  by  the  method  of  §  883,  to  consist 
of  faster  particles  filtered  out.  The  admission  of  0-1  mm.  pressure 
of  air  into  its  path  smothered  it  completely. 

Hence  we  can  add  to  the  properties  detailed  in  §  882  : — 

(e)  the  stream  can  penetrate  a  very  little  depth  into  a  solid,  or 
a  fraction  of  a  mm.  through  air  (the  lining  thickness  of  the 
purple  cap  in  §  895),  and 

(/)  the  faster  particles  penetrate  the  farther.  But  for  all  practical 
purposes 


the  Cathode  Stream  cannot  get  out  of  the  tube. 

This  investigation  had  followed  a  discovery  made  in  1895,  at 
the  mainly  medical  University  of  Wiirzburg,  by  Rontgen,  who 
also  had  been  looking  round  for  whatever  he  could  find  outside  a 
Crookes  tube,  that  some  sort  of  emission  from  it  could  fog  a  photo- 
graphic plate  through  a  wooden  door,  but  cast  a  shadow  of  the 
white-lead  putty  under  the  panel  moulding  of  the  door  ;  that  it 
didn't  blow  about,  but  travelled  straight  and  diminished  with 
distance ;  that  it  went  through  the  hand,  showing  the  bones. 
He  called  it  X-radiation,  and  it  immediately  caught  the  popular 
fancy  and  has  been  X-Rays  ever  since. 

§  912.  The  connection  between  these  two  utterly  dissimilar 
effects  soon  proved  to  be  this,  that  the  X-rays  start  from  where  the 
Cathode  stream  stops. 

The  tube  of  Fig.  398  was  devised,  and  held  the  field  for  25  years. 
The  cathode  stream  was  fired  by  40 — 120  kilo  volts  from  a  big 
sparking  coil,  §  826,  from  the  concave  aluminium  cathode  on  the 
left,  its  electrons  travelling  at  120,000 — 190,000  kilometres  per 
second  to  a  '  focus,'  made  small  to  get  good  definition  in  the 
'  shadowgraphs,'  on  the  '  anti-cathode.'  This  was  of  platinum 
(later,  tungsten),  partly  to  stand  the  intense  heat  of  the  focus, 
and  partly  because  it  was  found  that  targets  of  heaviest  atomic 
weight  gave  most  penetrating  radiation  :  it  was  backed  with  copper, 
and  that  with  various  conducting  contrivances,  to  get  the  heat 

742 


§  913]  X-RADIATION  743 

away.  The  focal  spot  was  inclined  to  wander,  but  the  Hcparate 
anode,  +>  steadied  it. 

The  tube  was  exhausted  to  as  high  a  vacuum  aa  wm  pomiblc  • 
in  action  the  X-radiation  from  the  anti-cathode  made  all  the  near 
hemisphere  fluoresce,  the  tube  looking  like  half  a  big  green  apple 

Unfortunately,  the  '  clean-up  '  effect  of  §  89()  soon  occuw,  the 
residual  gas  atoms  (mostly  hydrogen) 
getting  electrified,  dashing  into  glass  or 
metal  at  800  km. /sec,  and  sticking  there. 
As  the  cold  cathode  can  get  electrons 
only  from  them,  the  2-  or  3-milliamp^re 
current  diminishes — the  tube  getting 
'  harder,'  emitting  '  harder  '  and  more 
penetrating  X-rays,  but  seriously  fewer 
— until  the  discharge  finds  it  easier  to  ^®*  *•*• 

spark    round     outside.      The     by-pass 

Regulator  at  the  top  now  comes  into  play,  the  sparks  jump  to  the 
adjustable  bent  wire,  a  little  pad  of  asbestos  and  platinum-black. 
containing  lots  of  occluded  hydrogen,  gets  warme<i  up  by  the  dia- 
charge,  distils  out  enough  gas,  and  so  the  action  goes  on.  Or  see 
§  364  for  another  contrivance. 

The  introduction  of  the  hot  cathode,  with  the  demand  for  greater 
power,  has  put  these  capricious  '  gas  tubes  '  out  of  work,  while  the 
improvement  in  insulating  materials  has  enabled  a  static  trans' 
former  to  produce  high  driving  voltages  without  the  '  shock  tactics  * 
of  stuttering  '  break  '  machinery. 

For  all  that,  the  gas  tube  on  a  great  scale,  with  a  constantly 
running  pump,  §107,  and  a  million-volt  influence- machine.  §715, 
may  yet  make  a  triumphant  '  come-back,'  but  meanwhile  the 
principle  of  all  is  the  same — and  what  is  it  ? 

§  913.  Every  heavy  atom  of  the  anti-cathode  consists  of  a  minute 
massive  nucleus  guarded  by  a  flying  swarm  of  electrons,  78  for 
platinum  or  74  for  tungsten — fairly  comparable  to  bees  swarming 
round  their  queens,  over  an  acre  of  garden ;  swarm  after  swarm, 
all  through  the  village.  Into  this  community  dash  high-snecd 
electrons,  pulling  and  pushing  electrically  everywhere  :  999  of  them 
sooner  or  later  get  entangled  in  the  local  attractions,  and  do  no 
more  than  increase  the  general  activity,  i.e.  raise  the  temperature 
of  the  metal,  which  presently  radiates  away  their  energ>-  as  usele« 
radiant  heat  of  wave-lengths  between  5  and  0-5  microns,  Fig.  410. 

But  the  odd  one  chances  to  fly  straighter  for  the  centre  of  an 
atom,  and  is  captured,  and  its  energy  radiated  out  at  once  on  an 
extremely  short  wave-length,  somewhere  about  a  ten-thousandth 
of  those  just  mentioned,  a  splash  or  *  quantum  '  of  enerjo'  ^^^'!"f 
in  wave-form  (even  as  the  energy  of  a  struck  bell  does  through  air). 
but  at  the  universal  Speed  of  Light  or  Radio- waves  through  s|iace ; 
a  sort  of  light,  but  differing  from  it  as  does  an  mch-long  radio  wave 
from  an  ordinary  broadcasting  wave— that  is  an  X-ray. 


744  ELECTRICITY  [§  913 

We  shall  see  in  §  955  that  waves  only  half  the  length  of  visible 
light-waves  show  remarkable  differences,  they  pass  through  silver 
but  not  through  glass,  they  are  stopped  by  different  substances 
more  or  less  proportionally  to  their  Molecular  Weights ;  striking 
on  a  zinc  plate  they  wrench  electrons  from  it  which  ionize  the 
surrounding  air  ;  they  excite  fluorescence,  and  affect  a  photographic 
film  ;  they  are  bactericidal,  and  sunburn  the  skin. 

Naturally,  X-rays  '  go  one  better.'  They  pass  farther  through 
more  things.  They  are  stopped  more  intimately  according  to 
Atomic  Numbers,  e.g.  by  bone  with  Ca  20  and  P  15,  and  not  by  the 
C,  O,  N,  H  compounds  of  the  soft  parts  ;  by  '  paste  '  of  lead-glass, 
82,  and  not  by  diamond,  6,  nor  greatly  by  Al,  13. 

They  wrench  electrons  from  atom  to  atom  in  the  air  itself,  making 
them  Ions,  so  that  it  is  highly  conductive,  washing  out  all  electro- 
static charges  and  effects  promptly ;  producing  showers  in  the 
Wilson  expansion  chamber,  §  876. 

They  affect  a  photographic  film ;  they  excite  fluorescence,  on 
Screens  covered  with  barium  platino- cyanide  (or  calcium  tungstate, 
at  a  minute  fraction  of  the  price) — notice  the  heavy  atoms  in  both. 
The  screen  is  covered  with  black  paper  on  the  tube  side,  and  with 
dense  lead-glass  on  the  front,  to  protect  the  operator ;  for  though 
not  bactericidal,  they  have  deadly  cumulative  effects,  incurably 
burning  the  skin  to  start  with. 

One  property  they  have  to  themselves  :  wherever  they  go  they  go 
straight,  they  are  not  noticeably  refracted.  Violent  as  they  are, 
they  snatch  too  quickly  at  the  electrons  in  the  atoms,  in  glass  or 
water  or  wax,  and  the  electrons  are  not  drawn  into  the  movement : 
punch  an  open  door  with  your  fist,  punch  it  hard,  and  it  doesn't 
shut. 

§  914.  The  energy  of  the  Electron  flying  from  the  cathode  is  of 
course  its  charge  e,  multiplied  by  the  potential  difference  through 
which  it  has  fallen,  V :  if  these  are  in  coulombs  and  volts  their 
product  is  in  joules,  §  811,  and  must  be  multiplied  by  10^  to  give 
ergs.  This  energy  arrives  at  the  target  as  kinetic  energy  ^mv^ ; 
putting  in  the  known  values  e  =  15-9  X  lO-^o  and  m  =  0-d  X  10-^7 
gives 

0-5  X  0-9  X  10-27  X  v^==  15-9  X  IO-20  X  10^  X  V 
/.  ^;  =  6  X  10''-\/V  cm.  per  sec. 
OT  V  =  600\/V  km.  per  sec. 

which  would  give  210,000  km. /sec.  for  the  speed  of  electrons  in 
an  ordinary  125-kilovolt  tube,  but  above  100,000  km. /sec.  this 
increasingly  needs  correction  for  the  actual  increase  of  mass  of  the 
electron  with  speed  to  a  value  m'  =  w  -r\/(l  —  '^^/c^),  where 
c  is  the  velocity  of  light.  This  makes  the  speed  for  125  kV  about 
0-63,  for  250  kV  0-77,  and  for  a  million  volts  0-97,  of  the  unattain- 
able speed  of  light. 

The  hardest,  i.e.  the  shortest  wave-length  X-rays  obtainable 
from  a  tube,  have  wave-length  1-2345 /volts,  in  microns. 


915] 


X-RADIATION 


745 


§  915.  Let  us  now  examine  a  good   modern   X-Ray   outfit,  an 

constructed  by  Messrs.  Newton  and  Wright,  the  London  firm  claim. 
ing  descent  from  the  great  Sir  Isaac. 

The  '  Villard '  working  circuit  employed  is  shown  in  Fig.  399. 
A  high-tension  transformer,  drawing  current  from  the  A.C  muinB, 
charges  the  inner  plates  of  two  large  condensers,  and  these  charges 
induce  opposite  charges  on  to  the  outer  plates,  if  they  can  get  there. 
This  means  a  flow  of  electrons  from  side  to  side  of  the  circuit,  to 
the  left  through  V  or  to  the  right  through  X,  for  both  these  are 
highly  evacuated  tubes  with  cathodes  kept  incandescent  by  low 
tension  current  from  the  two  filament  transformers,  §  S81. 

They  differ  essentially  only  in  this,  that  V  has  yards  of  filament 
curled  up  in  ample  festoons,  while  X  has  only  one  short  little  curl : 
that  means  that  it  is  easy  enough  for  abundance  of  electrons  to 
flow  through  the  valve  V  to  pile  up  a  negative  charge  on  the  left, 


nrtrrro. 


-<I 


uuuuv 


Fig.  399. 


as  shown ;  sed  revocare  gradum  through  X  is  going  to  be  apua  and 
Idbor  of  the  hardest.  ^.       , 

It  is  performed,  for  the  transformer  not  only  withdraws  the  + 
charge,  leaving  the  -  unwanted  in  the  condenser,  but  proceeds  to 
pile  in  a  —  charge  on  its  own  plate  and  repel  it  out,  so  that  ultmiately 
It  rushes  X  under  nearly  double  the  nominal  voltage  of  the  trans- 

^^upiicated  circuits  would  utilize  the  return  wave  of  the  A.C.. 
which  this  simple  one  refuses  to  admit ;  but  as  it  likewise  refuses  to 
pay  for  it,  supplies  the  X-ray  tube  with  all  the  current  it  can  stand. 
and  gets  rather  better  quahty  X-rays,  the  makers  prefer  it. 

Fie  400  shows  the  general  appearance  of  the  apparatus  in /Aoul 
the  regulating  switchboard,  and  without  the  X-ray  tube,  the  leads 
to  which  would  be  plugged  into  the  black  ring  sockets  on  top  of 
the  condensers.     The  diagram  is  between  4  and  ;>  "•  ^'»«^;^  . 

The  three  transformers  occupy  the  three  rectangular  "•«"«*"»»• 
the  filament  transformers  are  quite  small  «^^P;«^^'"  .*^^*^;„*^^^ 
are  up  at  high  potential,  and  have  to  be  insulated  J"«^«f/^*^"  ^ 
the  main  onl,  so  that  outside  they  look  arge  and  >"^.P«^;"|  J^^ 
tanks  are  lined  with  layers  of  thin  ebomt^ ^^rrehilk  and^ed 
full  of  heavy  hydrocarbon  lubricating  oil,  very  carefull}  freed  irom 


746 


ELECTRICITY 


[§  915 


the  6%  of  dissolved  water  it  usually  contains,  and  thereby  incom- 
parably improved  in  insulating  character  :  tall  porcelain  insulators 
rise  from  the  bakelite  lids. 

The  H.T.  transformer  in  the  middle  is  the  10-kw.  one  figured  on 
the  right  of  Fig.  369.  It  has  a  primary  winding  of  two  layers  of 
4-mm. -square  cotton-covered  copper  wire,  wound  on  a  pressed- 
card  cylinder  surrounding  the  square  core  of  laminated  '  stalloy.' 
Over  this  winding  are  lapped  several  layers  of  1/16-in.  ebonite, 
and  then  a  secondary  of  some  few  miles  of  No.  36  silk-covered  copper 
wire  is  wound  on,  in  sections.  It  differs  from  the  older  trans- 
former of  Fig.  370  in  its  '  closed  iron  circuit,'  and  in  lesser  length 


Fig.  400. 

and  greater  thickness  of  secondary  wire,  since  heavy  power  output 
is  called  for  at  only  about  4  in.  spark-length. 

The  condensers  are  rolled-up  sheets  of  metal  and  micanite,  in 
oil,  and  stand  on  porcelain  insulators. 

All  conductors  are  thick  and  smooth  and  rounded,  to  avoid 
'  corona,'  §  895. 

Since  the  '  Hollwey  '  valve-tube — '  kenotron  '  type — ^provides 
abundance  of  electrons,  it  works  at  very  low  voltage  ;  it  never  gets 
overheated,  and  can  be  guaranteed  for  years,  and  any  X-raj^s  it 
generates  are  too  soft  to  get  out  through  the  glass. 

Both  these  and  X-ray  tubes  are  made  from  glass  selected  for  its 
comparative  freedom  from  retained  blowpipe  gases,  and  are  evacu- 
ated for  many  hours,  running  hot,  on  an  oil  condensation  pump, 
§107. 

§  916.  For  Radiography  with  this  outfit  the  modern  protected 
X-ray  tube  of  Fig.  401  would  be  used,  rated  3,  6  or  10  kw.,  and  110 
kv.  ;  it  is  about  2  ft.  long. 

In  the  lower  figure  you  see  the  hot-cathode,  a  single  straight 
close -wound  helix  of  tungsten  wire,  heated  by  a  current  coming 
in  by  the  slack  wires  sealed  in  at  the  '  pinch  '  on  the  left,  and  reaching 


i 


916] 


X-RADIATION 


747 


perhaps  5  amps.  It  is  backed  by  a  half -cylinder  metal  shield 
(only  faintly  suggested  in  the  diagram)  which  focusses  the  emitted 
cathode  stream  into  a  sharp  line  on  the  tungsten  face  of  the  copper 
target.  This  face  is  slightly  inclined,  so  that,  looked  at  through  the 
little  round  window,  the  line  appears  foreshortened  to  a  point : 
the  idea  is  to  get  the  desired  small  point  Radiant  which  will  ca«t 
sharp  shadows — for  that  is  what  radiograms  are — without  having 
the  whole  electronic  blast  concentrated  on  such  a  small  area,  for 
it  would  immediately  melt  a  hole  in  it. 

The  iron  sheath  projecting  from  the  cathode,  and  shown  cut  away, 
is  an  accessory  protecting  the  glass  tube,  close  outside  it,  from  Mtray 
destructive  gusts  of  electrons.  The  large  metal  glass  vacuum  seal 
on  the  right  is  part  of  the  fine  art  of  tube  making. 


Fig.  401. 

As,  with  the  best  intentions,  999  parts  of  the  energy  will  be  wastfd 
in  heating  the  anti-cathode,  this  must  be  got  rid  of  somehow: 
a  stout  copper  rod  leads  out  from  the  back  of  it,  and,  when  the  whole 
is  assembled,  has  the  gilled  air-cooHng  head  fitted  on.  Some 
tubes  have  a  copper  tube,  bearing  at  the  end  a  tumed-up  flaak  of 

That  completes  the  tube  :  a  6-kw.  size  will  carry  25  milliami*. 
for  20  sec.  on  110  kv.,  or,  by  heating  the  cathode  rather  hotter, 
8  881,  as  much  as  100  ma.,  but  only  for  1/lOth  sec. 

But  patient,  operator,  and  tube,  abke,  are  worth  pro'^'""*- 
A  brass  sheath  carrying  the  fixing  clamp  and  Imed  »"«h  3  mm^ 
of  lead  (and  shaded  heavily  in  the  middle  Ag"'*),  encases  the  acn»e 
middle  of  the  tube,  and  protects  the  operator  from  the  d«Mlb 
radiation  •  its  33-mm.  round  wmdow  carries  filters  of  Al.  K^.  or 
thin  Pb  to  top  the  soft  non-penetrating  X-rays  which  would  on  y 
burn  the  patient's  skin.  The  rest  of  the  .sheath,  shown  l«htlj 
shaded,  is  of  insulating  tubing,  to  the  brass  caps^  ^^ 

Such  tubes,  costing  about  £40  wil  give  Ave  y;*"  ^^^^y 

Tubes  for  higher  voltages,  used  in    deep  therapj ,  are  nece-aniy 


748  ELECTRICITY  [§  916 

larger  and  longer,  or  they  would  spark  over  outside  ;  consequently 
their  lead  sheaths,  1  cm.  thick,  are  bulky  and  ponderous.  They 
are  intended  for  continuous  running  :  cooling  arrangements  must 
be  good,  but  there  is  no  need  for  sharp  focus  on  the  target.  At  the 
time  of  writing,  300  kv.  is  the  most  they  stand  up  to  :  for  a  700-kv. 
tube,  see  §  940. 

The  X-Ray  Department  has  to  be  equipped  with  mechanical 
contrivances  for  moving  tube  or  patient  with  much  exactness  : 
they  will  remind  you  of  the  radial  drill,  planer,  etc.  that  you  have 
seen  in  large  machine  shops. 

§  917.  Fluorescent  Screens  for  visual  examination  are  usually 
of  calcium  tungstate,  which  shines  bluish- white.  They  are  backed 
with  black  paper,  and  in  front,  to  protect  the  operator,  is  a  10-mm.- 
thick  sheet  of  glass  containing  as  much  lead  as  possible,  equivalent 
to  3 -mm.  sheet  lead. 

Radiographic  films  are  usually  coated  with  emulsion  on  both 
sides  of  the  celluloid,  and  are  squeezed  in  between  Intensifying 
Screens  of  fluorescing  calcium  tungstate,  which  reduce  the  necessary 
exposure  to  a  twentieth. 

Every  solid  thing  X-rays  strike  on  gives  off  secondary  Radiations, 
see  §  985,  and  to  screen  these  off  from  fogging  the  picture,  a  '  Bucky 
grid  '  is  kept  moving  over  it  during  exposure  :  it  is  a  contrivance 
of  inch-wide  strips  of  lead  set  on  edge  close  together  ;  splaying  out 
exactly  as  the  outer  margins  of  the  leaves  of  a  half-opened  book 
all  radiate  from  the  back,  where  the  tube  is  placed  ;  it  works  on 
the  fly's-eye  principle  of  §  601,  reversed. 

When  flexible  protection  is  necessary,  the  operator  wears 
apron  and  gloves  of  2 -mm.  rubber  loaded  with  litharge  to  2  lb. 
per  sq.  ft. 

§918.  X-ray  dosage  for  treatment  is  usually  controlled  by 
Sabouraud  pastilles,  quarter-inch  discs  of  paper  coated  with  barium 
platino-cyanide.  These  are  greeny-yellow,  and  gradually  change 
to  orange-brown  during  exposure  :  the  operator  has  a  whole  scale 
of  tints  to  compare  them  with  :  they  are  restored  by  long  exposure 
to  daylight.  High-voltage  radiation  has  to  be  metered  by  leaking 
condenser  and  electrometer,  as  in  §  984. 

The  full  dose  causes  an  X-ray  erythema,  a  reddening  which 
develops  slowly  and  dies  away  still  more  so  :  no  second  dose  may 
be  given  for  a  week  after.  Successive  partial  doses  add  up,  and 
may  also  cause  deeper-seated  troubles  and  aplastic  anaemia,  or 
the  erythema  insidiously  develops  into  the  incurable  cancerous 
condition  which  led  to  the  repeated  surgical  mutilation,  and  ultimate 
decease,  of  some  of  the  X-ray  pioneers,  who  had  no  sufficient  warnii 
of  their  danger. 

The  eighth-inch  sheet-lead  protection  reduces  most  X-rays  to 
1 /10,000th  ;  and  with  its  equivalents  mentioned  above,  has  at 
last  made  Radiology  reasonably  safe. 


a 


§919] 


X-RADIATION 


740 


§  919.  X-Rays  find  daily  detective  employment  in  wav«  of  which 

a  medical  student,  have  only  to  go  across  to  vour  honpitki  JC-Rav. 
to  get  information  without  stint,  as  to  their' medical  ami  nurukMl 
uses,  straight  from  the  horse's  mouth.  Here  let  us  follow  thJm 
a  step  further,  and  see  how  they  have  disclosed  not  simnlv  th*» 
anatomy  of  the  body,  but  that  of  solid  matter  itself. 

Early  attempts  to  dififract  X-rays,  i.e.  to  throw  them  mmdo  by 
mterposing  regularly  spaced  structure  in  their  path,  were  faUure*  • 
until  It  occurred  to  Laue,  in  1912,  that  possiblv  their  wave-lemrth 
was  so  small  that,  to  them,  a  15,()0<)-line  diffraction  Krating  wm 
like  a  ploughed  field  to  waves  of  light.  Hi.s  suggestion  that  the 
natural  packing,  in  layers,  of  atoms  in  a  crystal,  was  a  reguhu- 


Fio.  402. 

structure  much  finer  than  anything  artificially  producihlc,  led  to 
immediate  success,  and  Fig.  402  is  sketcheil  from  the  photograj 
plate  on  which  were  thrown  the  short  '  spectra  '  diffracted  out 
a  crystal  of  rock-salt,  on  which  a  central  narrow  beam  fell. 

To  get  a  first  idea  of  what  this  figure  means,  look  at  a  distant 
bright  point  of  light  through  your  handkerchief  :  in  th©  prmmt 
case,  however,  it  is  not  caused  by  a  regular  spacing  across  the  line 
of  sight,  but  by  one  of  parallel  planes  of  {)acke<l  atoms,  im  depth. 

In  Fig.  403  let  pp,  qq,  rr  be  such  planes  at  distance  apart  d,  and 
PP'  a  X  wave-crest  aavancing  in  direction  PQ  at  angle  a.  Thr  ymrX 
P  of  it  is  reflected  to  an  observing  point  R  via  PQR.  P'  via  P'O  '^ 
and  so  on,  and  if  R  is  to  receive  a  conspicuously  bright  Haah  all 
these  miist  arrive  in  phase,  as  wave-crests  together,  at  H. 

Drop  perpendicular  QSN,  produce  P'Q'N  and  drop  perprwIictiUr 
QD.  The  extra  distance  P'  has  to  go  is  Q'Q  lew  the  bit  g'l)  by 
which  P'Q/  is  shorter  than  PQ. 


750 


ELECTRICITY 


[§919 


/> 

\      f^X 

/> 

Q 

\qO^ 

S 

Q 

r 

D%- 

N 

r 

Angles  being  equal  Q'N  =  Q'Q,  therefore  this  difference  is  DN, 
which  must  be  a  whole  wave-length,  X,  of  the  X-ray.  Plainly  the  angle 
DQN  =  a,  therefore  DN  =  QN  sine  a 

OT  X  =  2d  sine  a. 

The  angle  a  is  measured  from  the  position  of  the  spot  on  the 
photograph. 

As  yet  we  know  neither  d  nor  X. 

But  the  distribution  of  the  spots  has  told  the  crystallographer 
how  the  various  reflecting  planes  in  the  crystal  are  arranged ;  in 
this  case  they  cut  it  up,  in  the  simplest  possible  fashion,  into  a  mass 
of  equal  cubes. 

Further,  the  distribution  in  the  original  photograph  of  fainter 
and  darker  spots  has  suggested  that  these  planes  are  populated 

by  lighter  and  heavier  atoms, 
Na  and  CI ;  and  happens  again 
to  fit  in  with  the  simplest  pos- 
sible 'arrangement,  that  of  Na 
and  CI  alternately,  at  the  corners 
of  the  cubes,  all  three  ways. 

Make  your   own  diagram,    or 

stack  up  some  lump  sugar,  and 

you  will  see  that  8  cubic  corners 

meet  in   every   atom,  and  that 

every  cube  contains   8   corners ; 

consequently  each  cube  possesses 

one  atom,  either  Na  23,  or  CI  35-5,  with  an  average  mass  J(23  +  35-5) 

X  the  mass  of  a  hydrogen  atom,  or  29-3  X  5/3  X  10-24  =  49  X  10-2* 

gram. 

But  all  the  cubes  building  up  the  centimetre  cube  of  rocksalt 
together  make  its  mass  (its  density)  2-14  gm.,  therefore  there  are 
2-14  ^  (49  X  10-24)  =  0-0437  X  102*  cubes  per  c.c.  Taking  the 
cube  root  gives  the  number  along  any  edge  =  0-35  x  10^,  and  the 
distance  apart  of  the  reflecting  planes  is  the  reciprocal  of  this, 
d  =  2'S  X  10-^  cm.,  and  this  is  an  average  Diameter  of  an  Atom. 

For  the  most  prominent  X-radiation  from  platinum,  angle  a  was 
measured  as  11-4°;  hence  X  =  2  x  2-8  X  10-^  X  sine  11-4°  = 
1-10  X  10-^  cm.,  which  is  just  one  five-thousandth  of  the  average 
wave-length  of  light  used  in  our  calculations  of  §§  632,  635. 

Here  then  is  a  scale  and  a  weapon,  with  five  thousand  times 
increased  resolving  power,  by  which  one  can  attack  and  measure 
all  crystals,  the  geometrical  structure  of  their  '  crystal  lattice,' 
and  the  actual  size  of  it ;  and  by  continued  comparisons  can  find 
the  individual  sizes  of  common  atoms.  The  distortion  of  the  lattice 
caused  by  the  intrusion  of  foreign  atoms  can  be  watched,  as  in  alloys, 
and  be  linked  up  with  strength,  increased  hardness,  etc. 

In  10-®  cm.,  ten- thousandths  of  a  micron,  the  diameters  of  som©^ 
atoms  are,  approximately :  S,  Si,  2-0 ;  Ne,  Al,  Pb,  Bi,  2-8 ;  K  and 
O  ions  2-6 ;  (Na  2-0,  CI  3-4),  H  smallest  1-4,  Cs  largest  4-4. 


Fig.  403. 


§  921]  X-RADIATION  751 

§920.  Often  no  definite  crystal  can  be  obtained,  only  a  lump 
known  to  be  crystalline,  like  any  metal.  The  effectii  of  the 
individual  crystals  add  together,  not  into  a  smudge,  but  into  ring* 
round  the  axial  spot,  and  the  diameters  of  these  are  meamirad. 
The  same  technique  holds  for  a  fine  powder,  a  thread  of  X-ra\*t 
is  fired  at  it,  and  it  scatters  a  ring  picture ;  impalpable  lampblack 
proves  to  have  the  crystalline  structure  of  Graphite;  a  smear  of 
shoe-polish  discloses  what  particular  wax  it  is  made  of,  by  meanunng 
the  spacing  of  the  marks  due  to  the  CH-'s  of  the  paraflln  ;  the 
ultra-microscopic  particles  of  colloidal  gold  and  silver  prove  to  be 
crystalline  in  build,  though  containing  only  a  few  score  atoms  apiece. 

The  X-ray  Spectrograph  looks  not  unlike  the  visible-ray  Spectfo> 
meter,  Fig.  221.  The  X-ray  tube  is  its  source  of  radiation.  it« 
collimator  consists  of  fore-sight  and  back-sight,  narrow  slitu  in  plate* 
of  lead ;  its  grating  {not  prism)  a  slice  of  crystal ;  itu  *  telencofie  * 
is  an  ionization  tube  connected  to  a  gold-leaf  electrometer,  which 
will  leak  when  rays  enter ;  or  else  it  has  a  camera  with  light-tight 
shutter  and  no  lens. 

As  you  can  imagine  from  your  own  experience  with  the  Spectro- 
meter, it  would  make  very  much  for  clearness  and  accuracy  if  a 
'  monochromatic  '  X-radiation,  of  one  wave-length  onlv,  could  be 
used  in  making  measurements,  just  as  you  used  sodium  light. 

This  can  very  nearly  be  done,  for  while  the  X-radiation  in  mostly 
a  '  white  '  jumble  of  a  wide  variety  of  wave-lengths,  soft  long,  and 
hard  short ;  yet,  under  careful  conditions  of  working,  ever>'  metal, 
used  as  anti-cathode,  does  also  emit  preferentially  quite  well-marked 
wave-lengths  of  its  own,  having  in  fact  a  simple  Spectrum.  Th*iJi. 
using  a  molybdenum  target,  and  further  filtering  ita  radiation 
through  a  sheet  of  zirconia,  the  two  close  wave-lengths  0-707  and 
0-712  X  10-«  cm.  are  obtained  practically  pure,  and  this  makes 
the  '  powder  method  '  easily  workable. 

§  921 .  But  when  a  whole  series  of  anti-cathode  metals  is  examined. 
using  a  rock-salt  crystal  for  all,  much  more  than  this  appe^. 
It  is  found  that  they  all  radiate  characteristically  on  a  similar 
pair  of  spectrum  lines,  called  K  lines,  but  that  these  move  along  the 
spectrum  towards  shorter  wave-lengths  (higher  fwH^ueneie*)  with 
increase  in  Atomic  Weight.  And  another  simple,  but  weaker.  L 
group  of  lines  presently  appears  and  does  the  same  thm^. 

Moseley,  in  1913,  plotting  V(frequency  of  K  line)  against  Atomic 
Weight,  for  a  number  of  metals,  got  something  apprcmrhing  a 
straight  line,  but  next  year  he  plotted  it  simply  agamst  the  number 
of  the  metal's  position  in  the  Periodic  Table,  where  they  all  irtawl 
in  order  of  increasing  atomic  weight,  and  this  Atomic  NumMr  gave 
a  much  better  straight  line.  .,,_  # 

By  making  all  toe  this  line,  he  showe<l  that  it  ^"***7kI*I 
chemists  to  go  on  suspecting  some  new  elementa,  «^";«JJJ^ 
wasn't  room,  and  set  them  searching  instead  for  *»« '«T2?."5 
in  other  places,  and  Rhenium  and  Hafnium  were  i^mcorttta  ana 


752  ELECTRICITY  [§921 

fitted  in.     But  the  Atomic  Number  plainly  held  a  deeper  meaning 
than  this. 

§  922.  Stock-taking.  But  first  let  us  take  stock  of  where  we  are, 
and  be  clear  how  we  got  here. 

Somebody  sometime  weighed  a  litre  of  hydrogen  ;  no  theoretical 
difficulty  about  that. 

You,  knowing  the  value  of  a  Coulomb,  from  your  absolute 
measurement  of  H  and  your  tangent  galvanometer,  cf .  §  774, 
electrolysed  some  water  and  collected  a  volume  of  hydrogen,  and 
found  that  it  would  take  96,500  coulombs  to  produce  1-0078  gm.  of 
it ;  that  e/M,  elementary  charge /mass  of  atom  At.  Wt.  I,  is  96,500. 

[In  measuring  H,  you  took  a  '  moment  of  inertia '  from  a 
formula  ;  its  calculation  is  a  stock  little  bit  of  integration.] 

J.  J.  measured  e/m  for  electrons,  §  883,  and  found  176,000,000. 

Millikan  found  e  =  15-9  X  lO-^o  coulombs,  whence  m  =  0-905  X 
10-27  gm.,  and  therefore  the  hydrogen  atom.  At.  Wt.  1,0078,  which 
is  176,000,000/(96,500  -:-  1,0078)  times  as  much,  has  mass  1-66  X 
10-2*  gm.  (which  gives,  in  its  gram-Atomic  Weight,  0-606  X  10^* 
atoms,  Avogadro's  Number). 

[The  weight  of  his  oil  drops  was  got  from  Stokes's  Law  of  fall  of 
a  sphere  through  a  viscous  fluid,  another  little  bit  of  integration, 
if  one  has  mislaid  the  formula  book.] 

Both  these  e's  were  really  the  electrostatic  attracted  e,  but  the 
argument  of  §  839,  with  the  experimental  Speed  of  Light,  §  952, 
enables  them  to  be  quoted  in  coulombs. 

Chemists,  by  electrolysis  or  other  purely  analytical  methods, 
have  found  Na  23  and  CI  35-5  times  as  massive  as  hydrogen. 

Hence  we  know  what  mass  to  put  in  the  cells  of  the  '  crystal 
lattice  '  of  rocksalt,  which  Crystallography,  backed  by  X-rays, 
tells  us  are  cubical. 

You,  weighing  rocksalt  in  paraffin  oil  on  the  Hydrostatic  Balance, 
§  138,  deduce  that  I  cm.  cube  has  a  mass  of  2-14  gm. 

The  ratio  of  these  two  masses  is  the  number  of  cubelets,  i.e.  of 
atoms  ;  hence  the  average  diameter  d  of  these  atoms,  2-8  X  10-^  cm. 

Thence  the  wave-length  X  of  a  particular  X  radiation,  and  the 
experimental  application  of  this  to  other  crystals  gives,  by  the 
relation  7.  =  2d'  sine  6,  the  sizes  of  all  sorts  of  atoms. 

So  here  we  are,  with  no  mysteries  nor  mathematical  miseries, 
sorting  atoms  like  apples,  and  telling,  with  a  flash  of  a  pocket - 
torch,  just  how  they  are  packed  in  their  boxes. 

The  next  thing  is  to  bite  into  some. 

§923.  The  Atomic  Number.  To  resume  the  detective  story  of 
the  X-ray  :  what  is  this  in  the  atom,  which,  increasing  one  by  one 
from  I  to  92 — ^whereas  mass  increases  by  twos  and  threes  and 
fractions  from  1  to  238 — keeps  perfect  step  with  the  square  root  of 
the  frequency  of  a  radiation  which  they  all  emit  ?    .       ,   ;      :  ,, 


§  924]  X-RADIATION 


7M 


to 


We  have  seen  that  atoms  contain  electrons,  for  we  know  bow 
distil  them  out,  §871  ;    and  that  electrons  fired  into  gM  atooM 
m  the  electric  discharge,  cause  the  emission  of  charactmgtio  vinuii 
frequencies,  spectrum  lines  of  light.     How  many  decirons  jmr  atom  f 

Make  it  the  Atomic  Number;  so  that  the  atomic  charge  oT 
electricity  increases  unit  by  unit  from  1  to  92. 

But  the  Atom  is  not  charged  ;  so  that  somewhere  inxide  it  a 
neutrahzing  positive  Charge  must  also  be  increasing  from  1  to  92. 
And  a  Mass,  numerically  two  or  three  times  as  much,  is  i»traggling  up 
along  with  it,  whereas  all  those  electrons  don't  weigh  a  decimal 
point  in  the  atomic  table. 

Make  the  Atomic  Number  the  number  of  equally  positively 
charged  protons,  each  of  unit  atomic  mass,  contained  in  a  very  imall 
Atomic  Nucleus,  in  which  is  concentrated  practically  all  the  maM 
of  the  atom. 

Hydrogen  has  one  proton  and  one  electron.  Thia  electron** 
movements  produce  the  first  Series  of  Spectrum  Lines  ever  recogniied ; 
three  of  them  show  in  Fig.  223,  more  are  photographed  in  the  ultra- 
violet; Fig.  418  shows  the  whole  series,  and  Fig.  417  deacribea 
also  the  three  other  remote  series  subsequently  discovered. 

Helium  comes  next,  with  2  protons  and  2  electrons.  But  helium 
has  mass  4,  so  we  must  pack  into  its  nucleus  also  two  neatroiis, 
each  composed  of  proton  and  electron. 

The  Neutron  was  long  in  being  detected  wandering  at  large, 
for  it  travelled  with  no  electric  charge  to  attract  anything,  *  without 
lights,'  until  in  1933  it  smashed  into  collision. 

This  helium  structure,  2  neutrons  and  2  protons,  and  2  electrocM 
in  the  out-field,  forms  a  very  stable  unit,  for  the  chemist  can  do 
nothing  with  it,  and  terrestrial  physics  uses  it,  minus  ita  electrom, 
as  the  unbreakable  *  a  particle,'  §932,  wherewith  to  penetrate  the 
electronic  defences  and  batter  the  nucleus  of  bigger  atoma,  f  946. 

The  equally  inert  neon,  No.  5,  At.  Wt.  20,  mighi  Ik*  five  of  them ; 
but  nobody  cari  suppose  oxygen  is  four  heUum  units — and  now  oxy. 
gen  17  and  oxygen  18  have  turned  up. 

§924.  Isotopes.  But  Neon  isn't  20,  its  atomic  ma«i  ia  20-2; 
and  that  of  Chlorine  35-5,  and  of  Silver  107-9,  l)eyond  a  thadow 
of  doubt.     What  are  these  odd  bits  ? 

In  §885  there  were  found,  in  a  discharge  tube,  particlea  un- 
doubtedly  carrying  the  unit  protonic  positive  charge,  and  of  atomic 
masses  20  and  22.  They  were  found  by  their  tracks  on  a  photo- 
graphic plate,  and  that  of  the  20  waa  a  great  deal  the  blacker. 
There  was  no  20-2  track  :  neon  is  a  mixture  of  nine  atoma  of  maas 
20  and  one  of  22  ;  just  as  air  is  a  mixture. 

The  chemist  had  nothing  to  say  to  that,  for  neon  haa  nothing 
to  say  to  him  ;  but  chlorine  ?  u  •     k*  k 

Aston  took  up  the  work,  devising  his  '  Mass  Spectrocraph  which 
may  be  described  as  a  very  beautiful  instnimental  elabaration  « 
the  method  of  Fig.  395,  and  found,  for  Chlorine,  *  laotopea   of  nia«ea 


154:  ELECTRICITY  [§  924 

35,  36,  37  and  38 ;  for  Krypton,  the  inert  82-9  gas,  78,  80,  82,  83,  84 
and  86  ;  for  Silver,  107  and  109. 

By  Diffusion,  §  362,  he  separated  two  samples  of  neon  of  different 
density  ;  the  same  was  done  for  chlorine  ;  and  mercuries  have  been 
obtained  1 /4000th  lighter,  and  heavier,  than  ordinary  mercury. 

Isotopes  are  not  electrically  charged,  so  they  differ  from  one 
another  either  in  having  fewer  protons  in  the  nucleus  and  cor- 
respondingly fewer  electrons  outside  it,  which  would  mean  a  com- 
plicated re-adjustment  of  relations  for  each  ;  or  else  in  having 
fewer  neutrons  in  the  nucleus,  a  very  simple  matter  indeed.  Quite 
simple,  for  as  you  see  from  its  definition,  neutrons  don't  come  into 
count  at  all  in  the  Atomic  Number,  any  more  than  one  counts 
drones  in  a  hive. 

The  Atomic  Number  remaining  the  same,  i.e.  the  nett  nuclear 
charge  being  the  same,  and  the  planetary  electrons  being  unchanged 
in  number  and  disposition,  all  the  Isotopes  of  a  family  enter  into  the 
same  '  chemical '  relations  with  other  atoms,  i.e.  they  form  one 
chemical  element,  no  distinctive  qualitative  tests  apply  to  them. 
Only  by  weighing  them,  i.e.  by  an  accurate  determination  of 
atomic  weight  by  some  of  the  recognized  methods,  can  the  chemist 
distinguish  between  them. 

§  925.  The  discovery  of  1933  was  the  Isotope  of  Hydrogen, 
Heavy  Hydrogen,  or  Diplogen. 

In  1895,  chemist  and  physicist  had  joined  forces,  and,  discovering 
that  air  contained  1%  of  its  volume  of  a  strange  gas,  in- 
troduced Argon  to  a  incredulous  world  ;  thirty  years  later  a  similar 
combination  did  not  discover  that  water  contained  a  third  of  its 
volume  of  another  strange  gas  ;  for  it  was  outwitted  by  the  supreme 
skill  with  which  Nature  laid  a  trap. 

Chemistry  has  busied  itself  for  years  with  the  accurate  determina- 
tion of  the  atomic  weight  of  hydrogen,  reckoning  oxygen  16,  and 
obtained  1-0078.  Aston,  using  his  mass-spectrograph,  obtained 
perfectly  clear  tracks  of  oxygen  and  hydrogen,  and  deduced  1-0078. 
Therefore  the  chemists'  hydrogen  had  no  room  for  any  isotope 
of  different  weight. 

But  in  1930,  long  over-exposure  on  Oxygen  with  the  mass- 
spectrograph  showed  a  faint  trace  of  O  17,  and  a  plain  line  of  O  18. 

So  the  chemists  had  all  the  time  been  combining  their  hydrogen 
with  a  mixed  oxygen  which  was  16-  something,  while  Aston  had 
compared  it  with  the  plainest  line  only,  which  was,  of  course,  0  16  ; 
and  out  of  pure  cussedness  Nature  had  arranged  for  the  two  ratios 
to  be  equal.  And  there  had  been  a  (2H)  line  plain  enough  all  the 
time — presumed  just  molecular  hydrogen,  as  with  other  things. 

It  was  suspected  that  the  new  isotope,  which  must  be  at  least 
twice  as  heavy,  would  move  more  sluggishly  as  an  electrolytic  ion, 
so  that  in  old  battery  acid,  which  had  been  '  topped  up  '  and  '  gassed  ' 
alternately  for  years,  there  ought  to  be  an  accumulation  of  '  heavy 
water  '  ;  and  a  five-fold  enrichment  has  been  found. 


§  925]  X-RADIATION  755 

By  a  long  technique  of  repeated  electrolysis  of  caustic  soda  in 
nickel  cells,  heavy  water  has  been  prepared  as  almost  pure  H^ ; 
mixed  molecules  of  H^OH  may  not  be  all  eliminatc<l,  but  the 
equally  heavy  HgO^®  is  not  enriched  by  electrolysis. 

At  25°  C.  its  density  is  11056  that  of  common  water  at  the  same 
temperature.  Its  temperature  of  maximum  density  is  11-8'  C. 
It  is  less  volatile  than  water,  having,  e.g.,  a  v a ix)ur pressure  of  21-5 
mm.  Hg  against  25  mm.  At  its  boiUng  point,  101-8  C,  it  vaporizM 
with  latent  heat  2-6%  greater.  It  freezes  at  3-8®  C,  so  that  a  tube 
of  it  is  easily  frozen  by  immersion  in  melting  ice.  It  is  preaeDt  to 
the  extent  of  about  I /5000th,  2  minims  per  pint,  in  ordinan*  water. 

'  Heavy  hydrogen  '  does  not  possess  the  remarkable  Conduct  ivity 
for  Heat  which  the  mobility  of  light  hydrogen  confers  upon  it. 


EXAM   QUESTIONS,  CHAPTER   LIV 

You  must  be  perfectly  clear  about  §  911  and  the  befcinninff  of  912.  A 
little  theory,  which  you  will  understand  better  later,  starU  §  913;  and  lh» 
various  properties  of  X-radiation  which  follow  you  should  make  a  poini  of 
seeing  demonstrated  in  the  laboratory.  For  §§  915—918  you  muM  vMlmml 
explore  a  friendly  X-Ray  Department :  you  will  find  little  aifflculty  about 
that.  _  f   »^ 

There  you  can  leave  it,  or  you  can  plunge  into  the  Detective  *'****y®V '**• 
Atoms,  one  without  visible  gore,  indeed,  but  where  the  weapon  i«  <»**JJ* 
than  the  Spear  of  Lugh  :  §  922  will  show  you  how  deep  you  can  dive  •iremay* 
by  merely  putting  together  a  few  familiar  things;  then  go  on,  and  in  U»  coo- 
eluding  two  chapters  you  reach  the  limits  of  the  known. 

1  What  do  you  know  of  the  electrical  conductivity  of  air?  Under  what 
conditions  can  the  passage  of  electricity  through  air  produce  -^-^X'^ 

2.  Describe  apparatus  for  the  production  of  X-rays,  and  explain  how  it 

Give  a  brief  account  of  the  properties  of  X-radiation.     (  X  2) 

3.  Describe  the  production  of  X-rays  and  summarire  their  main  propcftiM. 
(  X  2) 

4.  How  may  X-rays  be  produced  ?  Give  a  brief  outline  of  thmr  prop.rti«t. 
How  are  they  related  to  light  rays  and  7  rays  of  radmm  7     (  x  .J 

5.  Describe  how  X-rays  may  be  produced. 

What  do  you  understand  by  '  hanl'  and  'soft  ray«T  how  mn  mm%^ 
length  and  penetration  related  ?     (  x  2) 

6    Describe  means  of  measuring  (o)  the  quantity,  and  (6)  the  quality  of 

""  7"  Describe  an  X-ray  apparatus      Mention  the  chief  phyaicjJJInot  ^^oU^) 
properties  of  X-rays,  and  show  where  they  come  m  the  .p^strum  of  n^^>oa 


CHAPTER   LV 
RADIOACTIVITY 


§  931.  It  was  by  a  marvellous  bit  of  terminological  good  fortune, 
which  none  could  foresee,  that  the  name  of  Uranus,  the  far-off 
father  of  the  Titans  and  the  Cyclopes,  dim  primogenitor  of  the  gods, 
was  bestowed  upon  the  heavy  metal  from  pitchblende,  with  its 
heaviest  known  atom,  for  Uranium  has  proved  to  be  the  prime 
ancestor  of  long  lines  of  descent  which,  if  gold  and  silver  are  '  noble,' 
may  fairly  be  described  as  divine. 

And  by  a  strange  parallel.  Thorium,  second  heaviest  of  atoms, 
name-child  of  the  second  of  the  gods  of  Asgard,  heads  an  independent 
lineage  of  its  own,  one  that  has  been  of  much  service  in  northern 
countries  in  whose  territories  the  mineral  traces  of  the  Mediter- 
ranean myth  are  not  so  freely  scattered. 

In  1896,  soon  after  the  discovery  of  X-rays,  Henri  Becquerel, 
whose  name  had  come  to  connote  phophorescence  to  all  men, 
was  astonished  to  find  that  salts  of  uranium  were  not  only  phos- 
phorescent after  exposure  to  light,  in  the  recognized  manner,  but 
that  without  any  prehminary  exposure  they  could  blacken  a  photo- 
graphic plate,  and  even  through  black  paper. 

A  few  years  later  he  found  that  by  precipitating  a  uranium  solution 
with  ammonium  carbonate,  he  could  concentrate  this  extraordinary 
property  tuto  a  thousandth  of  the  solid,  and  in  1902  he  brought  over 
a  little  packet  of  this  '  Uranium  X  '  in  his  waistcoat  pocket.  It  had 
reposed  there  during  a  six-weeks  holiday  tour ;  and  it  was  foolish 
of  Madame,  but  she  would  have  it  that  a  little  sore  spot  which  had 
developed  thereabouts,  doubtless  originating  in  the  varied  dietary, 
etc.,  of  such  a  trip,  was  due  to  that  nasty  new  chemical  material. 
Fortunately  Madame's  counsels  must  have  prevailed,  for  the  dapper 
little  white-haired  savant  was  spared  to  her  for  a  good  few  years 
into  this  century,  and  the  first  radioactive  sore  did  not  go  on  to 
develop  in  the  fell  fashion  which,  since  then,  men  have  learned  to 
dread. 

The  youngest  son  of  Uranus  was  Chronos,  and  the  youngest  son 
of  Chronos  was  Zeus,  a  bright  lad  who  took  the  gloomy  family 
affairs  in  hand  to  some  purpose ;  so  that  ultimately  the  gods  in 
concourse  in  their  high  stronghold  of  Oljnnpus  were  able  to  compose 
the  first  peace  on  record,  and  to  interest  themselves  in  pursuits 
of  which  les  chroniques  scandaleuses  are  all  that  most  of  us  recollect 
of  classical  mythology. 

So  Uranium,  after  a  ruminative  existence  of  thousands  of  millions 
of  years,  brings  forth  Uranium  X,  which  almost  immediately  suc- 
cumbs in  child-birth  of  Ionium  :  in  only  100,000  years  this  generates 

756 


§  932]  RADIOACTIVITY  757 

Radium,  and  he  so  brightens  things  up  that  the  ntudy  of  him  ami 
his  progeny  will  be  enough  for  us. 

§932.  In  the  Hesiodic  theogony  the  new  generation  invariablv 
took  an  early  opportunity  of  overpowering  their  parent,  who  had 
usually  given  them  good  excuse  ;  but  there  is  no  such  struggle  in 
what  might  perhaps  better  be  called  the  metamorphoiiij*  of  th« 
radioactive  atom,  only  a  violent  ecdysis  or  casting-oflF  of  an  un- 
wanted integument,  an  *  alpha  particle,'  leaving  each  time  a  umallcr 
atom  of  distinctly  different  character. 

Thus  Uranium,  of  Atomic  Number  92  and  Atomic  Weight  238. 
produces  Uranium  X,  No.  90,  At.  Wt.  234,  thence  Ionium,  abo 
No.  90,  and  therefore  an  isotope,  At.  Wt.  230,  and  from  it  Radium 
No.  88,  At.  Wt.  226,  and  then  these  losses  of  (mass  4,  charge  -f  2) 
go  on  faster  than  ever.     What  are  they  ? 

That  they  are  actual  particles  you  can  satisfy  yourself.  The 
self-luminous  paint  used  on  the  hands  and  figures  of  watchra 
consists  of  phosphorescent  zinc-blende  with  each  gram  of  which  ia 
intimately  mixed  0-1  milligram  of  radium  ;  this  quantity,  Hufficient 
for  6  sq.  inches,  costing  about  a  guinea  (and  retaining  only  a  quarter 
of  its  luminosity  after  ten  years,  for  the  zinc  sulphide'  get*  tired 
long  before  the  radium). 

Keep  the  watch  in  the  dark  all  day,  for  it  phosphoretwes  diffuaely 
to  ordinary  light,  and  this  partly  masks  the  effect  you  want  to  ««e. 
At  night,  in  the  darkness  of  your  room — for  the  whole  is  hardly 
brighter  than  the  starry  sky — examine  the  paint  with  your  pocket- 
lens,  or  even  your  microscope,  and  you  will  see  it  flickering  with 
rapid  little  splashes  of  light,  like  the  ripple  of  moonlight  on  the  am. 
Undoubtedly  the  zinc-blende  crystal  flashes  because  something 
hit  it,  not  on  account  of  an  '  aura,'  or  an  *  influence.'  or  any  such 
nonsense,  and  the  thing  that  hit  it  was  the  a  jmrticle,  and  he  hit 
hard. 

Using  a  plain  zinc-blende  screen,  and  a  speck  of  radium  on  a 
pin-point,  the  flashes  persist  to  6  or  7  cm.  distance,  after  which 
they  rather  suddenly  cease  ;  this  is  the  limit  of  flight  in  air  for  the 
a  particles. 

Cigarette  paper,  or  the  thin  aluminium  foil  from  the  packet. 
cuts  this  flight  down  by  half,  having  evidently  the  stopping  power 
of  an  inch  of  air  ;  double  thickness  stops  them  altogether.  It  look* 
as  if  the  a  particles  were  rather  large  and  clumsy.  

If  the  exposed  speck  of  radium  is  put  inside  a  gold-loaf  electroKOpe. 
the  charged  leaf  quickly  falls,  whether  its  charge  wero  +  **  "• 
With  a  very  sensitive  electroscope,  and  a  very  minute  trace  of 
radium,  the  leaf  falls  by  jerks,  evidently  as  each  a  jMirticle  it  sboi 
across,  and,  with  several  amplifying  valves,  a  loudspeaker  will 
bark  at  every  one.  . 

The  effect  is  incredibly  greater  than  a  single  electronic  ctiarge 
can  account  for  ;  in  fact,' knowing  the  capacity  of  the  ^Jf^^'^^^f*' 
it  is  found  that  the  a  particle  in  it«  blundenng  haate  bat  broken 


758  ELECTRICITY  [§  932 

200,000  air  molecules  into  pairs  of  ions,  and  this  is  the  charge  the 
leaf  loses. 

This,  therefore,  is  the  number  of  water-drops  that  compose  the 
shining  straight  rocket-path  of  an  a  particle  fired  into  the  wet  air 
of  the  Wilson  expansion  chamber,  §  887,  Fig.  396. 

When  shot  through  the  polar  gap  of  the  strong  electromagnet 
of  §  885,  a  particles  were  deviated  like  the  positive  particles  there 
described,  and  e/w  for  them  proved  to  be  half  96,500  ;  so  that, 
as  e  cannot  be  halved,  m  must  be  doubled,  they  must  be  atoms 
twice  as  heavy  as  hydrogen,  charged  +. 

Rutherford  sealed  up  radioactive  material  in  a  narrow  glass 
tube,  with  walls  only  1/100  mm.  thick,  the  sort  of  tube  one  gets  by 
drawing  down  a  test-tube  hastily  :  some  a  particles  got  through, 
because  a  zinc-blende  screen  sparkled  when  held  near.  The  little 
tube  was  put  into  a  good  vacuum,  and  in  a  week's  time  the  complete 
helium  spectrum  was  shown  by  electric  discharge  through  this. 
So  that  the  a  particle  is  a  Helium  atom,  mass  4,  therefore  carrying 
2  -f-  charges. 

Helium  gas,  even  at  atmospheric  pressure,  was  quite  unable  to 
get  out  of  these  thin- walled  tubes,  so  the  a  particle  must  be  crashing 
along  at  much  greater  speed  than  the  ordinary  molecular  movement 
of  about  1  km. /sec.  Knowing  both  charge  and  mass,  magnetic 
and  electrostatic  deviations,  as  in  §  885,  showed  a  speed  of  17,000 
kilometres  per  second,  or  1/18  speed  of  light.  This  particle  has  a 
5  cm.  range  in  air  :  there  are  many  characteristic  speeds  for  a 
particles  from  various  sources,  even  from  Radium  itself,  as  you  can 
see  in  Fig.  396,  a  ;  this  one  is  a  fair  average. 

The  range  in  air  is  (speed  in  km. /sec. )^  X  10-^^  cm. 

Counting  a  particles  is  evidently  simply  a  question  of  counting 
hits,  and  this  can  be  done  perhaps  most  easily  by  exposing  a  photo- 
graphic plate  to  them  for  an  exact  second,  developing,  and  counting 
black  dots  with  a  lens  ;  or  leaf -jumps  are  counted  in  the  electroscope. 
A  very  little  Radium  suffices,  for  from  1  milligram  of  Radium 
148,000,000  a  particles  are  shot  per  second  :  collected  for  a  whole 
year  these  amount  to  0-15  cu.  mm.  of  helium. 

§  933.  Since  the  actual  mass  of  the  helium  atom  (about  4H)  is 
6-6  X  10-2*  gm.  and  its  speed  as  an  a  particle  1-7  X  10®  cm. /sec, 
its  energy  ^mv^  is  10-^  erg.  Multiplying  by  the  148,000,000  of  them, 
one  milligram  emits  about  1500  ergs  per  second ;  which  is  a  good 
deal  more  than  the  small  fly  of  §  62  could  manage,  all  out. 

With  radium  in  bulk,  most  of  the  a  particles  cannot  penetrate 
the  mass  of  powder,  1 /1000th  inch  thickness  of  solid  being  quite 
enough  to  block  them,  and  those  from  the  surface  get  caught  in 
the  containing  envelope,  so  that  the  whole  of  the  a-  particle  energy 
gets  frittered  away  into  frictional  Heat. 

Multiplying  by  3600,  and  dividing  by  42  million,  the  quantity 
of  this  is  0-130  calorie  per  mg.  per  hour.  To  equal  this,  the  small 
fly  would  have  to  eat  its  own  weight  of  dry  sugar  daily,  a  poor 
lookout  for  bee-keepers  feeding  their  hives  over  winter. 


§  936]  RADIOACTIVITY  7M 

Several  hospitals  have  a  fair  stock  of  radium,  and  if  a  gnrni  could 
be  spared  from  service,  and  dropped  into  an  ordinar>'  vacuum  flaak 
its  130  calories  per  hour,  distribute<i  among  a  few  gramjj  of  matmak 
of  small  specific  heat,  would  soon  bring  about  a  riite  of  tcmraratuni 
of  100°  or  more. 

Roughly,  it  is  the  number  of  calories  produced  by  a  half- burnt 
match. 

§  934.  Radium  is  a  soft  white  easily  oxidizable  metal,  like  lianuia, 
and  is  never  used  as  metal,  but  alwjftvs  a«  soluble  bromide  RaBr,! 
or  insoluble  sulphate  RaSO^,  yet  we  have  been  referring  Nimply  to 
*  radium.' 

By  no  change  of  pressure  or  temperature  obtainable  on  mrth, 
nor  by  any  kmd  of  chemical  combination,  han  th©  radioactivity 
of  any  element  ever  been  altered. 

It  is  true  that  a  radioactive  gas  may  be  pumped  away  from  tlie 
interstices  of  a  powder  in  which  it  has  been  forming,  and'may  leave 
the  powder  comparatively  inert  for  the  time  ))eing.  but  that'i*  only 
a  mechanical  separation  of  diflferent  elements. 

Evidently  Radioactivity  is  a  property  of  the  Sucleus  of  Ike  Alam 
(where  alone  there  is  mass  for  disposal),  these  other  activities  being 
in  the  planetary  electronic  system  around  it. 

We  calculated  the  outside  size  of  this  system,  the  diameter  of 
the  Atom,  in  §  919  as  2-8  x  10"*  cm. ;  now  let  us  cakulate  Iht 
Size  of  the  Nucleus  of  an  Atom. 

Radium  has  Atomic  Number  88,  i.e.  positive  nuclear  charfe 
88e,  which  at  radius  r  causes  a  potential  88  «  r,  §  725. 

The  a  particle  has  charge  2e,  so  that  its  Potential  Energy  EV  at 
distance  r  from  the  centre  =  2€  x  88  e/r. 

The  unit  charge  e  is  15-9  X  10-«»  coulombs,  §883,  but  muat 
here  be  converted  into  electrostatic  units  of  charge,  of  which  a 
coulomb  is  0-1  X  3  X  W^,  §  839,  giving  this  potential  onerKy 
of  the  a  particle  at  start  =  4  x  lO-^jr  ergs. 

Assuming  this  all  converts  into  its  kinetic  energy  l(H  erg,  {  033. 
gives,  on  equating  them 

Nuclear  radius  r  =  4  X  10-"/!^  =  *  X  1^*  ^^' 
which,  you  see,  is  about  1 /3000th  the  half -diameter  of  the  oomplHe 
Atom,  1-4  X  10-«  cm.,  and,  really  rather  nearly,  U  the  reUUve  tin 
of  the  Sun  and  the  orbit  of  Neptune. 

As  the  a  particle  brings  away  2  +  charges,  2-eleotroiM  nitti« 
simultaneously  leave  the  atom  if  it  is  to  remain  n«'*«^^  ^°^ 
come  from  the  planetary  system  at  speeds  too  slow  to  enable  tlieai 
to  do  any  mischief. 

§  935.  This  incessant  conversion  of  the  pftreijV  >nj^  t!^^ 
else  means  that  after  some  length  of  time.  ^«"«J/««    ■^'^ 
only  half  the  original  mass  of  parent  substance  will  bo  left ;  •«^*"^ 
an  equal  time,  only  half  that,  and  so  on.     ^^^  "«  »^«rr^  ff*? 
old,  it  is  always  just  as  Ukely  to  explode  as  any  of  its  fellows,  so  ihrnl 


760  ELECTRICITY  [§  935 

the  rate  of  firing  always  remains  proportional  to  the  number  left ; 
and  the  measurement  of  the  relative  radioactivity  replaces  the 
usually  impracticable  one  of  relative  mass. 

The  Curve  of  Decay  of  Activity,  of  any  one,  is  the  universal 
logarithmic  curve  of  §§  231,  421,  etc.,  the  rate  of  loss  is  proportional 
to  the  amount  left  to  lose  it.  The  shape  of  the  curve  never  changes, 
only  the  horizontal  time-scale,  and  this  varies  enormously,  the 
uranium  curve  falling  to  half-value  in  4400  milUon  years,  radium 
in  1590  years,  and  some  things  in  a  matter  of  seconds. 

All  terrestrial  uranium  has  been  decaying  into  radium  at  that 
rate  since  the  Earth  began,  and  the  radium  produced  has  been  de- 
caying 4400  million/1590  =  2-8  million  times  as  fast,  into  a  gas 
which  soon  leads  to  inertness.  There  has  been  plenty  of  time  to 
arrive  at  the  '  Radioactive  EquiUbrium  Condition,'  that  each 
product  is  decaying  just  as  fast  as  it  is  being  made  ;  and  as  it  plainly 
takes  only  1 /2,800,000th  as  much  radium  as  uranium  to  do  that, 
this  is  the  proportion  of  radium  always  present  in  uranium  when 
mined,  or  360  milligrams  per  ton. 

There  is  no  possibility  of  striking  it  lucky  and  finding  more 
than  this  anywhere ;  uranium  ores,  pitchblende  and  carnotite, 
containing  from  1  to  30%  of  uranium,  are  pretty  uncommon, 
their  treatment  is  long  and  costly,  and  altogether  the  commercial 
production  of  radium  is  a  very  prosaic  affair. 

And  when  you've  got  it,  radium  is  a  wasting  investment :  still, 
1590  years  is  a  long  time ;  few  Roman  buildings  can  be  regarded 
as  half  as  useful  now  as  when  they  were  built. 

§  936.  The  puzzle  is,  what  makes  an  Atom  suddenly  and  spon- 
taneously explode  ? 

First,  what  makes  an  Atom,  with  its  dense  nucleus  and  its  tenuous 
envelope  of  planetary  electrons  ? 

It  is  calculable  that  in  the  interior  of  Stars,  even  our  own  Sun, 
which  is  past  its  youth,  §  560,  the  general  agitation  is  so  terrific — 
call  it  temperature  if  you  will,  and  write  it  in  tens  of  millions  of 
degrees — ^that  no  nuclei  can  retain  their  defensive  retinue  of  electrons 
in  good  order.  Sirius,  the  dog-star,  has  a  close  companion,  as  white, 
and  therefore  as  hot,  as  himself  ;  but,  from  its  feeble  light,  of  very 
small  bulk,  a  mere  pip  to  an  orange.  Yet  it  is  known  to  be  nearly 
as  massive,  its  density  is  70,000  gm.  per  c.c,  to  shift  a  piece  the  size  of 
a  lump  of  sugar  would  tax  your  strength  :  its  nuclei  have  lost  grip 
of  their  electrons,  and,  undefended,  have  fallen  to  pieces  themselves  : 
only  hydrogen  shows  in  the  spectrum  of  the  cool  shell ;  inside, 
protons  are  battering  one  another  '  with  bare  fists  '  as  a  '  super-gas,' 
packed  900  times  closer  every  way  than  in  hydrogen. 

As  things  cool  down,  they  will  begin  to  gather  together  into  twos 
and  fours  (helium),  which  will  at  first  have  a  hard  struggle  for 
existence  ;  then  into  larger  and  larger  groups,  new  chemical  elements, 
cf .  §  560,  as  they  can  impress  more  defensive  electrons  into  service  ; 


§  936]  RADIOACTIVITY  7^1 

but  not  for  a  moment  do  the  constituent  proton«,  neutrons   He 
of   the  nucleus   cease  struggUng ;    any  more  than  the  mofecuW 
of  water  in  §  293  cease  moving  because  they  happen  to  have  gmthciftd 
into  liquid  drops. 

Finally,  a  number  got  thrown  out  and  coolwi  in  a  hurry,  and  that 
was  the  Earth  ;  with  its  collection  of  different  associationji/'  elementa ' 
with  members  numbering  from  1  to  92,  all  trying  to  carry  cm  aa 
permanent  nuclear  societies  :   all  beyond  82  are  failing  to  do  ao. 

The  bits  of  floating  bread  in  §  367  possess  on  the  average  energy 
of  motion  equal  to  the  average  energy  of  the  minnows  anatdiiiw 
at  them,  but  you  see  them  moving  very  irregularlv,  now  ntoppum, 
now  rushing  off  particularly  fast,  when*^three  or  fo'iir  pull  the  Mine 
way  at  once.  Even  with  Brownian  particles,  where  the  numben  of 
molecules  battering  them  are  enormous,  you  watched  the  inrrwiiit 
variations  of  speed.  In  §  293  it  is  shown  how  speeds  ar«  distribated 
above  and  below  the  average  at  any  instant :  if,  instead  oif  1000, 
a  billion  had  been  taken,  it  would  have  been  possible  to  find  a  few 
at  speeds  well  over  double  the  average. 

In  that  paragraph  we  were  concerned  rather  with  keeping  eloee 
to  an  average  ;  here  we  are  seeing  how  far  away  we  can  get.  vVatch- 
oil  '  does  not  evaporate,'  an  eighth  of  a  drop  distributed  throusfa 
a  watch  lubricates  it  for  years,  yet  every  now  and  again  a  moleeole 
of  it  gets  such  a  happily  combined  succession  of  pushes  from  its 
fellows  that  it  gets  shot  out  into  vapour.  '  Its  vapour  presnre  is 
small '  means  that  the  probability  of  this  happening  is  enormously 
less  than  with  petrol,  but  there  is  the  chance. 


I 


The  Probability  of  somebody  scoring  70%  of  marks  in  your 
is  large,  that  of  getting  80%  is  not  small,  but  over  90%  is 
unusual.  Yet  if  the  exam  goes  on  in  scecula  sattulorum  the 
possibility  does  exist  that  one  day  all  the  questions  will  suit  one 
candidate  exactly,  and  everything  will  go  just  right  with  him,  and 
he  will  get  full  marks. 

So  in  the  turbulent  community  of  238  protons  and  146  eleotrooa, 
arranged  we  don't  yet  know  how,  which  together  w©  call  a  Uranium 
nucleus,  in  all  the  possible  combinations  of  arrangements  and  *pM<fe 
that  will  chance  to  be  gone  through  in  a  few  thousand  millkm 
years,  there  will  be  ome  in  which  a  combined  sucoession  of  nhoves 
will  set  up  a  violent  condition  of  resonance  which  will  hc*v»  a 
tetrad  clean  out  of  the  nest.  And  as  a  gram-molecule  of  Uranium 
(which  is  10  oz.)  contains  Avogadro's  0-6  of  an  Knglish  quadnilllioo 
of  atoms,  while  this  long  time  is  only  about  0-2  trillion  «^^^^* 
a  pound  of  uranium  nitrate  is  changing  3  million  atoms  per  ••«*|^- 

Or  a  fourth  of  a  farthing's-worth  of  Radium  makes  a  thousand 
splashes  of  light  on  a  zinc- blende  screen  in  a  second,  am  one  loses 
sight  of  the  smallness  of  the  probability  of  this  *  ^<*"*"^*°?'°' 
breakdown  '  of  its  3  million  times  more  uneasy  nucleus,  in  woodrr- 
ment  at  how  the  invisible  speck  can  go  on  doing  it. 


762  ELECTRICITY  [§  936 

We  have  bitten  into  our  apple,  and  measured  its  core — ^which  is 
the  important  part  of  the  apple — and  seen  a  pip  shot  out,  without  aid 
of  finger  and  thumb,  and  timed  and  traced  its  flight,  and  weighed  it. 
In  another  loculus  we  shall  find  a  super  X-ray  installation  complete. 

§  937.  The  no.  88,  at.  wt.  226,  atoms  of  '  alkaline-earth  metal ' 
Radium  each  eject  an  a-particle,  of  range  4-7  cm.,  at  such  a  rate 
that  half  are  broken  up  in  1590  years,  and  become  the  no.  86,  at.  wt. 
222,  atoms  of  Radon,  the  heaviest  of  the  series  of  chemically  inert 
gases  which  starts  with  Helium  and  Neon.  It  is  a  monatomic  , 
gas  which  glows  in  the  dark,  and  liquefies  at  —  62°  C.  to  a  colourless  jl 
liquid,  solidifying  10°  lower.  Usually  it  remains  clinging  in  the 
radium  powder,  the  maximum  bulk  of  it  '  in  equilibrium  '  with  1 
gm.  being  only  2/3  cu.  mm.  (this  is  called  a  '  curie  '),  but  it  can 
be  removed,  and  of  course  distended  to  any  extent,  by  a  vacuum 
pump  :  it  was  called  at  first  '  Radium  Emanation.' 

A  great  hospital  keeps  750  milligrams  of  radium  in  solution  as 
bromide,  inside  a  safe  lined  with  six  inches  of  lead,  and  twice  a 
week  pumps  off  the  gases  evolved.  Besides  the  valuable  trifle 
sought  for,  these  consist  of  hydrogen  and  oxygen  ionized  out  by  the 
a  particles,  helium  their  last  end,  possibly  HBr  cast  off  from  the 
inert  gas,  and  of  course  water  vapour.  The  oxygen  is  sparked 
away  with  excess  of  hydrogen,  the  radon  is  frozen  out  from  the 
Hg  and  He  by  aid  of  liquid  air,  evaporated  and  dried  and  passed 
into  thin  drawn  glass  tubing  about  1-5  mm.  diameter,  the  long  length 
being  then  sealed  off  in  halves,  quarters,  eighths  and  sixteenths, 
like  barley-corns,  so  that  each  of  these  Radon  Seeds  contains  from 
I  or  1-5  for  general  surgery  up  to  3  or  5  millicuries  for  eye  work, 
these  of  course  being  the  quantities  '  in  equilibrium  with  '  1,  1-5, 
3  or  5  milligrams  of  radium.  As  we  shall  see  in  the  next  paragraph, 
it  is  during  the  breakdown  of  Radon  that  the  curative  radiations 
are  emitted,  so  that  for  a  few  days  these  Seeds  do  all  the  surgical 
work  of  radium,  while  the  salt  itself  is  recovering  its  activity. 

These  are  issued  for  service,  and  returned  after  remaining  in  place  | 
for  a  week,  by  which  time  they  have  lost  3/4  their  strength  and  ' 
become  tinted  dusky  violet,  owing  to  a  particle  action  in  the  glass. 

What  has  happened  inside  them  is  this  :  the  Radon  atoms, 
no.  86,  Wt.  222,  each  emit  an  a  particle  at  such  a  rate  that  half 
are  changed  in  3-8  days,  into  a  solid  non-volatile  deposit  on  the 
glass,  Radium  A,  no.  84,  wt.  218.  In  3  minutes,  by  the  ejection  of 
another  a  particle,  this  has  changed  into  Radium  B,  no.  82,  wt.  214  ; 
and  then  something  different  happens. 

It  had  need,  for  although  the  emission  of  a  particles  takes  nine- 
tenths  of  the  energy  of  Radium,  they  are  quite  useless  in  hospital 
treatment ;  their  range  in  a  soUd  is  so  small  that,  even  if  one  could 
afford  to  paint  the  patient  with  radium  paint,  they  would  not 
penetrate  skin-deep  :  none  of  them  escapes  from  the  seed  tube. 


939]  RADIOACTIVITY 


7AS 


§  938.  The  something  different  is  the  ejection  from  the  nuckpus 
of  Beta  Particles.  These,  examined  in  the  uMual  way,  1 883.  prove 
to  be  free  Electrons,  travelling  at  tremendoun  speedji'  fr«nn  I  3  up 
to  0-96  the  velocity  of  light  (which  is  the  unattainable  maxiroum) 

They  differ  from  the  Cathode  stream  particles  of  the  hut  chapter 
only  in  their  speeds,  which  corre8i)ond  to  driving  volt«f{c«  (rtun 
300,000  to  5  million,  whereas  no  ordinary  X-ray  tube  yet  ulmntU  up 
to  more  than  the  smaller  figure.  These  great  Hpevdii  give  thmi 
more  power  of  penetration  than  cathode  electronji,  which,  aa  vou 
recollect,  cannot  get  out  of  their  tube. 

P  particles,  although  they  vary  enormously,  an  the  curve*  of  their 
paths  in  Fig.  396  indicate,  can  mostly  travel  from  inchen  to  fe»t 
in  air,  and  can  pass  through  thin  glass,  or  get  out  of  a  tin  boi ; 
a  few  can  traverse  2  or  3  mm.  of  aluminium. 

This  really  means  that  they  would  be  absorbed  in  timue  no  locmlly 
that  burning  would  usually  ensue,  and  might  excite  the  very 
mischief  that  radium  is  used  to  cure ;  the  radon  needa  are  tber»> 
fore  enclosed  in  platinum  needles  with  walls  0-3 — 0-5  mm.  thick. 

Before  dealing  with  what  does  get  out  of  these  retentive  *  ftlter*/ 
let  us  finish  the  Story  of  Descent : — 

RaB  fires  a  p  particle,  with  half-period  27  minutes,  producing 

RaC  which  fires  a  p       ,,  ,,  20 

RaC  fires  a  fast  a  particle  instantly,  thereby  bringing  back  the 

atomic  number,  which,  being  the  nuclear  -f  charge,  had  gone 

up  2  ;  and  produces 
RaD  fires  a  p  particle,  with  half-period  22  years,  which  in  no  tlow 

that  nobody's  stock  of  radium  has  yet  reached  full  equdibnuiii 

output  for  the  final  processes,  viz.  : — 
RaE  fires  a  p  particle  with  half-period  5  days,  producing 
RaF,  which  is  Polonium  no.  84,  at.  wt.  210,  and 

fires  an  a  particle,  with  half- period  140  days  producing 
Radio-Lead,  inactive,  no.  82,  at.  wt.  206,  isotope  of  lead  207. 

In  a  month,  practically  all  Radon  has  decomposed  ;  B  V  and  C 
have  followed  it,  and  the  seed  contains  only  D,  which  baa  a  alight 
laboratory  value  as  slowly  breaking  down  into  polonium,  •ourve 
of  a-rays  with  no  complications  to  follow. 

§  939.  Radium,  sealed  up  in  a  tube,  as  it  is  mostly  umd,  of  coune. 
goes  on  producing  Radon,  which  never  accumulates  in  quanlity 
because  its  life  is  brief  before  generating  the  whole  string,  so  thai 
'  Radium  '  commonlv  connotes  a  salt  of  radnim  metal  '  m  radio. 
active  equilibrium  with  '  its  long  family,  down  as  far  as  mi»rly 
D,  hanging  on  so  long  that  its  output  can  be  ignored  in  radium  as 
ordinarily  experimented  with.  ^^^.^^^^^, 

*  Radium '  therefore  is  firing  off  4  «  particles,  which  reprwrni 
an  energy  output,  per  gram,  per  hour,  in  calories,  i."*  2.  2»-H.  3HJ 
and  430,  total  129-9,  and  2  p-particles,  together  6-3  calones-  And 
now  comes  the  Sting  in  ite  Tail,  which  might  bo  expect«l  wtosn 


764  ELECTRICITY  [§  939 


particles  of  enormous  energy  are  in  association  with  atoms  heavier 
even  than  tungsten,  X-rays  of  the  utmost  '  hardness  '  and  penetra- 
ting power,  called  Gamma  Rays,  some  of  them  '  5-million-volt ' 
rays,  a  quarter  of  them  pierce  an  inch  of  lead,  1 /400th  can  pass 
through  armour  plate  :  true  radiation,  like  light  and  X-radiation, 
but  with  wave-lengths  from  that  of  the  X-ray  of  §  919  down  to  one 
ten-thousandth  of  it,  an  English  billionth  of  a  centimetre.  Their 
energy  totals  9-4  cals./gm.-hr. 

It  is  these  '  y  ^^Y^ '  that  escape  through  the  0-5-mm.  wall  of  the 
platin-iridium  Radium  Needle ;  and  the  rather  less  hard  ones, 
preferentially  absorbed  in  malignant  tissue,  inhibit  its  cell- division, 
and  kiU  it,  for  removal  by  natural  processes.  The  fewer  intensely 
penetrative  rays  escape  :  one  stores  radium  in  a  safe,  under  at  least 
3  in.  of  lead,  in  an  uninhabited  part  of  the  building. 

Thus  you  see  the  tail  is  the  business  end  of  radium,  radon  seeds 
and  radium  needles  are  merely  alternative  ways  of  working,  the 
former  involving  more  labour  but  less  risk  of  loss.  And  radium, 
with  the  radon  pumped  off,  or  ignited  off,  is  for  a  few  hours  almost 
inert,  and  safe  to  handle.     * 


I 


§  940.  Unfortunately,  the  energy  of  the  y-rays  is  only  about  ^ 
6-5%  of  the  total,  so  that  the  waste  of  effort  in  this  most 
costly  of  remedies  is  considerable,  especially  in  view  of  the  slow 
supply  and  the  extremely  small  size  of  these  ultra-high-pressure 
X-ray  laboratories,  radium  atoms. 

That  is  a  good  reason  for  pushing  on  with  '  million- volt '  X-ray 
plant,  of  engineering  scale  and  power.  A  25-ft.  tube,  built  up  of 
glass  sections  like  a  double-ended  telescope,  has  been  in  use  on 
700  kv.  in  the  California  Institute  of  Technology  since  1931,  on 
four  patients  at  a  time  (who,  of  course,  see  nothing  but  little  peep- 
holes in  a  vertical  column  apparently  supporting  the  ceihng),  and 
Millikan  assured  me  the  next  year  that  they  were  '  getting  results '  of 
a  quite  different  order  to  the  200-kv.  outfits. 

§  941.  The  other  natural  radioactive  series — abbreviated. 

There  is  evidence  that  3%  of  the  breakdown  of  Uranium  goes 
by  way  of  Actinium,  which  has  given  a  name  to  the  second  series, 
a  13-year  curiosity,  the  chemistry  of  which  is  unknown,  and  descent, 
through  protactinium  (of  some  stability)  still  doubtful.  It  produces 
radio -actinium,  an  isotope  of  thorium,  then  AcX,  isotope  of  Ra, 
then  actinon,  no.  86,  at.  wt.  219,  isotope  of  radon  ;  a  gas  of  half -period 
4  seconds.     The  series  ends  in  Lead,  207,  in  an  hour. 

Thorium,  no.  90,  at.  wt.  232,  is  of  course  the  basis  of  the  gas-mantle 
industry.  It  is  4  times  slower  in  breaking  up  than  uranium,  but 
a  mantle  will  print  itself,  through  black  paper,  on  a  photographic 
plate,  in  a  month.  Its  products,  meso-thorium  1  and  2,  have 
long  been  in  use  in  Europe  as  a  serviceable  substitute  for  radium. 
Thoron,  no.  86,  at.  wt.  220,  is  a  gas  of  half-period  55  sec.  The 
series  ends  in  Lead  208,  in  a  day. 

All  three  series  exhibit  minor  compUcations. 


§942] 


RADIOACTIVITY 


7«S 


§  942.  Comparison  of  quantities  of  radioactive  material.  All  thne 
radiations,  a,  p  and  y,  ionize  the  air,  and  therefon.  enaS;  the  chaj^ 
to  leak  away  from  a  gold  leaf  Electroscope.  It  in  charged  TX 
usual  way  and  then  watched  from  a  di.stance.  and  t l^^rf.  i 
fall  of  the  leaf  from  one  fixed  mark  to  another  are  comparS^tS 
quantities  in  action  are  proportional  to  them.  As  the^we  'mw 
possible  sources  of  error,  it  is  preferable  to  get  the  «peed«  ne«rlV 
equal  by  altering  the  amount  of  substance,  e.g.  the  numbeTS 
needles    used  as  standard.  »u«wuw  « 

Siiice  a-particles  have  only  a  short  range,  '  a-rav  comiMmoiM ' 
are  best  made  as  in  Fig.  404,  right,  the  material  being  gp^  v«T 
thinly  in  a  flat  tm  Ld,  and  causing  a  leak  from  the  meUl  pUta 
brought  down  close  above  it. 

This  is  immediately  convertible  into  a  p-ray  comparator  by  corer- 
mg  the  material,  as  shown  by  the  pecked  line,  with  ordinary  aln. 
mimum  foil,  which  is  about  001  mm.  thick ;   two  layers  wiU  stop 


Fio.  404. 

all  a-particles.  Different  thicknesses  of  metal,  up  to  1/8  in.  l«ul, 
enable  one  to  sort  out  slow  and  fast  ^-particles. 

Radium  needles  suspected  of  cracks,  which  will  let  out  radon,  are 
kept  in  a  box,  and  then  the  air  from  the  box  ia  carefully  poured  be- 
tween the  two  plates  ;  the  heavy  gas  will  cause  an  immeoiate  leak. 

Surgical  Radium  Needles  are  compare<l  by  their  y-TAy  output. 
The  electroscope,  with  all  the  gadgets  on  the  right  removecf,  i» 
shielded  by  eighth-inch  sheet-lead  (black),  and  the  needles  are  kid 
in  succession  in  a  fixed  notch,  as  on  the  left,  at  some  distance,  say 
20  cm.,  where  they  cause  a  conveniently  rapid  leak. 

An  ordinary  1-mg.  needle  is  shown,  half  size,  iK>inting  downward* 
on  the  left  of  the  Figure.  Needles  are  made  of  platinum  hardened 
by  10%  iridium,  for  stiffness  and  strength  ;  the  wall  U  0-5  mm. 
thick.  A  sharp  triangular  point  and  a  numbered  eye  are  hard- 
soldered  on.  The  radium  sulphate,  bulked  up  with  barium  sulphate. 
is  in  little  tubular  containers,  dropped  into  place  inside ;  a  S-mg. 
needle  will  have  3  in  succession. 

If  there  is  any  doubt  about  the  uniform  distribution  of  actire 
material  in  the  needle,  it  is  laid  on  a  photographic  plate  for  a  few 
seconds,  and  the  black  blur  which  develope  up  must  not  be  patcby. 


766  ELECTRICITY  [§  942 

Needles  are  left  in  place  for  a  week  or  a  fortnight.  Any  gone 
a-missing  must  be  searched  for  in  the  ordinary  way,  and  by  X-rays. 
An  electroscope  is  a  very  lame  help  to  finding  a  small  quantity  in 
an  awkward  place,  even  a  foot  away.  Burnt  radium  has  probably 
lost  its  radon,  and  emits  only  a  rays  of  no  penetrating  power ;  it 
is  therefore  safe  to  handle,  for  a  few  hours ;  and  until  it  has 
'  emanated  '  a  fair  amount  of  radon,  and  this  its  RaC,  say  in  24  hr., 
it  cannot  be  found  by  the  electroscope  :  suspected  ash  and  clinker 
must  therefore  be  kept  a  few  days. 

A  new  and  simple  application  of  the  delay -action  properties  of  the 
domestic  neon  night-light,  however,  has  resulted  in  a  radium 
detector  1000  times  as  sensitive  as  an  ordinary  portable  electroscope. 

§  943.  Fig.  405  epitomizes  the  distinction  between  a,  p,  and  y. 
On  a  strip  of  photographic  film  stands  a  diminutive  lead  cannon 
loaded  with  a  speck  of  radium  ;  an  inch  above  is  a  second  film. 
Since  thick  lead  absorbs  almost  all  that  strikes  it,  it  is  only  what 
travels  along  the  bore  of  the  gun  that  comes  under 
observation.  The  whole  stands  between  the  pole- 
pieces  of  an  electromagnet,  so  that  it  is  in  a  strong 
magnetic  field,  the  lines  of  which  run  straight  away 
from  you,  through  the  paper. 

Without  the  magnetic  field,  development  of  the 
films  after  a  short  exposure  shows  a  black  spot  in 
the  direct  line  of  fire  at  C. 

With  a  strong  field  there  appear,  in  addition, 
blackenings  at  A  and  B.     If  a  piece  of  thin  mica  or 
paper  be  laid  on  the  muzzle,  or  if  the  upper  film  be 
Fig.  405.  more  than  7  cm.  away  in  air,  spot  A  is  missing.     A 

thicker  plate  weakens  spot  B  also. 
A  is  therefore  caused  by  a  stream  which  is  deflected  like  a  current 
as  it  crosses  the  magnetic  field.  It  is  a  stream  of  positively  charged 
a-particles,  its  very  slight  deviation  shows  that  the  electromagnetic 
force  makes  but  little  difference  to  their  momentum,  they  are 
heavy  particles  travelling  fast,  but  they  are  big  and  soon  stopped 
by  molecular  collisions. 

The  sharp  curve  over  to  B  (really  many  hundred  times  as  sharp 
as  that  to  A)  the  opposite  way  is  evidently  the  negative  electron 
stream,  of  particles  possessing  comparatively  little  momentum, 
but  in  spite  of  that,  able  to  penetrate  some  thickness  of  solid,  so 
they  must  be  smaller. 

The  C  spot  is  unmoved  and  unchanged,  whatever  you  do  :  it  is 
the  y  stream  of  pure  radiation,  without  particles  or  charges,  travelling 
straight  through  everything. 

§  944.  The  internal  heat  of  the  Earth.  The  temperature  is  ob- 
served to  increase,  on  the  average,  1°  C.  per  32  m.  increase  in  depth 
underground,  and  the  rocks  in  which  this  rate  of  increase  has  been 
measured  have  an  average  thermal  conductivity  0-004.     The  area 


il 


I 


§  944]  RADIOACTIVITY  757 

of  a  sphere  40  million  m.    circumference  is  5-1  x  10"  «o    cm 
therefore,  applying  the  formula  of  §  238 

Loss  of  heat  outwards  by  Earth  =  0004  X  5-1  X  10"  x  I*  —  3200 

=  6  X  10*2  calories  per  sec. 
=  216  x  101*  calories  per  hr. 

Calculating  from  the  known  melting  points  and  specific  he«U 
of  materials,  Lord  Kelvin  informed  geologists  in  1870  that  this  lowi 
of  heat  could  only  have  been  going  on  for  40  million  years,  instca<l  of 
the  400  they  demanded,  and  shortly  afterwards  assurwl  them 
that  the  Sun  would  last  only  another  40  million. 

The  row  that  followed  lasted  the  rest  of  the  centiirj- ;  until 
radium  was  discovered,  and  found  to  be  producing,  in' all,  145 
calories  per  gram  per  hour.  So  that  if  the  Earth  contained 
216  X  1014  ^  145  =  15  X  W*  gm.  =  150  million  tons  of  radium 
m  the  whole  of  its  6  x  IO21  tons  mass,  its  loss  of  heat  would  be 
balanced  by  its  gain. 

Lord  Rayleigh,  having  discovered  that  radium  was  widcdpm&d  in 
rocks,  and  having  measured  amounts  between  0-7  x  10-**  gm. 
per  gm.  in  Rum  olivine,  and  10  times  as  much  in  Cape  granite, 
took  a  mean  value,  of  4  x  IO-12  gm.  per  gm.  of  rock,  which  in  a 
depth  of  30  km.  of  crust,  of  mean  density  2-5,  gives 

(area  5  X  lO^^  X  depth  3  X  10«)  c.c.  x  2-5  gm.  of  rock  x  (4  X  10"") 

the  required  150  million  tons.  This  looks  as  if  the  Earth  had  a 
better-than-electrically-heated  overcoat  quite  capable  of  keeping 
it  warm,  whatever  might  be  underneath.  (Iron  meteorites  contain 
no  radium,  and  from  the  earth's  mean  density  5-5,  its  core  is  pro- 
bably of  iron  or  highly  ferruginous  slags,  which  in  cooling  may  have 
gradually  floated  most  of  the  lighter  radiferous  minerab)  up  towanU 
the  surface.)  So  that  in  1906  geologists  were  invited  to  take  their 
own  time,  and  1000  million  years,  and  a  second  helping  if  they  wanted. 

Subsequently  the  radioactivity  of  thorium  was  diacovered, 
and  then  that  of  the  potassium  isotope,  slight  but  important  iMvauw 
of  its  greater  abundance,  and  some  geologists  began  l<»  fwl 
uncomfortably  warm. 

Of  course  it  was  an  Irishman  who  saw  the  way  out ;  the  Earth 
gets  hotter  and  colder  by  turns.  Trust  anything  in  Natun*  to  go 
quietly  on,  and  it  is  sure  to  start  swinging  pendulum  fashion : 
our  own  Sun  is  a  variable  star  with  an  11 -year  period,  why  not  old 
Earth  ? 

Just  now  it  is  getting  hotter,  the  underside  of  its  cnist  is  melting 
to  higher  and  higher  levels,  presently  will  come  a  breaking  through 
in  volcanic  eruptive  activity  all  over  the  surface,  then,  the  feTcr 
past,  and  chill  ensuing,  a  settling  down  to  a  reconstructed  map, 
until  the  next  outburst. 

Nay,  in  but  ten  years  geologists  have  found  evidence  oi  foor 


768  ELECTRICITY  t§  944 

major  upheavals  of  this  character  and  at  least  two  dozen  lesser 
ones,  about  30  million  years  apart.  Of  course  the  assumption  is 
made  that  there  is  always  plenty  more  where  that  came  from ; 
but  why,  every  time,  the  chill  should  be  so  profound  and  so  severe, 
does  not  seem  at  all  easily  explainable. 

Lord  Rayleigh  deprecates  introducing  this  into  a  physics  course, 
'  plausible,  but  speculative,'  he  calls  it. 

That  is  why  it  is  put  here  :  this  is  a  course  in  Natural  Philosophy. 
*  Down  to  twenty  miles,'  says  he,  in  effect,  '  and  then,  I  don't  know,' 
and  he  wouldn't  claim  that  as  infallible.  So  they  under-dig  him, 
and  erect  an  attractive  superstructure,  '  foundations  by  Rayleigh 
and  Co.' 

You  will  find  that  sort  of  thing  going  on  in  other  sciences ;   you 
will  find  it  everjrwhere,  wherever  you  go.     All  your  life — look  to>| 
the  foundations.  || 

§  945.  The  Age  of  the  Earth.  This  there  are  now  several  ways 
of  estimating,  and  they  do  not  greatly  disagree.  Radioactivity 
furnishes  two  : — 

The  uranium-lead  ratio.  Knowing  the  long  pedigree,  and  what 
each  member  does,  and  measuring  the  radioactivity,  of  uranium, 
it  is  easy  to  calculate  that  a  ton  of  uranium  annually  produces 
0-15  milligram  of  lead,  which  stays  with  it  as  non- volatile  and  in- 
soluble lead  uranate.  In  only  100  milHon  years  this  adds  up  to 
15  kilograms,  enough  not  only  to  determine  accurately,  but  to  do 
an  '  atomic  weight '  on,  to  make  sure  that  it  is  the  206  isotope,  and 
to  correct  for  any  contamination  with  common  lead,  207-2,  which 
might  have  been  there  from  the  beginning. 

From  the  various  proportions  of  lead  so  determined,  in  uranium 
minerals  from  the  rocks  of  different  geological  ages,  it  maj''  here  be 
summed  up  shortly  that  '  the  beginning  of  the  Tertiary  dates  back 
60  million  years  ;  the  Upper  Palaeozoic,  from  Permian  to  Devonian, 
covers  from  205  to  375  milHon  ;  and  the  Pre-Cambrian  gives  results 
grouped  about  600,  900,  and  1050  million  years,  and  there  must 
have  been  one  or  two  similar  eras  before  that,  to  provide  sedimentary 
materials  of  this  last  horizon.' 

Per  contra,  if  all  the  7-5  gm.  of  lead  originated  from  the  6  gm. 
of  uranium  and  15  of  thorium  per  ton  of  average  igneous  rock, 
the  age  cannot  have  exceeded  3000  million  years. 

Pleochroic  haloes.  '  Pleochroic '  means  changing  colour  in 
polarized  light. 

'  Black '  biotite  mica  is  one  of  the  commonest  constituents 
of  igneous  rocks  ;  glittering  flecks  of  it,  weathered  out  in  abundance 
from  the  ancient  rocks  of  the  Emerald  Coast,  form  a  dark  cloud 
which  lifts  and  settles  with  every  ripple  on  the  sands  of  the  Brittany 
shores  west  of  St.  Malo. 

Such  mica  is  often  spotted  with  tiny  black  dots,  which  under 
the  microscope  show  as  lighter  or  duskier  circular  areas,  with  radius 


§  946]  RADIOACTIVITY 


769 


0017  mm.,  surrounding  a  minute  inclusion,  whicli  from  itit cleAniew 
and  high  refractivity,  and  occasionally  crystalline  8luipe,  i»  idenii- 
fiable  as  Zircon,  a  mineral  containing  up  to  10%  of  uranium. 

It  was  mentioned  in  §  937  that  radium  darkcnji  gUum  :  thiw 
black  circles  have  a  radius  which  is  the  range  of  the  «  partk-lo  in 
mica,  and  100  or  more  particles  have  l)een  shot  out  ovcr>'  year  to 
produce  the  darkening;  this  can  be  mea«ure<l,  and  compiu«*l 
with  that  of  haloes  produced  experimentally  in  mica  by  known 
small  amounts  of  radium,  perhaps  100  million  timen  stronsvr. 
in  a  year  or  two. 

The  zircon  inclusion  is  micrometered  under  a  higher  power. 
and  its  mass  calculated,  and  credited  with  the  maximum  I0"o  of 
uranium  the  mineral  is  ever  known  to  contain.  From  that  you 
see  it  is  easy  to  compute  the  minimum  age,  around  which  the 
measurements  of  numerous  haloes  gradually  cluster  ;  and  the  remlu 
tally  quite  well  with  those  of  lead-content  from  roekH  in  the  Mine 
horizon. 

The  older  haloes  show  an  outer  penumbral  ring  of  about  7/4thii  the 
diameter,  due  to  the  longer-range  a  particles  from  Radium  C. 
Thorium  haloes  are  about  l/20th  mm.  diameter. 

§  946.  The  transmutation  of  the  elements.  Some  progrcM  haii 
been  made  with  this  Modem  Alchemy,  but  nothing  of  a  nature  to 
shake  the  Gold  Standard. 

One  throws  stones  at  the  Nucleus,  and  looks  for  the  pieces. 

This  is  a  very  different  problem  from  that  of  removing  a  few 
planetary  electrons,  which  is  effected  in  any  ordinary  electric 
discharge  tube  by  the  pull  of  the  field  :  here,  one  exceedingly 
small  thing  has  to  score  a  direct  hit  on  another,  which  in  repelling 
it  violently. 

Perhaps  one  might  put  it  better  in  a  way  which  will  n*mind  you 
either  of  Sisyphus  or  of  a  seaside  anmsement  jwirk  ;  you  have  to 
roll  a  ball  up  the  steep  '  potential '  slopes  of  a  model  Fujiyama,  in 
the  hope  of  getting  it  into  the  crater  and  provoking  an  exploetion. 

The  ball  is  the  swift  a  particle  from  Radium  C  :  in  1919  a  radium 
patch  was  removed  beyond  7  cm.  from  a  zinc-blende  8creen.  which 
abruptly  ceased  to  sparkle,  because  that  is  their  limiting  range; 
but,  watching  the  screen  through  the  microscoix*.  occasional  flaane« 
were  still  observed,  and  these  persisted  3  or  4  timon  w*  far,  an 
altogether  incredible  flight  for  an  a  particle.  In  oxygen  they  dis- 
appeared, in  pure  nitrogen  they  were  25%  more  frequent. 

The  experiment  was  transferred  to  a  Wilson  ex|Miiuiion  chnmlicr. 
§  887,  i.e.  a  speck  of  radium  was  stuck  inside  the  rim  of  thi*  mirt 
of  air-tight  Petri  dish  full  of  moist  nitrogen,  and  then  the  U^tt.mi 
was  suddenly  pulled  down  1/4  in.,  the  moisture  condenned  cm  the 
ions,  and  the  shining  straight  white  rocket-tracks  of  rrrent  « 
particles  were  photographed  with  a  stereoflcopic  camera. 

Out  of  270,000  tracks  examined  in  1925,  8  forkwi  at  the  lip.  mto 
a  right-angled  Y,  one  leg  short  and  bright,  the  other  very  long  and 
CO 


770  ELECTRICITY  [§  946 

thin,  Fig.  396,  collision.  Evidently  the  original  particle  had 
hit  something,  which  split  up ;  and  assuming  that  this  was  an 
explosion  (pushed  all  to  one  side  by  the  a  particle's  blow),  the 
momenta  of  bullet  and  gun  should  be  equal,  §  61  :  from  their 
lengths  it  was  calculated  that  the  masses  were  as  1  to  17. 

W,'  +  He^  =  0^^  +  HI 

Note. — The  superscripts  are  atomic  weights,  the  subscripts  atomic 
numbers,  which  are  the  numbers  of  +  charges,  protons,  in  the  nucleus. 

The  a  particle  coalesced  with  the  nitrogen  nucleus,  but  the 
total  energy  was  now  too  much  for  stability,  the  thing  blew  up 
with  a  violence  of  its  own,  much  greater  than  just  what  the  a  particle 
had  brought,  firing  off  a  proton,  i.e.  a  hydrogen  atom  minus  its 
electron,  therefore  charged  +  1j  small,  fast,  and  far-flying,  and 
leaving  an  atom  with  the  very  stable  figure  of  eight,  an  isotope  of 
oxygen,  not  actually  detected  in  the  ordinary  way  until  six  years 
later.     Magnetic  deflection  confirmed  the  +  H. 

BerylHum  shot  at  by  a  particles  did  not  produce  protons,  but  in 
1932  '  rays  '  were  detected  which  made  little  of  penetrating  2  in. 
of  lead,  and,  themselves  invisible  in  the  Wilson  chamber,  seemed 
to  give  rise  to  disconnected  splashes  in  it,  at  any  angle,  Fig.  396. 

Bel  +  He^  =  CI'  +  (J) 

This  was  the  neutron  of  §  923,  the  proton-electron  combination 
of  mass  1,  escaping  alike  electric  and  magnetic  temptations  to 
turn  aside,  because  carrying  no  charge ;  finally  accidentally 
knocking,  out  of  the  HgO  present,  a  proton  which  crashed  away 
with  much  making  of  ions — somehow  one  thinks  of  an  old  hen  hit 
by  a  stray  pellet,  shedding  squawks  and  feathers  as  she  goes. 

These  intensely  penetrative  Neutrons,  thus  obtained  via  Beryl- 
lium, have  now  been  employed  to  bombard  all  the  chemical  elements, 
and  every  one  has  exhibited  induced  radioactivity,  lasting  perhaps 
a  few  hours.  Those  below  atomic  no.  70  have  stepped  up  to  the 
next  higher  element  :   all  above  that  have  formed  isotopes. 

But  radium  is  an  expensive  gunpowder,  and  a  milligram  of  it 
provides  only  37  million  really  fast  shots  per  second,  the  a  particles 
from  Ra-C,  whereas  a  single  micro-ampere  current  in  a  discharge 
tube  at  about  0-1  mm.  pressure  can  provide  6  billion  '  positive-ray 
particles,'  i.e.  protons,  §  886.  It  is  true  that  the  Ra-C  a  particles 
carry  the  kick  of  4  million  volts,  but  remarkable  results  have  been 
obtained  at  the  Cavendish  Laboratory  with  one-seventh  as  much  : — 

Firing  protons  at  Lithium  has  produced  Helium 
Ul  +  HJ  =  He^  +  He^ 
and  from  an  Li^  isotope  He^  has  also  appeared. 

Firing  at  paraffin  wax  produced  an  isotope  of  Nitrogen 

c^^  +  HI  =  n;^ 


I 


947] 


RADIOACTIVITY 


S 


amplifi 


^ 


771 

but  this  is  unstable,  and  decays  with  half-pericxi  11  min  and  him 
been  observed  to  produce  something  which  coUided  with  a  curUnc 
electron  m  the  expansion  chamber,  and  both  vanished. 

N^3  =  CI'  isotope  +  (?)  the  long-sought  positron. 

A  simple,  though  bulky,  apparatus,  now  being  improved  upon 
IS  shown  m  Fig.  406.     By  30  kilovolts  a  red  stream  of  po«itivelv 
charged  hydrogen  particles  is  driven  through  the  minute  hole  in 
the  cathode,  and  is  then  accelerated  by  100  kv. 
in  the  high-vacuum  chamber,  and  falls  on  the 
target  at  the  bottom.     Particles,  knocked  out 
of  this,  pass,  through  a  very  thin  mica  win- 
dow,  into  a  catcher  connected  to  a  6-valve 
amplifier,  actuating  counters. 

[In  this  apparatus,  a  minute  leak  admits 
gas  enough  into  the  upper  part  to  supply  the 
ions.  It  flows  through  the  narrow  tubular 
cathode  into  the  larger  glass  cylinder,  which 
is  kept  much  more  highly  exhausted  by  a 
large  molecular  pump,  §  107,  so  that  no 
collisions  occur  :  the  tubular  metal  sheaths 
prevent  straying.] 

The  proton  bullet  is  rather  light  compared 
with  the  a  particle,  but  '  heavy  hydrogen  ' 
has  provided  the  diplon.  This,  shot  at 
'  diplogen  ammonium  sulphate,'  excites  a 
machine-gun  rattle,  as  compared  with  the  dropping  fire  of  the 
proton 

D?  +  Di  =  He.^  =  He.^  +  (S)  neutron 
because  the  normal  helium  produced  is  overloaded  with  energ\'. 

Alpha  particle,  proton ;  neutron,  diplon ;  electron,  positron ; 
gamma  ray  and  photon :  the  traditional  Philosopher's  Stone  has 
suddenly  become  a  handful. 

§  947.  Let  us  conclude  this  very  earthy  chapter  with  a  mention 
of  the  entirely  non- terrestrial  Cosmic  Rays. 

It  is  easy  to  contrive  a  gold-leaf  electroscope  so  that  leakage  through 
its  insulating  supports  cannot  discharge  it,  but  when  thin  haa  been 
done,  it  is  still  found  that  the  charge  gradually  disiippean*,  day  bv 
day,  even  when  the  electroscope  is  boxed  in  by  lead  thick  enough 
to  stop  every  y  ray  from  stray  radioactive  traces  in  the  soil,  etc. 

Hidden  in  a  cavern,  or  sunk  30  m.  deep  in  a  radium-free  lake,  the 
leak  is  small.  On  top  of  a  great  stack  of  ingots  of  lead,  which  might 
be  expected  to  stop  anything  coming  up  from  the  earth,  the  leak 
is  unaffected.  In  the  depths  of  a  canyon  open  to  the  sky.  but  into 
which  the  sun  does  not  reach,  it  goes  on  as  usual,  day  and  night. 
High  up  in  a  balloon  it  increases  many  score  times. 

Thus  the  exciting  cause  is  from  above,  not  from  below. 


tki 


\um.  tary«t 

Fio.  406. 


772  ELECTRICITY  [§  947 

A  trap  is  laid  for  it  by  putting  a  flat  Wilson  expansion  chamber 
in  between  the  broad  polar  faces  of  a  great  electromagnet.  At 
opposite  ends  of  a  diameter  are  contrivances  which  amount  to 
electrical  turnstiles ;  if  a  ray  gate-crashes  both  of  them  the  ex- 
pansion is  actuated,  and  a  photograph  taken  of  the  ion-drops  still 
marking  its  track  across  the  intense  magnetic  field,  Fig.  396,  cosmic. 

A  charged  particle  must  curve  round  in  this  field,  as  always, 
sharply  if  its  energy  is  small,  only  slightly  if  it  is  large.  The  field 
is  so  strong  that  electrons  started  up  by  traces  of  radioactivity 
curl  round  into  little  rings  scarcely  1  mm.  diameter ;  but  you  will 
find  quite  a  difficulty  in  measuring  the  trifling  deviation  of  the 
cosmic-ray  lines  against  a  straight-edge.  Their  energy  is  found 
to  correspond  to  a  driving  voltage  of  10^®  volts,  ten  thousand 
millions,  at  least,  which  puts  thunder-clouds  out  of  the  running  as 
a  possible  origin,  for  their  best  effort  is  only  a  tenth  as  much,  §  898. 

All  the  tracks  are  curved,  however,  one  way  or  the  other,  so  that  . 
the  shower  consists  of  both  electrons  and  positrons. 

They  come  down  at  all  angles,  more  from  east  than  from  west 
of  the  meridian,  as  the  Earth  rolls,  but  some  may  even  arrive 
horizontally  from  the  west.  All  are  curling  one  way  or  the  other, 
according  to  sign,  round  the  lines  of  her  far-reaching  magnetic 
field  ;  and,  like  the  auroral  particles  from  the  Sun,  they  spiral  down 
polewards  in  complicated  paths. 

Their  speed  is  almost  that  of  light,  and  their  energy  is  so  enormous 
that  it  can  only  have  come  from  the  complete  conversion  of  matter 
{v.  §  978,  at  the  rate  of  1  gm.  into  (speed  of  light)^  ergs),  and  we 
know  nowhere,  except  the  interior  of  stars,  §  936,  for  this  to  happen. 
There,  astrophysical  theory  can  permit  their  existence  only  on  the 
strict  condition  that  none  escapes  through  the  cool  envelope  that 
we  see  ;  certainly  there  are  no  signs  of  any  coming  from  our  local 
star,  and  it  is  unimaginable  that  life  on  earth  would  survive  any 
such  leakage. 

Whether  their  numbers,  now  scanty,  now  arriving  in  showers 
and  sudden  bursts,  started  from  the  unplumbed  depths  of  space 
before  the  Earth  was  born,  whether  they  originate  at  the  annihila- 
tion or  the  creation  of  matter,  or  whether  some  prosaic  explanation 
awaits  us  after  all,  we  do  not  yet  know. 


r 


RADIOACrrVlTY  77J 


EXAM  QUESTIONS,   CHAPTER   LV 

You  must  be  clear  about  the  distinctive  character  ami  propertira  of  «.  A 
and  y  :  this  involves  also  §§  942,  943.  See  942  domonKtratod  if  you  poaBibly 
can. 

In  all  the  radioactive  genealogies  the  important  things  to  you  aro  Radium 
complete,  radium  '  metal,'  rewlon,  and  ratlium  C. 

The  final  §§  are  off  the  exam  track,  but  much  in  the  public  «)'o. 

1.  Give  a  short  account  of  Radioactivity.     ( x  2) 

2.  Give  a  short  account  of  the  properties  of  Radium. 

3.  Compare  and  contrast  the  properties  of  X-rays  and  those  of  rsdiations 
from  radium. 

4.  Given  a  radioactive  ore,  how  would  you  detect,  and  determine  the  types 
of  its  radiations  ?     (  X  3) 

5.  Compare  Cathode,  X,  and  y  rays. 


RADIATION 


CHAPTER   LVI 


RADIATION 


§  951.  Radiation  is  energy  in  progress  through  space  in  directions 
that  radiate  from  its  source  ;  the  signals  from  the  radio-station,  the 
glow  of  warmth  from  the  cheery  red  fire,  the  sun's  rays  break- 
ing through  the  clouds  '  drawing  water,'  scorching  and  tanning 
our  skins,  X-  and  y-rays ;  energy  travelling  unattached  to  any 
material  particle,  sent  out  by  matter,  travelling  always  at  the 
utmost  possible  speed  unless  matter  intervenes,  ultimately  absorbed 
and  stopped  by  matter.     Read  again  §  471. 

By  it  we  live  and  move  and  have  our  being. 

Drop  a  lighted  match  into  water  and  it  hisses ;  drop  a  burnt - 
out  match  into  liquid  air  and  it  also  hisses — all  matter  contains 
energy.  But  now  comes  the  strange  thing  :  matter  seems  deter- 
mined not  to  retain  energy.  Bottle  up  some  hot  drink  in  a  vacuum 
flask,  and  forget  it  for  a  week,  and  it  has  given  away  its  energy 
through  the  vacuum.  Warm  and  cool  half-a-dozen  thermometers 
to  different  temperatures  and  put  them  into  a  flask  and  exhaust 
the  air  ;  the  hotter  give  heat  and  the  colder  take  it,  the  colder  give  ^ 
cold  and  the  hotter  take  it,  until  they  all  settle  on  a  common  tempera- 
ture which  gradually  approximates  to  that  of  a  thermometer  outside 
the  flask. 

A  complex  form  of  barter,  this,  intricate  to  work  out,  until 
Prevost  of  Geneva  simplified  it,  in  1792,  by  his  Theory  of  Exchanges, 
that  everything  is  giving  heat  all  the  time,  the  hotter  the  faster,  not 
only  the  thermometers  one  to  another,  but  they  to  the  surroundings, 
and  the  surroundings  back  to  them,  giving  all  they  can  and  taking 
all  they  can  get ;  a  true  Socialism  which  forgets  neither  aspect, 
and  so  attains  equality  in  the  end. 

No  matter  lives  for  itself  alone,  it  perpetually  gives  of  whatever 
sort  of  energy  it  has — and  the  kind  is  defined  by  the  temperature — 
at  a  rate  settled  by  temperature,  and  surface — the  internal  activity, 
and  the  frontier  conditions,  of  the  atomic  aggregate. 

Why    should    there    be    this    incessant    interchange     between 
quiescent  massive  matter  and  utmost  speed  in  otherwise  empty  ^ 
space  ?     Is  not  one  the  complete  antithesis  of  the  other  ? 

No  :  by  1870  it  was  plain  to  all  that  matter  is  not  quiescent,  it 
is  permeated  by  energy  in  the  form  of  vibratory  motion  in  its  atoms. 

774 


§952]  RADIATION  775 

Half  a  century  later,  and  light  is  proved  to  have  weight,  hitherto 
an  exclusive  property  of  matter ;  the  light  from  a  star  i^Hng 
near  the  sun  in  the  1919  eclipse  fell  in  toward**  it,  »)ent  out  of  iu 
straightforward  course  more  than  a  second  of  arc. 

By  1874  Dalton  in  Manchester  had  packe<l  up  matter  into  Atooui. 
of  various  sizes,  and  Young  in  London  had  spread  light  out  into 
waves  :  a  century  later  it  became  necessary  to  pack  up  light  into 
Quanta,  of  various  sizes ;  and  by  1928  the  Hying  dot  which  is  st 
once  the  frontier  guard  of  matter  and  its  means  of  communication 
with  outer  space,  the  Electron,  had  taken  on  a  wavc-struoturv 
itself,  and  been  pressed  into  service  for  making  diffraction  patterns, 
just  as  X-radiation  is,  §  919. 

More  :  the  hydrogen  atom  is  nothing  but  a  proton,  dcfemled 
by  an  electron  of  mass  00005,  yet  it  weighs  1-0078 ;  whereas  4 
protons  and  4  electrons  pack  to  form  helium,  §§036,  946,  a  verj- 
stolid  and  unbreakable  atom  of  reduced  internal  energy,  weighing  4  ; 
the  rest  of  the  mass,  0-0312,  has  become  energ>'  of  radiation,  in 
modern  spite  of  the  chemists'  Conservation  of  Mass  and  the 
physicists'  Conservation  of  Energy. 

We  reckon  now  that  grams  x  (spee<l  of  light)*  =  ergs,  Cosmir 
Rays  coming  in  from  Space  claim  a  quality  higher  than  any  hut 
this,  we  surmise  that  the  condensation  of  hydrogen  into  helium 
helps  maintain  the  high  heat  of  stars,  cf.  §  560 ;  calcuUtion  goes 
even  further,  §  978. 

And  at  the  time  of  writing,  it  has  just  been  claime<l  that,  per 
contra,  a  y-Ray  has  been  captured,  and  become  matter. 

These  deep  affairs  do  not  greatly  concern  you  ;  but  you  see  how 
the  sea-wall  between  space  and  matter  has  been  breached,  and  how 
the  tides  of  energy  flow  through  both. 

§952.  How  fast  does  Radiation  travel   through   spaee?     We 

have  outlined  one  method  of  measuring  this,  for  the  ver>*  slowest 
frequency,  in  §  838.  The  speed  of  radio-waves  is  measured 
by  methods  which  are  the  electrical  counterpart  of  the  resonance 
tubes  of  §§  443, 444  ;  that  of  X-rays  by  specialized  means.  For  all. 
the  speed  is  the  same,  that  of  visible  light  and  infra-red,  for  lioth  o( 
which  the  following  method  has  served,  a  development  of  (ialileo's 
endeavour  to  ascertain  the  Speed  of  travel  of  Light,  by  stationing 
two  observers  with  dark  lanterns  a  long  way  apart.  B  to  uncorer 
his  light  when  he  saw  A's  opened,  and  A  to  judge  the  interval 
between  opening  his  own  and  seeing  B's.  But  the  swed  of  light 
far  exceeds  that  of  sound,  and  we  know  now  that  the  uncertain 
small  results  in  this  experiment  measured  only  '  personal  equation. 
in  this  case  a  double  interval  between  the  eye  seeing  a  signal  and 
the  hand  making  an  effective  response. 

In  1849  Fizeau  replaced  A's  hand  and  shutter  by  a  rotating 
cog-wheel,  which  gave  a  succession  of  shutters,  and  B  by  a  mirror. 
and  so  evolved  a  method  exactly  analogous  to  that  for  the  !«prrd 
of  sound,  §  413. 


776  RADIATION  [§  952 

Fig.  407  shows  a  cog-wheel  which  had  720  square  teeth  separated 
by  720  spaces,  teeth  and  spaces  being  of  equal  width.  From  an 
arc  lamp  (the  asterisk)  light  is  sent  through  a  lens,  and  is  then 
reflected  by  the  inclined  mirror  to  form,  among  the  teeth,  a  bright 
image,  which  lies  at  the  principal  focus  of  a  second  lens.  '  Parallel ' 
light  therefore  travels  hence  several  miles  to  the  reflector  on  the 
left,  returns  to  the  second  lens,  and  is  formed  by  it  into  a  return 
image  among  the  teeth.  This  is  inspected,  through  a  hole  in  the  in- 
clined mirror,  with  a  magnifying  eye-lens.  The  teeth  of  the  wheel 
are  bevelled,  and  highly  polished,  so  that  the  outgoing  illumination 
which  falls  on  them  as  they  pass  is  thrown  away,  and  not  reflected 
back  to  trouble  the  eye. 

Now,  if  the  wheel  turns  at  a  certain  speed  it  will  happen  that  the 
flash,  sent  out  through  one  of  its  gaps,  travels  to  the  reflector  and 


^s 


Fig.  407. 

back,  while  a  tooth  moves  in  and  blocks  up  the  sending  gap.  At 
this  speed  the  observer  sees  no  return  image  at  all.  Speeding  up 
the  wheel,  the  image  reappears,  at  twice  the  speed  it  reaches  a 
maximum  brightness,  for  the  next  gap  has  moved  into  the  stead 
of  the  sending  gap  ;  at  3  times  it  disappears,  at  4  times  is  bright 
again,  and  so  on.  From  a  series  of  speeds  taken  like  this  the  speed 
which  just  brings  the  first  tooth  over  the  gap  can  be  accurately 
found ;  if  this  is  n  turns  per  second  the  time  occupied  is  1/1440^1 
second,  and  in  this  time  light  has  travelled  to  the  reflector  and  back, 
a  distance  2D. 

.*.  Speed  =  distance  -^  time  =  2D  X  1440ri. 
In  an  actual  determination  in  1874,  D  was  23  km.  and  n  4-53 

.-.  Speed  =  46  X  1440  X  4-53  =  300,000  km./sec.  =  186,000  mi./sec. 

=  3  X  10^^  cm.  per  second 

The  best  mean  of  modern  results  is  (2-998  ±  0-001)  X  10^^  cm. /sec. 

§  953.  Let  us  put  together  therefore  a  Great  Spectrum  of 
Radiation,  starting  from  where  Kugby,  in  splendid  isolation,  shouts 
to  the  world  on  a  wave-length  of  30  km.,  and  carrying  right  through 
to  those  competitors  of  armour-piercing  shell,  the  hard  y-rays  of 
Radium. 

It  is  drawn  in  Fig.  408  on  the  logarithmic  scale,  with  which 
you  are  perfectly  familiar  on  the  piano  keyboard,  equal  distances 


§953] 


RADIATION 


777 


meaning  always  equal  multiples  of  frequency,  and  with  the  high 
frequencies  on  the  right.  Only  we  have  to  cramp  our  keyboani 
rather,  for  we  want  about  56  octaves,  8  ordinary  piano«  on  end ; 
so  that  the  graduations  reading  to  the  right  represent  octaves  of 
frequency. 

Equally,  read  the  other  way,  they  represent  doubUngs  of  wave- 
length  ;  it  is  wave-lengths  that  are  measured  and  quoted  in  all 
short-wave  radiation,  and  their  figures  increase  towards  the  left. 

Radio  or  Wireless  we  have  dealt  with  already,  let  U8  give  them 
nearly  16  octaves,  down  to  2/3  of  a  metre  ;  that  ought  to  be  ultra- 
short enough  for  commercial  purposes  awhile  yet.  Set  a  mark 
there  in  honour  of  Heinrich  Hertz,  for  that  was  the  short  est  wave- 
length he  reached  when  investigating  the  practical  possibility  of 
electric  waves  in  1888,  and  give  him  the  next  8  octaves  as  his 


INFRA-RED 


ILTRA-I 


^^--..    ReFLecTeoi  TRANSMISSION       rviOLETI 
1000^         FROM  LIMITS  OF   ToUlAlri     VAC.  I 


R       A 

SOFT  I 

MjCK)      wa)i 


Y     S      I 


IA«» I ^ 

WW*/ 

ak  I      / 


w 


Ml 


m 


_^ , I I 

iPHOTOGf^APHlC  INFRARgQllvi's'lBLEllTHgaAPfcUTtC  jLTKAVtOLtTl  KoWtOCVJ 

EarthImax^        sunJmax. 

RUGBY  BROAOCASTINQ  BEAN  HERTZ  ^     ^\ 

30,000)n..  /600-m5PO,300m. lOm |i^|a/3T»\. lo  '~' 


x 


,  .»        ,1    DIVISION.  I  OCTAVE    OF   FREQUENCY—* 

^  xtJ.L«3Klo'*cn«/$ec.  * —  ••  wAve-LeN^TM 

Fig.  408. 


memorial,  the  laboratory  range  of  experimenUl  electric  waves, 
down  to  about  2  mm. 

Then  comes  a  no-man's-land  of  2|  octaves,  and  a  complete 
change  of  technique,  which  now  is  based  on  thermal  mcasureracnU. 
originated  in  the  red  of  the  Visible  Spectrum,  still  distant  8i  octaves 
of  Dark  Radiant  Heat,  or  Infra-Red. 

The  Visible  Spectrum,  of  Light,  roughly  from  0-8  to  0-4  |a  wave- 
length, occupies  one  octave,  and  has  already  been  dealt  wth  in 
Chapter  XXXVII.  It  is  followed  by  3i  octaves  of  Ultraviolet : 
and  then  another  gap  of  nearly  5  octaves,  in  which  some  oiitl>-ing 
ultra-soft  X-ravs  are  being  investigated,  brings  us  to  0-OUI-W 
micron,  an  L  X-ray  from  zinc,  and  0-0010,  a  K  X-ray  from  mag. 
nesium,  still  far  too  weak  and  soft  for  any  practical  "«©.?•»  •Jt? 
the  000014  L  from  tungsten,  technical  radiolog)'  only  starUng  ai 
about  0-00005  ^x,  corresponding  to  25  kv 

Volts  X  wave-length  in  microns  =  1-2345  ;  at  60  ^v  Jhe  hard  K 
radiation  of  tungsten  is  excited,  but  fortunately  it  has  plenty  of 


778 


RADIATION 


[§953 


'  continuous  spectrum '  beyond,  the  usual  125  kv.  bringing  in  0-0001  \i, 
and  the  present '  deep  therapy  '  250  kv.  half  as  long  ;  which  reaches 
the  bulk  of  the  y-rays  from  radium,  though  not  by  any  means  their 
limit,  nor  their  most  effective  constituents. 

§  954.  The  Ultra-Violet  section  of  the  Spectrum  extends  from 
wave-length  X  =  0*4  [l  (micron),  the  right-hand  end  of  Fig.  223, 
or  a  little  beyond,  say  the  H  and  K  lines  of  the  solar  spectrum, 
due  to  calcium,  of  which  H  0-397  is  visible  and  K  0-393  much  less 
so,  while  both  are  the  strongest  lines  of  the  whole  photographic 
spectrum,  which  reaches  X  =  0-2  micron.  The  great  sensitivity  to 
it  of  the  '  ordinary  '  photographic  plate  long  since  led  to  accurate 
records,  in  spite  of  its  invisibility. 

The  yellowish  flint  glass  of  ordinary  spectroscope  prisms  is,  how- 
ever, very  obstructive  to  ultra-violet  radiation,  crown  glass  fails 
at  0-3,  and  ultra-violet  work  has  to  be  carried  on  with  quartz 
lenses  and  prisms,  cut  so  as  to  dodge  double-refraction,  §  653. 

At  0-2  [L  comes  another  difficulty, 
air  itself  becomes  opaque  to  it,  and 
so  does  the  gelatine  of  the  photo- 
graphic film  :  vacuum  spectro- 
graphs have  to  be  employed,  and 
a  very  minimum  of  gelatine  in  the 
emulsion,  and  this  has  enabled  the 
spectrum  to  be  pushed  to  0-1,  0-06, 
and  0-036  [i,  by  successive  investi- 
gators. 

Visual  observation  of  the  ultra- 
violet spectrum  is  made  possible 
only  by  fluorescence.  In  the 
little  spectroscope  Fig.  409  the  slit- 
images  formed  by  a  quartz  prism  and  quartz  cylindrical  lens  fall, 
in  focus,  on  a  strip  of  uranium  glass,  and  are  seen  as  fluorescent 
green  lines  and  bands,  neither  brilliant  nor  very  sharply  deflned, 
through  an  adjustable  magnifying  eyepiece  with  a  light-excluding 
rubber  collar.  The  short  coloured  visual  spectrum  is  just  in  sight  at 
the  right-hand  end,  between  0-40  and  0-45  on  the  wave-length  scale, 
which  lies  alongside  and  is  lit  from  a  porthole  beneath. 

§  955.  Sources  of  Ultra-violet.  As  we  shall  see,  to  get  an  econo- 
mical amount  of  ultra-violet  from  an  incandescent  soHd,  it  has  to 
be  heated  very  hot  indeed,  even  the  bare  carbon  arc  emits  only  a 
small  fraction  of  1%  of  its  energy  as  ultra-violet. 

Much  better  are  electrically  excited  metallic  vapours,  which  often 
have  very  strong  bright  lines  in  the  ultra-violet,  emitting  the  bulk 
of  their  energy  on  those  frequencies,  and  wasting  little  on  lower- 
frequency  radiant  heat. 

For  small  local  applications  the  'jar'  spark  of  Cadmium  or 
Magnesium  can  be  used,  a  compressor  plate  of  ice,  which  is  trans- 


FiG.  409. 


§  955]  RADIATION  779 

radiable,  squeezing  and  chilling  obstructive  blood  out  of  tiia 
tissues ;  but  these  sparks  are  small  and  noisy  and  demaod  tpeeiAl 
transformers  and  condensers. 

Such  fusible  metals  burn  up  too  fast  in  the  Arc,  with  copiout 
fumes,  poisonous  in  the  case  of  CdO.  But  the  almoHt  invisibtoiuro 
itself  becomes  a  brilliant  white  flame,  full  also  of  ultra-violet  jturt 
beyond  the  visible,  when  cerium-cored  carbons  are  employed,  thii 
light  producing  a  pleasant  sunburn  tan  on  the  skin. 

Small  arcs  are  also  run  with  one  or  both  poles  of  Iron,  for  trtm 
gives  thousands  of  lines  all  through  the  spectrum  ;  hence  the  uie  of 
iron  filings  for  sparks  in  fireworks.  You  see  theee  arcs  in  arr- 
welding,  for  which  ultra-violet-proof  goggles  are  indispensable. 

To  a  little  Tungsten  Arc  two  minutes'  exposure  is  a  full  dose : 
they  make  little  smoke  and  are  much  used  to  counter  infection  of 
surgical  wounds,  and  in  the  treatment  of  lupus. 

The  Mercury  Arc  gives  the  most  abundant  yield  and  is  rtry 
largely  used  :  its  whole  spectrum  is  shown  in  Fig.  418.  It  must 
of  course  be  bottled  up,  and  in  fused  silica  glass,  as  dencribed  in 
§  880.  The  reek  of  ozone,  poisonous  in  any  quantity,  calls  for  good 
ventilation,  but  the  skimpily  clad  patients  call  for  warmth  during 
their  20  min.  exposure,  so  '  infra-red '  radiation  is  supplied  them  in 
addition,  from  common  red-hot  electric  *  bowl  *  heaters,  sold  at 
a  fancy  price.  Mercury  lamp  treatment  gives  the  skin  a  duaky 
greyness  :  there  is  no  doubt  about  its  tonic  effect. 

Dosage  can  be  controlled  by  an  *  lonto-quantimeter,*  which  is 
simply  an  electrometer  fed  by  a  zinc  or  cadmium  plate,  from  which 
ultra-violet  extracts  ions  photo-electrically,  §  984. 

Ultra-violet,  excluding  the  vacuum  stuff,  passes  through  air. 
distilled  water,  glycerine,  ice,  quartz,  fluorite.  cellophane,  and 
some  other  things,  as  readily  as  light,  but  not  mica  and  many  oiU. 
Thin  glass,  and  spectacle  lenses,  are  not  proof  against  it,  but  window- 
glass,  largely  on  account  of  contamination  with  iron,  to  which 
its  greenish  tint  is  due,  is  rather  opaque  to  it ;  and  '  Vita  '  and  similar 
glasses,  from  melts  very  free  from  iron,  are  consequently  recom. 
mended  for  nursery  windows  :  for  plant  nurseries  the  R.H^. 
experiments  show  no  significant  effect. 

Incidentally,  for  most  part  of  the  year,  the  spcctrowope,  Fig. 
409,  gleans  nothing  at  all  from  London  skies. 

Further,  nobody  gets  sun-tanned  in  England  before  lata  April. 
and  early  morning  and  late  afternoon  sun  doesn't  tan ;  they  tmy 
the  sun*^raust  be  more  than  40°  up  in  the  sky.  Definite^  the 
spectrum  of  the  solar  ultra-violet  ends  in  winter  at  0-315  ji.  white  in 
summer  it  reaches  to  0-29  y.  (after  which  see  §  a57). 

Direct  experiment,  made  bv  throwing  an  mtense  spectrum  on 
the  skin,  has  shown  that  tanning  is  caused  onlv  by  the  nAm« 
band  of  spectrum  between  0-31  and  0-29  ^,  so  the  rt^Bon  for  Um 
difference  between  low  and  high  sun  is  self-evident. 

The  amazing  thing  is  that  the  radiation  can  penetrate  only  0- 1  mm- 
of  skin,  and  yet  this  brings  about  the  beneficial  production  and 


780  RADIATION  [§  955 

absorption  of  the  anti-rachitic  vitamin  D  (see  also  §  958)  :  you  must 
wait  for  Physiology  to  tell  you  how. 

The  lethal  effect  on  the  unicellular  motile  Euglena  viridis  seems 
to  be  due  to  a  higher  range,  0-28 — 0-25  \i,  and  there  is  room  for  a  lot 
of  exploration  yet  in  the  wide  region  still  lumped  together  as  Ultra- 
violet. 

§  956.  Daylight  ultra-violet.  In  a  clear  atmosphere,  §  568  has 
told  you  that  blue  light  is  '  scattered '  up  to  5  times  as  much  as 
red,  forming  the  blue  sky.  Using  a  sensitive-paper-under-glass 
photographic  exposure-meter,  inside  and  outside  the  shadow  of 
your  head,  you  will  find  that  (sun  +  sky)  amounts  to  about  double 
sky  alone,  so  that,  of  the  blue  light — mainly  effective  on  the  meter — 
half  comes  from  the  blue  sky.  Applying  the  inverse  fourth-power 
law  of  scattering,  again,  to  blue  of  0-47  micron  and  ultra-violet  0-32, 
the  ratio  of  scattering  ultra-violet/blue  =  (47/32)*  =  (3/2)*  =  5,  so 
that  while  half  the  sun's  blue  reaches  us  as  scattered  light  from 
the  sky,  the  great  part  of  its  ultra-violet  gets  scattered  out  of  the 
direct  beam,  and  is  diffused  to  us  from  the  vault  of  sky. 

Sun-bathing  in  this  country  is  great ;  but  a  very  little  experience 
of  the  Sun  in  lands  where  he  is  a  maUgnant  devil  who  burns  up  the 
crops,  and  the  pitiful  sore  blisters  and  sun-cracks  lying  open  to  all 
sorts  of  stray  infection,  common  on  every  English  beach  these  fine 
summers,  advise  caution.  Granting  that  it  is  the  ultra-violet  that 
does  the  good,  you  see  that  just  sheltering  from  direct  sunshine, 
under  a  small  shade,  will  still  leave  more  than  half  the  beneficial 
effect,  while  conferring  complete  safety  :  the  thicker-skinned 
areas  can  chance  it.  This  is  a  point  worth  considering  for  tender 
skins  :  undoubtedly  ultra-violet  is  bactericidal  (hence  its  use  in 
sterilizing  water),  and  undoubtedly  it  is  easily  absorbed  and  only 
goes  skin  deep,  but  Nature  doesn't  keep  a  stock  of  tan  for  pro- 
tection against  an  altogether  unmixed  blessing. 

'  Snow-blindness,'  caused  by  the  glare  of  ultra-violet  from  the 
mountain  snow-fields,  is  just  as  likely  to  befall  anyone  who  has 
been  gazing  at  an  arc  or  other  ultra-violet  lamp.  One  attack 
is  enough ;  with  glued-up  eyes  and  '  red-hot  needles,'  and  rags 
wet  with  sulphate  of  zinc  solution  (up  to  2%),  getting  hot  a  few 
minutes  after  every  application,  for  24  hr.  on  end. 

The  preventive  is  a  pair  of  goggles  of  Crookes'  '  reddish-amber  ' 
soda-glass  containing  10%  ceria,  which  blots  out  the  whole  range 
of  ultra-violet ;  but  the  mercury-lamp  attendant  will  wisely  make 
still  more  sure,  while  the  arc -welder  wants  a  triple  protection,  from 
ordinary  light,  and  heat,  as  well. 

§  957.  Ozone.  The  ordinary  carbon-arc  yields  more  ultra-violet 
than  is  contained  in  sunshine,  whereas,  according  to  temperature, 
the  sun  should  be  a  score  times  the  more  efficient  source,  §  972. 
Evidently  there  are  losses  in  the  atmosphere,  apart  from  scattering, 
and  on  examination  with  the  quartz  spectroscope  it  is  seen  that 


§  969]  RADIATION  7gl 

the  celestial  supply  shuts  down  suddenly  at  wave-length  0-29  ^. 
while  laboratory  sources  go  far  beyond. 

We  feel  confident  that  the  doubly  energetic  quanta,  or  phot*— 
J)  and  q  mean  the  same  thing  for  once — of  the  double- frequency 
ultra-violet,  are  responsible  for  part  of  the  ioniuition  of  the  upper 
atmosphere,  producing  probably  the  lower  layer  of  the  lonmtpherv 
which  makes  long-distance  wireless  possible,  §  984 ;  and  thU 
seems  bound  up  with  the  production  of  Ozone,  for  it  in  ozone  which. 
in  the  laboratory,  is  found  to  be  the  cause  of  the  intenae  abaoqption 
commencing  at  0-29.  Based  on  that,  daily  determinationii  of  thr 
total  content  of  ozone  in  the  atmosphere  are  being  ma^lc  by  a  spectro- 
photometer which  measures  the  steepness  of  the  foot  of  the  nidden 
absorption  hill  at  0-29  \l  (for  the  top  is  always  too  black  to  meaaore) : 
the  box  of  quartz  prisms  and  lenses  and  amplifying  valves  it 
wheeled  out  in  its  perambulator,  its  collimator  pointed  towmfda 
the  sun,  and  on  a  dial,  turned  to  bring  the  galvanometer  pointer 
to  zero,  the  amount  of  ozone  in  the  upper  atmosphere  ia  reaa. 

Reckoned  as  concentrated  into  a  layer  of  pure  ozone,  meaaured 
at  76  mm.  pressure,  it  varies  from  2  to  4,  averaging  3  ram.  thick. 
and  this  is  as  opaque  beyond  0-29  (x  as  eighth -inch  »heet  lead. 
Knowing  the  effects  of  overdoses  of  ultra-violet,  we  can  «ay  that. 
but  for  this  protection,  vegetable  and  animal  life  woukl  acaroely 
outlast  the  day. 

§958.  Against  what  little  we  get,  chlorophyll  and  hemoslohin 
possess  protective  absorption  bands ;  exposed  bacteria  are  killed. 
and  the  use,  both  natural  and  artificial,  of  ultra-violet  for  producing 
vitamin  D,  is  the  subject  of  innumerable  advert inemcnta.  The 
brown  Diatoms  which  form  the  abundant  plankton  of  the  cokier 
seas — they  dominate  our  inland  waters  until  Spirog>Ta  uahen  in 
the  Spring— are  browsed  on  by  the  Copepoda,  which  are  tngestad 
by  the  little  fishes  that  are  the  prey  of  the  Cod,  and  the  now  pale- 
tinted  Liver  Oil  of  the  latter  is  tested  for  vitamin  content  by  lU 
spectrum  absorption  of  ultra-violet :  the  film  clinging  on  a  ailioa 
strip  dipped  in  the  oil  almost  blots  it  out,  and  it  haa  ^  *»  FJ^ 
metered  in  dilute  chloroform  solution,  on  wave-length  0-3W  (•. 
using  a  small  copper  arc,  and  a  fluorescent  glass  screen. 

§959.  Ultra-violet  can  be  filtered  free  from  admixture  with 
visible  light  by  a  silvered  quartz  plate,  for  a  silver  film  reflects 
very  little  of  it  and  transmits  a  good  deal.  A  cheai»er  filter  w  a 
plate  of  Uviol,  a  potash  glass  stained  very  deep  violet  ^»«»' '^«;j;;'; 
and  this  enables  its  remarkable  powers  of  exciting  Pluorweenct 
to  be  observed.  ,       ,         ...      .  ^^^  «,i.w»K 

The  arc,  seen  through  it,  is  in  the  midst  of  a  pale  blue  ^-^"'^ 
window-glass  cuts  off  :  this  is  the  fluorescence  o  «»|f  /«^  ^^ 
lens  of  your  own  eye.  The  skin  also  shines  light  ^lue  the  mn. 
standing^out  dark  and  non-fluorescent :  the  teeth  (natural)  fluor«* 
white,  and  almost  all  unpigmente<l  ammal  cells  and  tissue,  fluore** 


782  RADIATION  [§  959 

in  colours  so  distinctive  that  a  branch  of  ultra-violet  microscopy 
has  been  developed  from  this. 

The  French,  always  interested  in  phosphorescence,  have  powders 
and  paints  of  every  hue,  whereby  a  daylight  scene  is  suddenly 
changed  to  the  tints  of  evening,  or  gleams  out  as  an  entirely  different 
picture  when  invisibly  illuminated  by  ultra-violet.  Sulphate  of 
quinine  was  the  first  substance  employed  in  studying  it ;  its  dilute  • 
solution,  acidulated  with  a  drop  or  two  of  sulphuric  acid,  shining 
the  same  heavenly  blue  as  it  does  in  the  visible  violet. 

Signalling  lanterns,  masked  to  invisibility  by  this  glass,  but  kept 
constantly  in  view  by  officers  of  the  watch  armed  with  telescopes 
having  focal  plates  of  fluorescent  uranium  glass  in  their  eyepieces, 
have  been  used  in  keeping  convoys  together  '  without  lights.' 

§  960.  The  Infra-Red.  Now  we  must  go  to  the  other  end  of  the 
visible  spectrum,  and  start  to  explore  the  Infra-Red,  the  region  of 
Dark  Heat  which  stretches  away  towards  the  electrically  produced 
radiations  of  the  laboratory. 

As  long  ago  as  1840  John  Herschel  coated  thin  paper  with  gum 
and  Indian  ink,  and  dried  it.  Moistening  again  with  alcohol,  he 
exposed  strips  of  it  beyond  the  red  end  of  the  solar  spectrum  and 
found  rapid  drying,  with  indications  of  narrow  bands  which  did  not 
dry  so  readily  ;  the  '  infra-red,'  and  its  cool  '  dark  lines.' 

Nowadays  the  first  beginning  of  it  is  being  generally  rediscovered 
by  the  photographer,  who  sticks  over  his  lens  a  screen  of  the  very 
deepest  ruby  glass,  loads  up  with  plates  made  sensitive  to  this  radia- 
tion by  special  dyes,  exposes  a  little  longer  than  usual  and  develops 
in  the  dark,  and  is  rewarded  as  likely  as  not  by  a  background  of 
hills  which  he  didn't  know  were  there,  instead  of  the  usual  disap- 
pointment that  the  distant  delectable  mountains  are  unprintably  dim. 

For  appealing  again  to  the  inverse  fourth-power  law  of  scattering 
of  §  568,  if  he  works  on  a  wave-length  only  half  as  long  again  as 
the  average  0-55  y.  assumed  for  white  light  in  §  632  he  will  get  only 
l/5th  as  much  '  atmosphere  '  as  ordinarily  veils  the  view  to  the  eye,, 
and  only  1/lOth  as  much  as  in  the  blue,  which  is  the  effective  agent, 
when  using  orthochromatic  films  and  a  light  yellow  screen  to  cut 
out  the  violet  beyond. 

The  '  sky  '  depends  for  its  existence  on  scattered  light,  so  on  the 
0*84  y.  wave-length  optimum  for  these  plates,  only  a  little  beyond 
the  visible  red,  it  will  be  only  1/10  as  bright  to  the  plate  as  it  was 
to  the  light-yellow  screened  ortho,  and  therefore  prints  out  dark ; 
while  as  most  things  refiect  long  waves  better  than  they  do  short 
(to  which  their  surface  roughnesses  are  more  troublesome),  the 
landscape  comes  out  light ;  so  that  sometimes  these  extremely 
clear  views  have  a  suggestion  of  '  negative  '  about  them. 

A  quite  successful  determination  of  the  speed  of  travel  of  infra- 
red has  been  made  by  the  method  of  §  952  over  the  whole  length  of 
New  York  and  back,  the  down-town  reflecting  station  being  in- 
visible in  the  haze. 


§  961]  RADIATION  7^ 

But  up  to  the  present  only  half  the  first  octave  of  thU  long  nuMv 
has  been  opened  to  the  photographer,  and  glass  lensM  and  piinnii 
fail  by  the  end  of  the  second  octave,  and  quartz  soon  after  ;  fluahte 
carries  through  a  third  octave,  and  rock-salt  lenses  and  priifm»  a 
fourth,  leaving  only  sylvine,  KCl,  the  most  widely  trannparmt 
of  all  substances,  capable  of  completing  the  fifth,  to  23  {i. 

Even  beyond  that,  however,  by  using  silvered  concave  mtrrofv 
instead  of  lenses  in  a  spectrometer,  and  a  fine  wire  diffiactioo 
grating  instead  of  a  prism,  isolated  patches  of  radiation  have  bevo 
measured.  We  shall  see  that  the  trouble  is  the  extreme  wcaknnH 
of  the  radiation  generally  obtainable  on  long  wave-lengths:  it  is 
like  pointing  a  pocket  spectroscope  at  a  tea  kettle. 

Fortunately  dry  air  gives  no  trouble,  it  is  practically  com. 
pletely  *  diathermanous,'  an  oldish  word  meaning  *  radiant -beat- 
transmitting.'     But  water  vapmir  absorbs  heavily,  §  976. 

§  961.  This  is  the  main  Heat  region  of  the  Spectrum,  and  ihu^ 

is  no  general  way  of  measuring  the  quantity  of  the  radiatioo  except 
catching  it  on  a  lamp-blacked  surface  and  measuring  it  a*  heat. 

For  experiments  in  bulk,  the  Blackened  Bulb  of  a  Thennometcr 
serves,  and  rate  of  rise  measures  rate  of  input.  More  senntiTe 
is  the  blackened  bulb  of  an  Air  Thermometer,  which  has  just  a 
long  narrow  tube  with  a  short  index  thread  of  liquid  in  it. 

For  putting  straight  into  the  spectrum,  and  measuring  its  inten- 
sity line  by  line,  Langley  used  his  Bolometer,  a  platinum  t henna* 
meter,  §  778,  the  '  coil '  of  which  is  a  straight  strip  of  blackened 
platinum  *  the  size  of  a  spectrum  line '  and  about  0-2  mg.  in  maii :  it 
reaches  a  sensitiveness  of  a  millionth  of  a  degree  C. 

Or  the  spectrum  is  focussed  on  a  plate  with  a  narrow  slit  in  it, 
and  travelled  along,  so  that  line  after  line  falls  through  tiie  slit. 
on  to  the  blackened  receptive  plate  of  one  of  the  ThemiopUee  of 
§  799,  actuating  a  sensitive  galvanometer,  in  which  an  elaborate 
astatic  system  records  its  swings  on  a  photographic  film  muving 
pro  rata  with  the  spectrum.  The  measurementa  can  be  made 
'  absolute  '  by  heating  the  plate  eauallv  by  peonng  •  eurreot 
through  it,  and  noting  the  power  employed. 

But,  for  all  that,  not  one  of  these  instruments  approaches  the 
sensitivity  the  Eye  possesses  in  the  middle  of  its  limited  range. 

The  whole  visible  spectnim  can  of  course  l>e  gone  over  in  tlie  Mne 
way  :  there  is  no  discontinuitv  in  the  record  as  it  P**^  ^'^ 
visibility  limit,  0-8  [i,  and  when  drafted  out  spectnim.faijhioo.  a 
strong  line  for  a  strong  radiation,  etc.,  infra-red  spectra  look  much 
like  any  others.  .  ^      .       ,^^, 

The  isolated  patches  beyond  23  y.  were  got  by  rr  flee  ting  beat. 
from  a  gas-mantle,  from  various  crystals,  which  h«Pj«pJ^  ^^ 
strong  absorption  bands,  and  consequently  h'^fjjjf*  i?S^ 
reflective  powers,  §  562  :  such  were  NaCl  52  ji,  KU  63  »* .  K »  «  »*• 
AgBr  113  {X,  Til  152  {x.  Even  beyond  these,  at  218  and  «=:  nua 
third  of  a  millimetre,  lucky  radiations  from  a  mercur>-  arc 


784  RADIATION  [§  961 

found  and  focussed  by  lenses  of  quartz,  which  had  woke  up  to  trans- 
parency again  with  a  big  refractive  index. 

At  Flagstaff,  where  Lowell  sought  clear  air  at  7000  ft.,  by  planting 
his  observatory  in  the  midst  of  the  Arizona  desert,  with  a  20,000- 
sq.-mile  pine  wood  to  keep  the  dust  off,  a  42-inch  reflector,  crouching 
under  a  bubble  of  canvas,  and  wrapped  in  mummy -cloths  to 
steady  its  temperature,  spends  its  nights  measuring  the  heat  of  the 
stars,  or  patiently  exploring  the  surface  temperatures  of  the  planets, 
while  the  refractor  beside  it  photographs  them  in  violet,  and  yellow, 
and  infra-red. 

§  962.  Heat  radiation,  in  bulk.  Thus  far  we  have  been  examining 
in  detail  the  radiation  from  various  sources,  finding  out  exactly 
where  it  is,  and  making  some  comparative  estimate  of  its  strength. 
But  we  light  a  fire  for  its  warmth,  not  its  wave-length,  and  we 
make  a  very  practical  study  of  radiation  by  wholesale. 

Pick  a  hot  Coal  from  the  fire  by  the  tongs,  and  look  at  it,  i.e.y 
examine  it  by  means  of  radiation. 

You  see  its  shape  because  it  shines  bright  red ;  it  is  emitting 
radiation,  of  good  red-hot  quality. 

It  scorches  your  Jiand,  held  six  inches  away  from  it  in  any  direc- 
tion, underneath,  or  at  the  side,  or  anywhere.  Less,  however,  on 
the  dark  side  than  on  the  bright ;  its  radiation  is  not  only  of  poorer 
quality  there,  making  little  or  no  appeal  to  the  eye,  but  is  scantier 
in  quantity.  Holding  your  hand  over  the  top,  you  feel  warmth 
gradually  enfolding  it,  and  that  you  know  to  be  the  convection 
current  of  hot  air  rising  ;  but  the  scorch  radiated  out  sideways  was 
instant. 

Daylight,  which  is  diluted  radiation  of  solar  quality,  strikes  the 
rest  of  the  coal,  and  some  is  reflected  off — irregularly,  as  they  say — 
and  tells  your  eye  that  it  is  gray,  or  brownish  ;  but  dark,  not  so 
good  a  reflector  as  a  white  tile,  or  your  own  hand,  i.e.  much  of  the 
incident  radiation  is  being  absorbed,  and  done  away  with. 

Now,  if  there  happens  to  be  a  thin  Flame  rising  in  the  fire,  you 
notice,  further,  that  you  can  not  only  see  it,  but  see  through  it.  In 
the  same  way,  if  you  drop  a  bit  of  broken  Glass  in  the  fire,  and  fish 
that  out  when  hot,  you  can  see  it  shining — rather  wanly — ^red, 
you  can  see  the  glint  of  the  window-light  on  it,  and  you  can  see 
through  it.  So  these  radiators  can  also  transmit  radiation  :  the 
coke  couldn't. 

Putting  back  the  now  dull  ember,  you  see  it  gradually  growing 
into  brightness,  as  its  neighbours  kindly  radiate  to  it,  until  it  is 
hot  enough  to  join  in  the  general  combustion  again. 

So  a  body  can  emit,  reflect,  absorb,  or  transmit,  radiation,  and  bodies 
differ  quite  a  bit  in  the  relative  proportions  of  these  four. 

Now,  suppose  you  took  a  cinder,  a  scrap  of  paper,  a  micro  slide,* 
a  penny,  and  a  bright  shilHng,  and  put  them  all  together ;    in  a 
vacuum,  if  you  like,  where  there  can  be  no  draughts,  nor  combustion  ; 
or  just  together  on  the  table,  where  they  will  not  be  interfered  with 


§963]  RADIATION  ^^ 

by  radiation  from  sun,  or  fire,  or  lamp,  but  are  free  to  aeUle  mattm 
among  themselves ;  they  will  undoubtedly  all  arrive  at  the  mal 
temperature  before  long,  by  mutual  radiation  accortlimr  to  PreroBi. 

But,  visibly,  most  of  the  radiation  that  falla  on  the  dark  ciimW 
or  the  brown  penny  goes  mto  them,  most  of  that  reaching  the  white 
paper  or  the  shmmg  shilling  is  thrown  back  immediately,  and  nuMt 
of  the  received  radiation  passes  clean  through  the  altm ;  ao  that 
these  last  three  don't  get  a  chance  of  absorbing  much. 

Yet  if  any  one  emits  more  than  it  absorbs,  it  must  grow  colder 

We  won't  debit  a  body  with  what  it  refuses  at  the  door,  eo  r^ 
fleeting  power  doesn't  come  into  the  balance  sheet ;  nor  aak  the 
glass  to  account  for  what  transparently  goes  through  it  and  out  the 
other  side ;  so  we  are  left  with  this,  that  for  each,  individually 
Absorption  and  Emission  have  got  to  balance  :  *  Any  body's  radia- 
ting power  is  equal  to  its  absorbing  power.* 

Now  that  is  a  thoroughly  mischievous  old  half-truth. 

When  you  dropped  that  cinder  back  in  the  fire  among  his  fellowa 
he  grew  hotter,  he  wasn't  giving  as  much  as  he  got.  And  be  grow 
brighter,  and  through  a  pocket  spectroscope,  or  any  priim,  yoa 
would  see  green  growing  in  his  spectrum,  and  perhaps  eren  a  tua- 
picion  of  blue,  ftesently  he  comes  to  a  new  and  higher  equilibrium 
with  his  surroundings,  and  changes  no  more  : — 

Any  body  radiates,  at  the  same  rate  as  ii  absorbs,  radiation  of  UU 
temperature-quality  characteristic  of  its  temperature. 

How  the  activity  increases  with  temperature  we  shall  look  into 

later,  §  968. 

§  963.  At  the  same  temperature,  different  bodies  differ  rvy 
much  in  their  rate  of  radiation  ;  the  quahtv  is  the  same,  but  not  Im 
quantity.  The  glass  fetched  out  of  the  nre  glows  with  the  «iiie 
tint  of  red  as  the  coal,  but  with  far  from  the  same  fierceneet,  only 
thinly :  for  it  habitually  sticks  to  only  10%  of  radiatioo  ttriking 
it,  so  how  can  it  be  expected  to  radiate  on  more  than  a  10%  haw  f 

Another  transparent  radiator  is  a  bunsen  flame  :  if  you  hold  one 
hand  in  front  of  it,  and  open  and  close  the  air-holes,  vou  feel  thai 
the  luminous  flame  sends  out  more  radiant  warmth  than  the  noo* 
luminous,  in  spite  of  its  less  perfect  combustion  and  lower  tMDpvm- 
ture.  The  almost  transparent  blue  flame  absorl>s  little,  and  coiia»- 
quently  can  radiate  but  little  :  the  luminous  flame  is  raor©  opaooe, 
and  can  absorb  light  and  heat  strongly,  so  it  must  in  turn  radiat* 
strongly.  In  fire-place  gas-stoves  an  opaque  solid,  aahattos,  it 
heated,  and  radiates  much  more  than  could  the  hotter  elear  flame 
alone  :   so  also  in  incandescent  mantle  bunieri. 

A  transparent  bead  of  fused  borax  remains  clear  and  almoit 
invisible  while  the  encircling  wire  of  opaque  platinum  is  glowiof  led  : 
the  solution  in  it  of  a  trace  of  copper  makes  it  at  once  a  fienr-r»d 
mass,  which  cools  to  a  glass  partially  transparent,  but  greenish. 
i.e.  absorbing  just  that  red  light  which  it  emiU  vigorously  when 


786  RADIATION  [§963 

hot  enough.  So  also  a  bit  of  dark-green  bottle  glass  fetched  out  of 
the  fire  glows  furiously  red. 

Transparent  '  diathermanous  '  air  gains  no  radiating  power  from 
being  heated,  it  remains  invisible  as  ever  inside  a  white-hot  tube, 
or  right  up  to  the  carbons  of  the  arc.  The  arc  itself,  like  the  blue 
bunsen  flame,  has  a  little  luminosity  ;  but  there  ionization  is  active, 
and  that  is  a  different  story,  v.  §  877. 

Curious  confirmation  of  this  parallelism  between  absorptive 
power  and  radiating  power  is  afforded  by  a  crystal  of  tourmaline, 
§  653  ;  cold,  it  polarizes  light  by  absorbing  the  one  beam  of  it ; 
bright  red  hot,  it  emits  polarized  light. 

The  highly  reflecting  shilling,  lifted  out  of  the  hot  fire  quickly 
in  a  loop  of  iron  wire,  glows  noticeably  less  fiercely  than  the  rough 
coke.  It,  too,  always  deals  with  less  than  half  the  quantity  of 
radiation  possible. 

The  exact  application  of  the  good  absorber  good  radiator 
principle  to  bright  and  dark  spectrum  lines  has  already  been  dealt 
with  in  §  559. 

§  964.  But,  of  course,  for  him  who  announces  broad  generaliza- 
tions in  a  loud  voice,  Nature  has  her  traps  ready,  as  always.  '  Once 
a  good  reflector  always  a  good  reflector,  and  therefore  a  poor  ab- 
sorber or  radiator  '  is  a  non  sequitur.  The  vast  range  of  temperature 
radiation  has  room  for  differences.  Polished  silver  reflects  80%  of 
light,  98%  of  infra-red,  and  5%  of  ultra-violet.  A  white  crock  with 
a  dark  pattern  on  it  may  fairly  be  expected  to  show  a  bright  pattern 
on  a  dark  less-radiant  ground  when  heated  up  to  redness,  and  so 
it  does ;  but  it  doesn't  follow  that  because  a  polished  metal  tea 
kettle  is  a  good  reflector  of  daylight,  which  is  dilute  radiation  of 
6000°  temperature  quality,  it  will  necessarily  emit  less  of  the  low- 
temperature  radiation  of  tea  kettles,  than  a  dingy  one.  Give  it 
an  invisible  coat  of  lacquer  and  up  goes  its  radiation  loss  threefold — 
though  anjrway  the  radiation  loss  from  the  blackest  of  kettles  is 
insignificant,  v.  §  233.  Poker  and  tongs,  though,  lying  in  the  hearth 
before  the  glowing  fire,  heat  up  more  quickly  if  the  brass  is  lacquered 
than  if  its  brightness  is  due  to  hand  labour  ;  and  the  shining  glass 
bulb  of  a  thermometer  heats  faster  in  the  sunlight  than  the  brilHance 
of  the  quicksilver  inside  the  glass  would  lead  one  to  expect. 

§  965.  The  great  differences  that  exist  in  the  power  of  transmitting 
radiation  at  different  wave-lengths  have  been  noticed  already  for 
glasses,  quartz,  rocksalt,  etc.  :  things  have  '  absorption  bands  ' 
covering  several  octaves  during  which  they  cannot  transmit.  The 
longer  waves  of  infra-red  disregard  the  obstruction  which  paper  and 
other  fibrous  fabrics  present  to  light,  in  the  repeated  alternation 
of  refractive  fibre  and  air,  and  pass  through  them  with  ease.  A  slow- 
combustion  stove  going  fairly  strong  is  radiating  abundantly 
between  4  y.  and  6  (jl  ;  sitting  in  front  of  it  reading  a  newspaper 
your  face  will  scorch  as  if  the  paper  wasn't  there,  a  doubled  sheet 


966]  RADIATION 


Ul 


of  cardboard  is  little  protection,  and  you  can  warm  your  hands 
through  1/8-in.  ebonite — but  not  through  the  pane  of  glaM  which 
would  transniit  plenty  of  0-5—2  y.  from  a  bright  open  fire. 

Our  own  tissues,  including  the  blood,  owe  their  opacity  to  light 
to  a  similar  change  of  refractivity  between  liquid  and  soft  aoud, 
repeated  over  and  over  again  on  a  small  scale.  Thia  again  the  long 
waves  can  largely  overcome,  and  from  any  source  betwvrn  bare 
visibility  and  a  full  red  heat,  radiation  like  that  of  Fig.  410  pourt 
through  clothing  and  penetrates  the  body  much  more  than  skin- 
deep.  Call  it  '  Infra-red  '  or  *  Radiant  Heat '  Treatment,  or  what- 
ever you  like ;  it  is  that  immediate  hearty  comfort  we  all  get  from 
sunshine,  or  a  glowing  hearth,  or  a  welf-stoked  stove,  at  opponcd 
to  the  slow  convective  thawing  in  rooms  kept  uniform  by  hot -water 
air-and- wall-warmers ;  rooms  which  everyone,  feeling  somethuig 
lacking,  keeps  10°  F.  too  hot. 

Water  is  a  great  absorber  of  infra-red,  a  glass  of  it  standing  in  the 
hearth  readily  gets  hot,  the  flasks  of  Fig.  244  concentrate  a  cold 
light,  a  tank  of  it  is  a  safeguard  in  lantern  microscopes  etc.,  and  can 
be  made  even  more  eflFective  by  dissolving  a  little  fcrrous-ammooium- 
sulphate. 

A  lime-soda  glass  containing  3 — 5%  of  ferrous  iron  is  not  con- 
spicuously green,  but  stops  96 — 98%  of  radiant  heat,  and  is  marketed 
as  Calorex,  etc.,  for  roofing  workshops  where  the  sun's  scorching 
heat  is  unwelcome.  On  the  other  hand,  P>Tex  glass  transmits 
solar  heat  very  freely. 

Roughly  speaking,  and  ignoring  the  absorption  of  thin  glass,  a 
half-Utre  spherical  flask  filled  with  water,  and  backed  by  window- 
glass,  concentrates  Visible  Light ;  the  flask  alone,  filled  with  a  dark 
1/5000  water  solution  of  nitroso-di-methyl-aniline,  concentrates 
Ultra-Violet ;  and  full  of  a  violet-black  solution  of  iodine  in  carbon 
disulphide  it  is  '  very  diathermanous  to  *  Infra- Red,  the  liquid  re- 
maining cool. 

§  966.  There  is  a  theory  that  the  glass  of  a  Greenhooi*  leU  in 
solar  heat  readily — as  it  does,  down  to  2  pi,  and  that  is  quite  » lot — 
and  then  refuses  to  let  out  the  10  (x,  or  longer,  low-temperature 
radiation  from  the  warmed  plants  and  soil. 

Now,  a  well-exposed  greenhouse  in  this  country  may  perlis|is 
average  two  hours  of  sunshine  a  day,  and  all  through  the  Mnnn 
of  high  sun  the  gardener  will  do  his  best,  with  screens,  or  Wmds. 
or  '  summer  shading,'  to  keep  most  of  it  out,  and  will  open  erery- 
thing  up.  And  R.  W.  Wood  exposed  thermometers  m  miniature 
greenhouses  roofed  with  glass  and  with  rock-salt,  and  found  no 
significant  difference  between  them.  ^^ 

In  fact  a  greenhouse  mostly  works  by  the  strong  check  it  ^j^Bp°*^ 
on  ventilation,  for  the  air  movement  inside  is  never  m  the  •ugli*» 
likely  to  reach  l/20th  of  that  outside  ;  although  <*'^P«™^*V^ 
by  the  R.H.S.,  on  the  effect  on  plant  growth  of  [ncreMmgihe  pro- 
portion  of  CO2  in  the  air,  had  to  be  given  up,  because  two-Uuids 


788  RADIATION  [§966 

of  it  succeeded  in  escaping  from  the  best-closed  greenhouse  every 
hour. 

Yet  on  a  clear  night,  when  all  the  radiation  is  that  travelling 
up  from  earth,  then  doubtless  the  long- wave- stopping  action  of 
glass  does  come  in  effectively  to  the  assistance  of  the  gardener's 
other  precautions. 

§  967.  Infra-red  is  reflected  by  polished  metal  and  concentrated 
by  concave  mirrors  just  exactly  as  is  light,  in  fact  there  is  less  need 
for  high  polish  :  the  focus  is  the  same  for  both,  as  you  can  soon 
find  in  the  sunshine. 

Glass  lenses  refract  it  less,  with  lower  refractive  index,  to  a  longer 
focus  :  the  brown  paper  catches  fire  under  the  burning-glass  quicker 
just  a  little  beyond  the  distance  at  which  the  most  dazzling  spot 
of  light  on  it  is  formed.  A  hundred  years  ago,  when  there  were  no 
really  bright  lights  for  lantern  shows,  §  478,  people  would  submit 
to  be  shut  in  a  dark  room  while  the  sun  shone  in,  through  14-inch 
condensing  lenses,  to  light  up  objects  in  a  Solar  Microscope,  and 
project  them  on  the  screen.  No  protection  against  concentrated 
infra-red  heat  was  provided  for  the  object,  for  it  was  found  that, 
when  this  was  best  lighted,  the  heat  was  still  too  far  from  being 
focussed  to  do  it  any  harm. 


RADIATION   AND   TEMPERATURE 

§  968.  It  was  pointed  out  in  §  231  that  Newton's  Law  of  Cooling 
was  derived  from  experiments  on  convective  cooling  in  a  draught, 
and  in  §233  that  radiation- cooling  at  hot-water  temperatures, 
as  in  a  vacuum  flask,  is  notoriously  trifling.  Consequently  this 
law  is  of  no  use  here. 

Tyndall,  in  the  course  of  a  vast  amount  of  radiation  research  in 
the  1860's,  had  experimented  with  a  platinum  wire,  the  resistance 
of  which  gave  its  temperature,  and  Stefan,  twenty  years  later, 
observed  that  an  old  result  of  his,  that  at  1200°  C.  the  wire  radiated 
11-7  times  faster  than  at  525°  C,  agreed  with 

(1200°  +  273)V(525°  +  273)*  =  11-6 

and  by  1884  sound  theoretical  basis  was  found  for 

Stefan's  Law. — The  amount  of  heat  energy  radiated  per  second 
from  a  fully  radiating  surface  is  proportional  to  the  fourth  power  of 
the  absolute  temperature. 

Ergs  radiated  per  sq.  cm.  per  second  =  0-00005725  X  T* 
.*.  Nett  interchange  between  two  radiators  =  0-00005725  X  (T*— ^*)^ 

§  969.  The  earlier  experiments  made  to  test  this  law  and  obtain 
the  numerical  factor,  and  unfortunately  the  bulk  of  Tyndall's^ 
work,  were  hampered  by  the  absence  of  a  *  full '  radiator.    This 


§  969]  RADIATION  799 

is  a  surface  best  described  by  its  converse  property,  thmt  it  ii  a 
complete  absorber  of  all  kinds  of  radiation,  reflecting  dom;  a 
•perfectly  black  body.'  Hence  it  would  give  out  evwy  •art  of 
radiation  in  full  proportion  according  to  it8  temperature,  without 
partiality,  a  smooth  continuous  spectrum  without  lines  or  ibadowB. 

Of  '  red-heat '  Radiation,  lampblack  reflects  1-2%,  platinum  bUck 
1-7%  ;  neither  is  '  perfectly  black.'  But  the  tiny  deep  cavitiM 
between  the  fibres  of  velvet  are  dark,  the  pupil  of  the  eye  iji  black, 
the  bunghole  of  an  empty  barrel  is  black  as  the  mouth  of  a  railway 
tunnel.  That  is,  a  deep  hollow,  or  a  small  hole  in  the  side  ol  a 
large  closed  cavity,  acts  as  a  perfectly  black  body ;  for  light  that 
gets  in  is  diffusely  reflected  from  wall  to  wall,  losing  by  alMiorption 
every  time,  until  practically  none  is  left  to  leak  out  (unlem,  of  course, 
it  is  deliberately  directed,  as  in  the  ophthalmoscope). 

Consider  a  closed  cavity,  with  radiating  walls  kept  all  at  one  tem- 
perature, such  as  a  gap  between  glowing  coals  deep  down  in  the 
fire,  and  think  of  a  body  inside  it,  a  bit  of  hanl  glass,  for  instaooa. 
It  is  soon  heated  up  to  the  full  temperature  by  radiation  for  whieh 
there  is  no  escape  ;  and  now,  in  any  particular  direction,  it  radialM 
on  its  own  account,  but  not  fully,  for  it  has  reflective  and  trana- 
missive  powers  discounting  its  absorptive  =  its  radiating  noww. 
But  both  these  are  now  kept  fully  occupied,  the  one  in  reflecting 
radiation  from  the  walls,  and  the  other  in  passing  forwanis,  from 
the  wall  behind,  just  as  much  as  it  loses  through,  backwards,  of 
the  radiation  of  the  wall  in  front. 

A  little  clear  thinking  will  show  you  that  all  deficiencies  in  walk 
and  body  are  made  up,  and  that  through  a  minute  observation  hole, 
like  that  speck  that  is  the  far  end  of  the  tunnel,  *  full '  radiatioQ 
of  that  temperature-quality  will  pour  out. 

In  fact,  things  become  indistinguishable  :  sit  by  the  ftre,  and 
drop  into  its  depths  nails,  chips  of  crockery  or  glass,  of  any  coloar, 
and  watch  how  they  fade  out  of  view  when  shut  in  on  all  sides  by 
the  glowing  coals  that  themselves  have  lost  their  outlines.  It  is 
exasperating,  when  one  has  taken  advantage  of  a  good  fire  to  drop 
in  a  lathe  tool  that  wants  hardening,  to  have  to  pull  the  fire  to  bits 
to  find  the  wretched  invisible  thing. 

You  may  object  that  you  can  see  things  in  the  room  ;  but  sbul 
out  the  6000°  radiation  the  window  admits,  and  admit  no  lamp, 
or  heater,  or  cigarette,  or  radium  button,  and  let  tropical  weather 
have  heated  evervthing  to  98-5°  F.,  and  your  eyes  and  your  t«^P»- 
ture  sense  become  perfectly  useless— unless  somebody  suddenly 
develops  a  malarial  temperature,  or  a  firefly  lights  up. 

Further  instances  of  the  contrast -destroying  effect  of  too  unifonn 
an  illumination  are  the  invisibiUty  of  dust  floatmg  m  the  open  air, 
§  642,  the  futility  of  spotlights  in  fog,  §  612,  and  the*aie  covering 
the  field  of  your  sixth-inch  when  your  substage  condenser  is  loo  wiae 

open,  §  636.  ,        .  *  _  •  :«  .a^^^ 

If  unconvinced,  consult  §568,  reading  .  t«npe«tiJ«    jnPj^ 

of  '  potential '  and  '  lines  of  flow  of  radiaUon    inslaad  of    el0Otno 

lines/ 


790  RADIATION  [§  969 

The  modem  full-radiating  perfect  black  body  is  therefore  a 
square  centimetre  aperture  in  a  water-cooled  shutter,  through 
which  the  receiving  thermopile  (lamp -blacked,  and  1-2%  allowed  it, 
and  radiation  back  proportional  to  its  T*)  looks  down  a  long  por- 
celain or  carbon  tube,  towards  lumps  of  similar  material  at  its 
middle,  the  whole  electrically  heated  to  a  high  uniform  temperature. 

§  970.  Stefan's  Law  appears  in  Fig.  410,  where  the  Areas  under 
the  curves  represent  the  full  radiation  of  a  perfect  black  body,  and 
are  proportional  to  the  fourth  powers  of  the  temperatures  marked. 
The  interchange  from  one  body  to  another  is  the  area  contained 
between  their  curves. 

In  'Whole-radiation  Pyrometers,'  based  on  this  law,  a  gilded 
concave  mirror  faces  a  hole  in  the  side  of  the  furnace  and  forms  an 
image  of  it  to  cover  a  thermo- junction,  which  then  actuates  a  gal- 
vanometer proportionally  to  the  heat  received.  They  are  subject 
to  troubles,  not  the  least  of  which  is  that  careful  enclosure  of  the 
hot  body  to  make  it  a  '  perfect  black  body  '  cannot  always  be  con- 
trived. Surfaces  vary  in  '  emissivity ' ;  while  the  black  oxidized 
surface  of  a  nickel  sheet  is  almost  perfect  up  to  1300°,  an  exposed 
surface  of  clean  metal,  undergoing  *  heat  treatment,'  with  perhaps 
only  50%  emissive  power,  may  be  hundreds  of  degrees  hotter  than 
its  effective  black  body  temperature,  with  disastrous  results  for 
the  process. 

Special  instances  of  '  black  body '  radiation  will  be  considered 
in  §  975  ff.  :  we  will  take  first  the  other  laws  of  radiation. 

§971.  Radiation  pressure.  In  §  110  it  was  shown'  that  a  fluid 
under  pressure  contained  energy  equal  to  the  product  of  pressure 
and  volume.  Conceiving  of  Radiation  as  such  a  fluid,  let  us,  for 
instance,  open  a  1  sq.  cm.  window  to  bright  sunshine  for  one  minute. 
In  that  time  a  sq.  cm.  stream  of  it  of  length  [(3  X  lO^*^  cm. /sec.)  X 
60  sees.]  cm.  passes  through,  filling  volume  1-8  X  10^^  c.c.  Energy 
=  the  Solar  Constant,  §  976,  is  contained  in  it,  1-93  X  4-2  x  10^  = 
8  X  10^  ergs,  and  this  divided  by  the  volume  gives  0-000045  dyne 
per  sq.  cm.  pressure. 

This  is  small,  but  capable  of  pushing  aside  a  hanging  gold  leaf 
under  the  microscope,  and  the  Radiation  Pressures  of  much  less 
brilliant  sources  have  been  measured  to  within  2%. 

When  the  collection  of  a  few  million  tons  of  meteoric  stone  and 
dust,  travelling  in  company,  which  constitutes  the  head  of  a  Comet, 
gets  near  the  Sun,  the  intense  Radiation,  falling  on  a  small  particle 
proportionally  to  its  area,  presently  exceeds  the  gravitational 
attraction  which  is  proportional  to  its  volume.  Dust  and  distilled 
vapour  and  gas — for  atoms  of  course  absorb  radiation — are  there- 
fore driven  out  as  the  Comet's  Tail,  which  always  points  away 
from  the  sun.  Larger  particles,  having  much  greater  mass  per  unit 
^rea  of  surface,  are  of  course  little  affected. 

This  is  the  close-home  instance  of  the  force  which  counters  gravita- 


§  973]  RADIATION  7«l 

tion  in  controlling  the  distribution  of  matter  in  the  Universe. 
Gravitation  calls  upon  a  vast  volume  of  finely  divided  matter  to 
close  in  towards  a  centre  :  as  it  does  so,  coUisions  increase,  and 
heat  is  generated  and  radiated,  and  this  presses  back  the  incoming 
particles,  forbidding  further  contraction.  It  radiates  away,  in 
time,  and  contraction  is  resumed,  so  that  from  cold  dark  dust, 
such  as  makes  those  very  conspicuous  black  blots  on  the  Milky 
Way,  a  nebula  slowly  gains  luminosity,  and  goes  on  condensing, 
most  probably  slowly  pulsating  in  brightness,  until  it  shows  a  dis- 
coid centre  as  a  '  planetary  nebula '  ;  or  produces  starH,  like 
Antares,  fifty  times  as  massive  as  the  sun  perhaps,  but  fifty  million 
times  more  bulky  ;  or  shining,  as  do  the  stars  of  the  Pleiades  to  the 
patient  eye  of  the  camera,  on  the  vast  volumes  of  dust  from  which 
they  were  born. 

§  972.  From  this  conception  of  Radiation  Pressure  flowed,  in 
1893,  Wien's  *  frequency  '  Law. — In  full  radiation,  the  frequency 
of  the  radiation  which  is  being  emitted  in  greatest  quantity  is  proportional 
to  the  absolute  temperature. 

Inverting  into  wave-lengths  in  microns,  30  years  later  it  became 
possible  to  write  this  law  definitely  as 

T  =  2885 


'^max. 


And  from  this,  and  Stefan,  follows  Wien's  energy  Law  that  the 
Quantity  of  energy  being  radiated  on  maximum-intensity  frequency 
is  proportional  to  the  fifth  power  of  the  absolute  temperature. 

Both  these  laws  are  contained  in  Fig.  410  (which  happens  to 
be  on  a  wave-length  scale,  and  the  other  way  round  to  Figs.  223 
and  418).  Measuring  the  wave-length  of  the  peak,  and  multiplymg 
by  the  temperature  of  that  particular  radiation  you  get  4  X  725. 
2-7  X  1100,  2  X  1450,  etc.,  all  about  2885.  Notice  how  the  dotted 
curve  on  which  the  maxima  are  strung  is  drawing  nearer  to  the 
visible  spectrum  RV  :  it  means  that  the  feet  of  these  curves  (too 
small  to  draw),  already  just  inside  it,  are  pushmg  further  along 
towards  the  blue  end,  the  very  small,  but  self-assertive  visiUe 
part  of  the  radiation  is  changing,  from  the  dullest  red,  to  r©a 
1100°  A.,  'bright-red'  1300°  A.,  'yellow  hot'  1500°  A.,  and  so  on, 
S  192,  the  march  being  shown  plainly  by  a  pocket  spectroscope. 

Measure  the  heights  of  the  peaks  and  you  find,  omittmg  a  stnng 

Height  of  peak  =  T^  x  constant  (about  7) 
so  that  with  the  slow  progressive  Whitening  of  the  Colour  comae 
a  very  rapid  Brightening  of  the  intensity  of  the  Light. 

§973.  With  these  two  effects,  taken  together,  you  have  be« 
familiar  all  your  life,  if  not  at  the  country  blacksmith  s.  at  any  rate 
bv  vour  o^^  fireside,  particularly  near  tea-time     ,9^^  ^^  ,.\^"  "^ 
S^  an  all-electric  community,  like  one  of  those  c^.htfulv^^ 
dotted  along  the  King's  Highway  through  the  peacli  grove-  wert 


792 


RADIATION 


[§973 


of  Niagara ;  where,  with  a  wood  fire  crackling  and  roaring  in  wel- 
come of  the  visitor  from  '  home,'  they  prayed  him  pardon  the  absence 
of  toast  because  the  current  was  cut  off  ? 

And  will  you  try  the  experiment  of  §  816,  and  find  how  very 
much  more  light  you  get  from  half  a  square  millimetre  of  tungsten 
at  2800°  or  so,  in  a  pocket  lamp,  than  from  half  a  square  foot  of 
good  red  glowing  coke  fire  at  1000°  or  more  ? 

The  topmost  curve  of  Fig.  410,  1650°  A.,  is  only  a  sallow  tint, 


Fig.  410. 


Fig.  411. 


-A 

COMPAKAnVE 
HEIGHT  or 
PEAK,<xT^ 

TOTAL 

OUTPUT 

(AREAocT^ 

6ooo 

5,Zoofloo 

y4  000,000 

3ooo 

toopoo 

4,600,000 

f6So 

Sooo 

4)io,ooo 

lloo 

65o 

80000 

30Q 

1 

V60 

6    LJave-Lc-n^m  tn  ■microns. 


for  if  you  try  to  fit  in  that  pocket  lamp,  you  find  that  2-8^  is  174, 
so  you  want  1-2  m.  of  height,  the  peak  still  to  the  right  of  1[jl  ; 
and  a  sun  at  5750°  or  at  6000°,  wants  44,  or  55,  m.  beyond  the  top 
of  the  page,  the  sharp  peak  at  last  standing  above  the  middle  of  RV. 

The  area  under  the  curve  is  the  total  output  of  radiation,  increasing 
asT*. 

The  very  lowest  curve  is  getting  near  to  being  visible  in  the  dark, 
so  you  can  get  an  idea  of  the  difficulty  of  measurements  like  those 
of  §  870  on  radiation  of  5  (x  or  longer  ;  the  intensity  attainable  is 
so  exceedingly  small. 


§  9'^]  RADIATION  7«t 

You  see,  too,  that  a  temperature  only  half  725**.  practically  Um 
temperature  of  boiling  water,  would  give  a  peak  only  l/500Ui 
inch  high  :  there's  more  heat  in  the  thickness  of  the  baae-line  than 
there  was  in  '  the  burning  deck.' 

So  when  you  want  to  tell  your  examiners  about  Radiation,  and 
liave  been  given  a  free  hand  in  the  matter,  don't  for  goodnMi  aaks 
make  us  shiver  with  a  tale  of  a  tea-pot.  Think  how  you  till  a  vacuum 
tlask,  and  it  radiates,  and  radiates,  and  goes  on  radiating,  hour  after 
hour,  and  then  you  scald  your  mouth  l>ecau8e  it  hasn't  radiated 
enough  :  see  §  233,  things  at  temperatures  you  can  licar  to  touch 
hardly  radiate  worth  mentioning  ;  if  you  must  talk  of  them,  make 
it  up  with  area  ;  take  the  whole  garden.  Think  rather  of  something 
that  is  hot^that  coal  in  §  9G2  glowed  sensibly  hotter  below  than 
above,  radiation  is  beginning  to  come  into  its  own — gather  rouml 
the  fire,  and  tell  us  tales  of  your  own  experience,  in  the  warm  light 
of  the  vacuum  lamp  that  can  but  radiate ;  not  ghost  stories  of 
'  bodies  '  radiating  in  the  dark. 

All  the  curves  in  Fig.  410  are  of  essentially  the  same  shape ;  you 
see  this  more  completely  in  Fig.  411,  which  is  the  same  thing  over 
again,  but  with  the  scales  altered  to  fit  the  curve,  instead  of  the 
other  way  about :  four  definite  scales  are  given,  Sun,  tungsten 
wire  lamp,  red  heat,  and  the  ordinary  temperature,  e.g.  tbegardso 
radiating  to  the  sky.  The  Visual  Spectrum  is  marked  VrC,  and 
lines  are  carried  up  from  the  part  of  it  (on  a  wave-length  scale) 
to  which  the  eye  is  really  sensitive.  Figs.  223,  414;  between  them 
lies  the  radiation  of  that  temperature  quality  by  which  we  see  the 
radiator. 

Earlier  theories  disagreed  with  experiments  on  the  shape  of  the 
curve,  and  it  was  not  until  Planck  packed  up  the  radiate<l  eoeq^r 
into  Quanta,  §  982,  which  had  to  be  emitted  or  absorbed  complete, 
that  he  was  able  to  calculate  the  curve  shown,  which  subsequent 
experiments  have  abundantly  confirmed. 

§  974.  Optical  Pyrometry.  The  mercury-pressure  thcnnooieter, 
§  198,  the  thermo-junction  §  799,  and  the  platinum  resistAQce 
thermometer,  §  778,  all  come  to  an  end  of  their  usefulness  below  ths 
temperatures  required  in  manv  modem  manufacturing  proossses, 
and  far  beneath  that  of  ordmary  flame.  The  full- radiation  pvro. 
meter  is,  as  we  saw  in  §  970,  often  too  exacting  to  accommodate. 
but  Wien's  Laws  now  open  up  two  other  valuable  methodsof  measur- 
ing to  the  highest  temperatures,  and  both  are  widely  used. 

From  Figs.  410  and  411  vou  can  gather  how  the  foot  of  ^n^J^^^ 
tion  curve  invades  the  visible  spectrum,  stepping  along  ^^''JJ™*^ 
blue  as  the  temperature  rises,  the  curve  sUnding  higher  and  bigtier 
in  the  mid-red.  i,.iu 

Series  of  dye-solutions  can  be  prepared,  and  scalwl  up  in  iimic 
cells,  to  be  held  before  the  eyes,  which  absorb  and  blot  «>*»«/*»*\^^* 
as  far  as  definite  places  along  the  spectnim  (it  is  casv  ^oooimm 
two  reds  which  look  just  alike  but  together  h\ack  each  oUier  oaf 


794 


RADIATION 


[§974 


DIMMER 


AMMETER 


BATTERY 


Fig.  412. 


completely)  so  that  the  radiator  has  to  reach  a  particular  high 
temperature  before  its  light  passes  beyond  the  absorption  band  and 
becomes  visible.  The  workman  is  told  to  heat  up  until  through 
filter  so-and-so  he  can  see  light  after  half-a-minute's  watching. 

The  Disappearing  Filament  Pyrometer,  Fig.  412,  is  a  simple  telescope 
with  the  fi  filament  of  a  little  lamp  in  its  focal  plane,  and  a  red  glass 
over  its  eyepiece.     The  user  points  it  at  the  incandescent  surface 

and  adjusts  a  shding  resistance  un- 
til the  filament  becomes  invisible  in 
the  bright  field,  both  emitting  red 
light  at  the  same  level  on  Fig.  411, 
and  then  reads  the  ammeter  in  the 
pocket-battery  lamp  circuit. 

Little  calculation  is  indulged  in 
with  these  instruments,  they  are 
calibrated  on  nickel  and  palladium 
at  their  melting  points,  1725°  and 
1825°  A.,  and  the  ammeter  is  gradu- 
ated to  read  temperatures  direct. 
For  very  high  temperatures,  up  to 
3000°,  the  briUiant  field  is  dimmed 
by  a  neutral- tint  dark  glass,  of 
measured  effect  (down  to  1/200) 
which  avoids  any  risk  of  burning- out  the  lamp. 

These  convenient  pyrometers  hold  the  field  in  industrial  work. 
For  the  very  Highest  Temperatures  one  has  to  form  a  spectrum, 
study  the  distribution  of  brightness  in  it  with  a  Spectro-Photometer, 
and  apply  Wien's  Frequency  Law  of  §  972,  X  max.  X  T  =  2885. 
In  this  way  temperatures  are  arrived  at  of 

2160°  A.  for  a  candle  flame  or  an  old  carbon  filament  lamp, 

2670°  for  a  gas  mantle  or  a  50-watt  wire  lamp, 

3300°  for  a  1000-watt  gas-filled  lamp  or  the  boiling  point  of  iron, 

3820°  for  the  carbon  arc, 

6140°  for  the  Sun, 

10,000°  for  Sirius  and  the  white  stars, 

19,000°  to  26,000°  for  the  blue-white  Rigel,  etc. 

These  are  above  their '  black- body  temperatures' — a  cold  gas  mantle 
visibly  isn't  black  ;  they  are  selectively  radiating '  Gray  Bodies '  which 
have  to  be  rather  hotter  to  equal  the  output  of  a  '  full  radiator.' 

Astronomers,  however,  have  just  begun  to  peer  more  closely 
'  between  the  absorption  lines,'  and  are  making  allowance  for  the 
obscuring  smokiness  of  the  chromospheric  envelope,  which  of  course 
obstructs  the  high-temperature-indicating  violet  most,  §  568. 
Stellar  faces  thus  theoretically  washed  prove  much  more  shining  ; 
the  Sun's  photosphere  is  credited  with  6450°  A. ,  and  Sirius  with  as 
much  as  18,000°. 

Perhaps  someone  will  figure  out  for  us  just  how  much  whiter  and 
brighter  London  would  be  without  its  smoke. 

Stars  are  classified  by  wholesale  for  temperature  by  photographing 


§975]  HADIATION  795 

in  blue  light  and  red  light,  and  then  comparing  the  intensities  of  the 
images  by  the  micro-photometer  :  increasing  strength  of  blue  image 
means,  of  course,  increase  of  temperature. 

New-made  lamps  are  similarly  classified  by  a  machine  which 
lights  them,  looks  at  them  with  two  photo-electric  cells,  §  984,  one 
red-sensitive  and  the  other  blue-sensitive,  and,  by  the  ratio  of  the 
currents  these  produce,  knows  into  which  bin  to  drop  the  lamp. 

§  975.  Lamps.  This  all  has  a  great  bearing  on  the  wasteful  way 
we  obtain  the  artificial  light  in  which  we  spend  about  half  our 
working  lives.  We  heat  things  hot  and  they  radiate  energy  pro- 
fusely, and  the  little  cream  that  comes  to  the  top  of  the  curve  is 
all  we  get  to  see  by. 

Look  at  Fig.  411  :  the  first  scale  has  to  do  with  nocturnal  radiation 
from  the  soil,  and  is  discussed  elsewhere  ;  the  second  is  red  heat, 
and  the  trifle  by  which  it  oversteps  the  visibility  boundary  accounts 
for  §  557  ;  and,  between  that  and  the  third,  you  see  it  was  not  until 
metal  lamp  filaments  were  made,  §  816,  which  would  stand  higher 
temperatures,  without  hastily  disintegrating,  then  the  2200°  or  so 
of  carbon  filaments,  and  the  2800 — 3300°  A.  range  came  in,  that  any 
common-sense  fraction  of  the  area  of  the  radiation  curve  lay  between 
the  visibility  lines  at  all. 

The  Carbon  Arc  has  its  special  uses  on  account  of  the  great 
local  intensity  of  its  light  radiation,  170  c.p.  per  sq.  mm.,  a  sixth 
that  of  the  sun ;  and  sunlight  is  utilized  extremely  well,  which  means 
of  course  that  eyes  have  evolved  to  make  the  best  use  of  the  most 
abundant  radiation  received  :  in  §  632  we  took  X  0-55  [i  as  fairly 
representative  of  'white  light'  and  now  we  find  Wien's  formula 
XT  =  2885  giving  X  =  2885/5750°  =  0-50  for  sunlight ;  the  differ- 
ence may  be  due  to  the  shift  in  the  maximum  sensibility  of  the  eye 
from  0-55  in  bright  light  to  0-50  in  twilight. 

Carrying  up  the  0-6 — 0-45  [i  limit  lines  for  the  tungsten  lamp, 
you  see  they  enclose  only  about  2-5%  of  its  total  radiation. 
Illuminating  engineers  have  nothing  better  in  sight  for  indoor  use, 
but  for  street  lighting,  where  colour  is  not  objected  to,  there  are  the 
mercury  etc.  lamps  of  §  890.  These  you  now  see  are  more  highly 
efficient  because  they  are  not  in  the  slightest  '  full  radiators,*  but 
radiate  selectively  on  useful  wave-lengths  only — in  fact  the  green 
mercury  line  is  in  the  ideal  position — though  subsidiary  actions 
inside  their  tubes  still  waste  the  bulk  of  the  energy  as  heat. 

The  Glow-worm,  the  Fireflies,  and  Deep-sea  fishes,  found  out  all 
about  this  long  ago.  They  produce  an  oxidizable  substance,  which 
glows  like  phosphorus,  or  fresh-cut  potassium,  and,  though  they 
differ  in  tint,  not  one  has  made  the  mistake  of  radiating  far  from 
the  middle  of  the  visible  spectrum — Nature  seems  to  have  standard! - 
ized  her  visual  purple  for  everybody.  Thus  they  are  lOO^/o  efficient . 
and  large  fireflies,  emitting  perhaps  a  third  of  a  candle  power 
forward  only,  over  about  unit  solid  angle,  and  between  0-52  and 
0-62  (ji,  are  probably  emploving  only  1 /25,000th  the  energy  a  candle 
would  have  to  put  into  similar  brief  flashes. 


796  RADIATION  [§976 

§  976.  The  rate  at  which  we  receive  radiation  from  the  Sun,  called 
the  Solar  Constant,  was  measured  simply  enough  by  turning  the 
black  bolometer  to  face  the  sun,  at  various  heights  on  Mt.  Whitney 
in  California,  and  deducing  what  correction  to  apply  for  the 
atmosphere  still  above  its  15,000-ft.  summit.  It  appeared  that  the 
Sun  sends  us  1-93  calorie  per  sq.  cm.  of  surface  directly  facing  him, 
per  minute.  Of  this  it  is  allowed  that  water  vapour  in  clear  air 
absorbs  a  tenth,  that  Clouds  reflect  an  enormous  amount  (no  wonder 
the  Stratosphere  above  them  keeps  warm),  and  with  direct  reflection 
from  sea  and  land,  1  calorie  per  sq.  cm.  per  minute  is  left  to  be 
absorbed  at  the  surface. 

Over  the  eleven-year  Sunspot  Cycle,  §§  698,  944,  the  '  constant ' 
varies  by  8%  in  all,  being  greater  at  periods  of  greater  spot  develop- 
ment. At  such  times,  the  increased  heat  reception,  together  with 
increased  ionizing  activity  of  solar  radiations,  commonly  results 
in  raising  more  cloud,  bringing  warmth,  but  wind  and  wet,  good 
growing  years  for  vegetation.  The  11 -year  period  has  been  traced 
back  3000  years  in  sections  of  Sequoia  trunk  ;  dating  Indian  dwellings 
by  the  annual  ring  sequences  of  posts  found  in  them  is  commonplace 
in  Arizona,  and  fossil  trees  are  being  examined. 

Settled  conditions,  rather  than  furious  solar  activity,  seem 
conducive  to  fine  summers  in  England,  though  of  course  most  of 
us  judge  these  by  holiday  impressions  :  I  seem  to  recollect  1887  and 
1897  as  delightfully  fine  warm  summers,  and  1911  and  1921  were 
years  of  hot  sunshine  and  drought ;  all  four  were  towards  the  end  of 
the  diminishing  phase  of  spottiness  ('97  rather  early),  while  1932, 
1933,  and  1934,  with  a  very  marked  and  prolonged  scarcity  of  spots, 
were  real  old-fashioned  fine  dry  summers,  1933  indeed  being  the 
finest  and  driest  on  record,  changing  the  outlook  and  the  ways  of 
all  of  us. 

The  total  radiation  we  receive  from  all  the  Stars  amounts  in  200 
years  to  the  one  minute's  '  solar  constant.' 

§  977.  The  Sun,  with  his  lined  and  shaded  spectrum,  cannot 
be  a  '  perfect  black  body,'  but  taking  him  as  if  he  was,  let  us 
calculate  his  *  black-body  temperature.' 

His  mean  angular  diameter  is  32  minutes  of  arc,  so  that  his  angular 
radius  is  16'  =  16/(60  X  57-3)  =  1/215-5  radian;  which  means 
that  the  radius  of  the  Earth's  orbit  is  215-5  solar  radii. 

The  *  Solar  Constant,'  §  976,  is  1-93  calories  received  per  minute, 
per  sq.  cm.  of  a  surface  facing  the  sun,  here,  at  a  distance  of  215-5 
of  his  radii ;  therefore  applying  the  inverse  square  law,  he  emits 
1-93  X  (215-5)2  =  90,000  cals./min.  sq.  cm.  of  his  surface,  or  per 
second  90,000  X  42  million/60  =  6-3  X  lO^o  ergs  =  0-00005725^*, 
by  §  968. 

/.  T  =  5750°  A. 

You  see  it  is  lower  than  the  6140°  A.  deduced  from  the  colour 
law  :  that  dealt  with  not  a  '  black '  but  a  '  gray'  surface,  which 
has  to  be  rather  hotter  to  emit  at  the  same  rate. 


§  979]  RADIATION  707 

§  978.  The  total  output  of  radiation  from  tlie  Sun  is  easily  cal- 
culated on  the  basis  of  the  Solar  Constant,  for  this  is  the  wiipply 
of  energy  per  minute  to  every  sq.  cm.  of  a  hollow  sphere  of  radius 
=  the  mean  distance  from  sun  to  earth. 

This  total  area  is  4tc  X  (1-495  x  lO^^  cm.)2  =  2-8  x  10*'  sq.  cm. 

and  1-93  calorie  per  minute  =  (1-93/60)  x  4-2  x  10'  =  1'35  x 
10^  ergs  per  sec. 

.-.  total  output  =  2-8  X  1-35  X  lO^s  =  3-78  x  lO^^  ergs  per  sec. 
and  the  problem  is,  how  is  it  kept  up  ? 

The  blue-white  stars  are  the  most  intensely  brilliant  things  we 
can  see,  and  therefore  the  hottest,  but  their  26,000°  connotes  no 
fundamental  alteration  in  the  properties  of  a  hot  fluid.  Astro- 
physical  calculation  finds  that  the  hidden  interiors  of  stars  must, 
however,  be  hotter,  or  they  would  crush  in ;  the  common  critical 
temperature  is  40,000,000°,  and  at  that,  '  energy  appears  to  issue 
from  matter  like  steam  from  boiling  water.' 

Modern  theory,  which  we  cannot  go  into  here,  indicates  that 
the  rate  of  exchange  is  1  gram  of  Matter  =  (speed  of  light)*  ergs 
=  9  X  1020  ergs. 

If  therefore  the  Sun's  heat  is  maintained  by  conversion  of  matter 
into  energy  in  its  interior,  such  energy  being  carefully  filtered  out 
through  its  surface  as  6000°  radiation,  it  is  losing  mass  per  second 
3-78  X  1033/(9  X  1020)  gm.,  or  about  4  million  tons ;  and  we  don't 
know  how  else  it  can  be  done  for  any  length  of  time. 

As  its  mass  is  2  X  102'  tons,  it  could  last  at  that  rate  15  billion 
years,  if  all  convertible  ;  but  if  it  is  the  hydrogen  into  heUum  change 
of  §  951  that  is  providing  waste  matter  for  disposal  as  energy,  this 
would  come  down  to  about  50,000  million  years.  This  is  not  a 
score  times  the  life  of  our  little  chip,  §  945  ;  and  besides,  the  sun  is 
elderly  as  stars  go,  but  we  see  no  vast  amount  of  helium  on  him, 
though  plenty  of  hydrogen  :  it  looks  as  if  the  complete  conversion 
process  is  the  one  mainly  in  operation. 

§  979.  Now  we  are  about  it,  what  is  the  mean  black-body 
temperature  of  the  Earth  ? 

Notice  that  the  1-93  calories  are  received  on  a  surface  flatly 
facing  the  sun  :  the  sum  total  of  this  is  the  area  of  the  earth's  flat 
circular  shadow,  nr"^,  whereas  it  radiates  from  its  whole  spherical 
surface  4:Tzr^.  The  loss  per  second  is  therefore  1-93  X  42  X  10*/ 
(60  X  4)  ergs  =  0-00005725^,  giving  ^  =  59  X  10*  or  <  =  277**  A., 
=  4°  C. 

Statistically,  the  average  is  288°,  so  again,  the  earth  is  not  quite 
a  full  radiator. 

In  no  way  is  the  immense  effect  of  Stefan's  Fourth  Power  Law 
of  Radiation  more  strikingly  illustrated  than  in  this  :  a  little  disc 
puts  in  a  half-time  appearance  in  a  sky  100,000  times  its  apparent 
area,  and  by  its  radiation,  at  5750°,  it  maintains  the  supply  of  enerev 
that  the  earth  radiates  by  day  and  by  night  to  that  whole  cold 
vault. 


798  RADIATION  [§979 

It  was  noted  in  §  312  that  dust  in  the  atmosphere  prevented  solar 
radiation  reaching  the  earth's  surface,  and  within  living  memory 
'  years  without  summers '  have  followed  exceptional  volcanic 
eruptions.  The  calculation  has  indeed  been  made,  though  one 
doesn't  know  how  much  salt  should  be  taken  with  it,  that  only 
*  1 /700th  cubic  mile  of  fine  dust  flung  into  the  upper  atmosphere 
once  every  two  years  would  have  sufficed  to  account  for  any  of  the 
five  known  glacial  epochs.' 

§  980.  Planetary  temperatures.  And  if  we  pry  into  our  neigh- 
bours' affairs,  Mercury  is  at  36/93  our  distance  from  the  Sun,  Venus 
at  67/93  and  Mars  at  142/93 ;  their  Solar  Constants  are  of  course 
our  1-93  multiplied  by  the  inverse  squares  of  these  ratios,  and  their 
average  temperatures  come  out,  by  the  same  calculation  as  above, 
to  445°,  326°  and  224°  A.,  or  172°,  53°  and  -  49°  C. 

Mercury  is  only  2-5  the  mass  of  the  Moon  and,  like  it,  must  be 
an  airless  mass  of  dull  grey  or  reddish  lava ;  its  reflecting  power 
is  only  a  quarter  that  of  white  cloud. 

Venus  is  nearly  as  big  as  the  Earth,  and  has  a  dense  atmosphere, 
which  by  its  refractivity  upset  the  accuracy  of  observations  made 
at  her  '  transits  '  of  1874  and  1882  ;  she  shows  no  fixed  markings, 
and  has  the  reflecting  power  of  cloud,  in  which  she  is  probably 
perpetually  enfolded.  On  the  outside,  this  must  stave  off  a  good 
deal  of  the  incident  radiation,  as  it  does  locally  with  us  ;  and, 
beneath,  it  must  maintain  a  vigorous  atmospheric  circulation  on 
the  lines  of  §  321,  doing  much  to  equalize  climatic  temperatures, 
which,  at  the  cooler  parts,  can  scarcely  exceed  those  of  damp 
tropical  jungle.  Luxuriant  vegetation,  such  as  we  know,  ought  to 
be  perfectly  possible,  producing  oxygen  for  animal  life.  Into  the 
lives  of  the  inhabitants  cold  or  arid  conditions  have  never  entered, 
the  sun  to  them  is  no  more  than  a  moving  brightening  of  unbroken 
cloud,  minor  celestial  bodies  they  have  never  seen,  and  towards 
extra-mundane  ideas,  or  exact  measurement,  their  attention  has 
never  been  drawn  :  one  imagines  that  a  dinosaurian  brata  would 
be  adequate  to  their  needs,  one  cannot  credit  them  with  the 
development  that,  through  many  millenia,  these  stimuli  have  been 
working  in  our  own. 

Stray  clouds  in  the  atmosphere  of  Mars  occasionally  conceal 
tracts  of  his  red  deserts,  his  polar  snows  soon  melt  in  summer,  and 
a  green  vegetation  develops  in  the  surrounding  moistened  areas. 
The  rapidity  with  which  these  things  happen,  with  a  solar  constant 
only  two-fifths  our  own,  and  the  clearness  with  which  we  can 
often  see  his  surface,  prove  amply  that  he  is  but  thinly  clad  with 
atmosphere  or  blanketed  with  aqueous  vapour.  Warm  enough 
in  the  sun,  his  unprotected  night  frosts  must  be  terrible,  —  49°  0. 
is  just  about  the  lowest  Canada  or  Siberia  can  touch,  and  that  is 
his  all-round  average. 

One  can  credit  the  Martians  with  plain  living  and  high  thinking ; 
Lowell,   and  now  G.   H.   Hamilton,  have  maintained  insistently 


981]  RADIATION 


799 


that  the  linear  markings  to  be  seen  on  its  surface,  under  favourable 
telescopic  conditions,  are  tracts  of  fertile  country,  like  the  Nile 
valley,  bordering  the  water-courses— whether  natural  or  artificial— 
which  bring  the  water  of  the  circumpolar  swamps  down  through 
its  arid  deserts.  They  form  a  '  test  object '  in  the  sky  on  which 
amateur  astronomers  are  much  given  to  exercising — and  not  seldom 
stretching — the  Resolving  Power  of  §  632. 

Jupiter  and  Saturn  are  enveloped  in  floating  cloud,  the  spectro- 
scope speaks  of  ammonia  and  methane  in  their  atmospheres,  even 
their  most  stable  features  are  not  perennial,  the  Red  Spot  of  the 
one  and  the  White  Spot  on  the  other  are  probably  the  fumes  of  vast 
volcanic  eruptions  ablaze  beneath. 

§981.  What  does  the  1-2  calorie  amount  to,  absorbed  by 
atmospheric  aqueous  vapour,  or  fairly  reaching  the  earth's  surface, 
§  976,  all  over  that  '  diametral  disc'  of  area  tt  X  (6-4  x  10*  cm.)* 
=  128  X  10i«  sq.  cm.,  on  which  the  Sun  is  shining  all  the  time ; 
which  throws  the  round  dark  shadow  of  the  Earth  ? 

It  is  (1-2/60  sec.)  x  4-2  joules  x  128  x  10i«  =  10-8  X  W*  joules 
per  sec.  or  108  billion  kilowatts  ;  which,  divided  among  a  supposed 
population  of  2400  millions,  gives  45,000  kw.  or  60,000  h.p.,  per- 
petually working  day  and  night,  for  every  one  of  us,  as  our  human 
Heritage  of  Power. 

Much  research  is  being  directed  to  the  problem  of  unlocking  the 
vast  stores  of  energy  which  radioactivity  disclosed  as  hidden  in 
all  atoms,  §  933,  and  some  success  has  been  achieved,  §§  923,  946 ; 
but  just  as  we  have  hitherto  found  the  practicable  way  of  getting 
at  the  energy  of  coal  is  to  raise  it  to  a  high  temperature,  in  preference 
to  eating  it,  so  Nature  stokes  the  interior  of  stars  up  to  40,000,000° 
as  '  energy-gas  producers,'  while  the  best  she  can  do  in  the  radio- 
active line  appears  to  be  the  extremely  trivial  heat  output  of  Earth 
itself,  §  944.  It  really  looks  as  if  coming  generations  may  well 
consider  how  to  catch  and  harness  some  of  those  60,000  horses. 

Something  always  has  been  done,  in  sails  and  windmills  and  water- 
mills,  and  now  in  great  hydro-electric  schemes ;  but  for  the  most 
part  we  spendthrifts  rob  the  bank  of  hydrocarbons,  which,  in  past 
ages,  the  oil-storing  diatoms  and  other  alga?,  and  the  lignin  producing 
land  plants,  won  from  the  carbonic  acid  of  the  volcanoes,  when 
by  that  same  action  they  were  preparing  an  atmosphere  of  oxygen 
for  more  active  animal  life — for  the  crust  of  the  Earth  contains 
a  score  thousand  times  as  much  oxygen  as  lies  above  it,  and  it  is 
unthinkable  that  Nature  stayed  her  hand,  just  at  that  last  narrow 
margin,  for  our  coming  breathing. 

When  accessible  coal  and  oil  are  exhausted,  in  a  thousandth  of 
the  time  it  took  to  lay  them  down,  and  water-power  is  exploited  to 
the  full,  when  such  comparatively  tractable  volcanic  eflForts  as  the 
Tuscan  soffioni  are  all  blowing  their  boric  breath  through  steam 


800  RADIATION  [§  981 

turbines,  and  the  day  is  being  lengthened  by  draining  the  rotational 
energy  of  the  earth  through  tidal  barrages,  there  will  remain  the 
cultivation  and  combustion  of  the  timber  of  forests,  and  the 
production  of  alcohol  from  the  fermentation  of  softer  vegetable 
materials ;  and,  in  addition,  windmills,  and  the  direct  collection  of 
solar  radiation. 

Equating  the  1-35  million  ergs  per  sec.  of  the  solar  constant  to 
0'00005725  T*,  for  a  black  sq.  cm.  facing  the  sun,  shows  that, 
apart  from  convective  cooling,  it  may  attain  392°  A.  or  119°  C. 

This  goes  to  explain  why  you  carefully  step  over  the  strip  where 
quarter-inch  ripples  are  idly  lapping  the  black  margin  of  comminuted 
vegetable  debris  on  a  sunny  West  Indian  beach  ;  and  it  also  shows 
how  it  has  been  possible  to  raise  steam  of  small  pressure  in  air- 
tight double-glazed  '  garden  frames,'  for  use  in  a  turbine  ;  but  while 
such  frames  are  quite  serviceable  for  distilling  drinking-water,  long 
parabolic  reflectors  of  polished  sheet  metal,  stretching  east  and  west, 
facing  up  to  the  sun,  and  focussing  its  heat  on  a  water  pipe  running 
the  length  of,  the  E.  and  W.  axis, — reviving  the  invention  of  Archi- 
medes,— have  proved  their  power  of  raising  steam  at  more  practicable 
pressure  from  the  persistent  sunshine  of  California. 

Looking  down  from  Mount  Wilson  by  night  over  the  wealthy 
residential  plain  of  that  province,  one  sees  how  its  two  or  three 
millions  have  lit  their  three  score  townships  with  innumerable 
lamps ;  league  upon  league  there  lies  outspread  a  shimmering 
cloth  of  gold.  Probably  our  own  great  metropolitan  area  is,  on 
the  whole,  less  lavish,  unless  in  this  gay  week  of  Royal  Jubilee, 
but  does  either  of  them  get  this  vast  outpouring  of  radiation,  some- 
where about  as  bright  as  full  moonlight  perhaps,  from  the  million 
times  greater  supply  of  sunlight  they  have  had  by  day  ? 

Little  but  from  oil  and  coal,  300  million  years  in  bond. 

London,  besides,  for  half  the  year,  burns  maybe  a  ton  of  coal  per 
head  to  keep  its  brick  boxes  of  rooms  a  very  few  degrees  above  the 
wind  without  :  in  winter  the  whole  area  is  about  3°  F.  warmer 
than  the  surrounding  country.  If  her  winter  warmth  came  from 
direct  sunshine,  instead  of  largely  from  warm  Atlantic  wind  and 
vapour,  she  would  soon  learn  to  dispense  with  a  smoke  screen  that 
kept  off  more  heat  than  the  coal  that  made  it  gave. 

When  such  expenditure  of  energy  becomes  inadmissible,  when  the 
film  of  population  floats  to  summer  climes,  as  most  of  us  do  when 
we  get  the  chance,  when  the  struggle  for  mechanical  power  is  almost 
the  chief  material  concern  of  civilization,  then,  unless  the  tight- 
closed  energy  strongholds  of  the  atoms  have  been  unlocked  with  a 
far  more  economical  key  than  we  can  foresee  at  present,  it  must 
be  by  some  such  now  little-thought-of  devices  as  those  outlined 
above,  that  the  arrogant  fifty-billionth  of  its  mass  that  is  the 
Human  Race,  will  seek  to  maintain  what  it  deems  its  dominion  over 
the  affairs  of  Earth. 


§  982]  RADIATION  qqI 

§982.  The    Mechanism    of   Emission    of    Radiation.    Fig.    382 

suffices  to  illustrate  this  for  radio-waves ;  electrons  in  billions  are 
rushed  up  and  down  a  wire,  and  their  long  line-of-force  tails  lash 
out  in  all  directions,  dancing  up  and  down  just  as  would  waves 
on  a  dozen  of  slack  ropes,  radiating  like  the  spokes  of  a  spider's 
web,  if  you  stood  at  the  middle  and  shook  them  all  up  and  down 
together,  sending  out  waves  '  polarized  in  a  vertical  piano  of  vibra- 
tion.' 

Where  are  the  jumping  electrons  doing  this  on  a  lO^^*  times  smaller 
scale,  to  emit  light,  etc.  ? 

That  ratio  lands  us  inside  the  atom,  cf.  §  919. 

As  we  have  said  already,  the  Atom  consists  of  a  small  central 
massive  positively  charged  nucleus,  and  round  it  are  circulating 
planetary  electrons,  from  1  to  92  in  number,  carrying  in  all  a  negative 
charge  equal  to  the  positive  on  the  nucleus  ;  bound  to  it  by  electric 
attraction,  on  the  inverse-square  law.  This  compels  them  also 
to  keep  the  peace  with  one  another,  in  spite  of  their  mutual  repul- 
sions, and  they  can  do  so  only  by  adopting  certain  definite  configura- 
tions, K,  L,  M,  etc.,  orbits,  compUcated  as  yet  for  the  mathematician, 
who  is  worrying  at  this  astronomical  problem  by  successive  approxi- 
mations as  best  he  can. 

In  general,  the  outer  electrons  seem  not  too  happy  in  their 
allegiance  to  the  central  attraction  so  far  away,  and — ^well,  you  know 
how  a  little  pig  who  has  left  it  a  bit  late  runs  round  trying  to  squeeze 
in,  and  if  he  can't  he  wanders  off  to  the  next  trough  and  fills  a 
vacancy  there — effecting  a  chemical  combination  of  atoms — or 
maybe  in  desperation  routs  out  another,  who  hasn't  a  foot  in  the 
trough,  and  he  flees  to  the  third  mess,  and  so  on — that  is  conduction 
in  a  metal.  If  the  outer  ring  contains  eight  electrons  you  get  the 
self-satisfaction  of  the  inert  gases. 

By  the  electrical  force  in  any  form  of  discharge,  or  by  the  electrical 
force  of  chemical  action  in  the  bunsen  flame,  or  by  the  reception 
of  energy  in  the  form  of  radiation,  one  electron  is  dragged  out  tem- 
porarily into  some  outer  orbit,  as  if  Mars  were  displaced  into  Saturn's 
orbit,  the  atom  being  now  said  to  be  '  excited.' 

A  definite  amount  of  work  has  been  done  in  dragging  out  the 
electron  this  definite  distance  against  the  nuclear  attraction.  The 
arrangement  is  not  permanently  stable ;  and  presently  it  drops 
back  into  its  home  orbit,  flinging  back  that  energy  out  into  space 
as  a  Quantum  or  Photon  packet  of  Radiation. 

One  can  imagine  how  :  the  electron  rushes  in  with  momentum 
which  carries  it  too  far,  and  has  to  swing  back,  and  so  oscillates, 
bouncing  along  in  its  orbit  for  some  time  as  it  gradually  loses 
this  energy,  in  radiation  of  some  particular  characteristic  frequency. 

There,  you  see,  is  the  electron  dancing  up  and  down  the  aerial. 

This  may  also  throw  a  little  light  on  the  question  which  the  solitary 
electron  of  hydrogen  particularly  raises  :    why  must  it  submit  to 

D  D 


S02  RADIATION  [§  982 

the  same  orbital  restrictions  as  are  desirable  in  big  families  ?  Be 
content  with  this  :  the  Electron  has  a  wave -structure  of  its  own, 
it  slips  from  one  orbit  to  the  other  more  like  a  wriggling  chain  than 
a  pellet,  and  any  orbit  must  fit  its  undulations  like  a  Melde  string 
fits  its  fork,  §  436. 

It  can  drop  from  any  one  orbit  to  any  other,  and  a  different, 
but  perfectly  definite,  quantum  of  energy  is  associated  with  each 
of  these  possible  falls  :  just  as  on  a  particular  chff  you  can  fall  from 
any  higher  to  any  lower  ledge,  you  can't  remain  stuck  in  the  air ; 
nor,  if  the  wind  does  happen  to  blow  you  up  again,  can  you  retain 
any  more  energy  than  was  necessary  to  lift  you  to  the  particular 
ledge  on  which  it  drops  you.  And  each  fall  calls  forth  a  particular 
yell,  the  bigger  the  drop  the  shriller  the  note. 

The  result  of  all  this  is  that  regularities  appear  in  the  hitherto 
indefinite  and  almost  infinite  complexity  of  Spectra.  These 
forced  themselves  into  attention  in  the  X-ray  spectra,  §  919,  and 
their  meaning  was  first  deciphered  there  :  the  falls  happen  to  be 
few  and  simple,  there  are  only  4  levels  instead  of  the  9  of  Fig.  417, 
which  illustrates  the  question  more  fully. 

This  discovery  that  Radiant  Energy  is  carried  and  handed  over 
to  Matter,  and  handed  out  again,  only  in  complete  sealed  packets, 
quanta,  or  phota,  or  photons  as  it  is  become  the  fashion  to  call  them, 
is  due  to  Max  Planck,  and  the  factor  h  which  relates  the  size  of  the 
quantum  to  the  frequency  of  the  radiation  is  6-55  X  10-^7,  and  is 
called  Planck's  Constant. 

Quantum,  or  photon,  in  ergs  =  6-55  x  10-^^  X  Frequency, 
e.g.  a  quantum  of  green  light  contains  6-55  X  10~^^  X  500  X  10^^  _ 

3-3  X  10-12  erg. 
e.g.  a  quantum  of  tungsten  K  X-ray  contains  6-55  X  10-^7  x  15  X 10^^ 
=  10-7  erg. 

One  begins  to  see  how  it  is  that  while  the  great  bulk  of  Radiation 
comes  to  us  in  long-wave  guise,  of  lower  frequency  than  appeals  to 
the  eye  ;  the  short  waves  on  the  other  side,  the  ultra-violet,  X,  and 
Y  radiations,  of  increasing  frequencies,  possess  potencies  unknown 
in  the  infra-red. 

As  we  still  most  commonly  quote  wave-lengths,  and  as  the  fre- 
quency is  the  number  of  waves  that  fill  their  travel-distance  in  1 
second,  3  X  10^^  cm.  =  3  X  10^*  microns,  we  can  write  it  Q  =  h  x 
(3  X  101*) /X  -  6-55  X  10-27  X  3  X  101* A  =  l-965/10i2x  ergs,  so 
that,  nearly  enough  to  remember  : — 

One  Quantum  or  Photon  contains  (2  ergs  divided  by  a  billion  times 
its  vMve-length  in  microns). 

For  average  light  X  =  0-55  jx,  while  a  250-kv.  deep-therapy 
X-ray  has  X  =-  1-23/250,000  =  0-5  (x/100,000,  and  therefore  110,000 
times  the  energy.  Comparing  a  light  photon  to  a  cigarette,  the 
other  is  the  Exposicion  cigar  of  Seville,  9  ft.  long,  guaranteed  to 
contain  good  tobacco  enough  to  make  11,000  cigars;  but  some 
of  the  longest-wave  infra-red  quanta  would  have  hardly  a  shred  of 
tobacco  apiece. 


984]  RADIATION 


803 


§  983.  Thus,  insubstantial  radiant  energy,  flying  about  in  space 
is  only  allowed  to  cross  the  frontier  into  matter,  and  become  kinetic 
energy  of  a  moving  particle,  or  sensible  caloric  energy,  if  it  is  duly 
packed,  not  indeed  in  cartons,  but  in  photons,  and  contents  aii 
declared. 

It  is  just  as  if  the  invisible  energy  of  the  wind  is  only  allowed 
to  be  transferred  to  the  water,  where  it  becomes  visible  wave 
motion,  in  definite  pufiFs  :  one  breath  to  cool  the  hot  liquid  in  the 
saucer,  one  cat's-paw  to  ruffle  the  calm  of  the  bay,  one  squall  to 
spoil  the  hopes  of  a  quiet  sail. 

The  parallel  is  close,  the  photon  is  a  gust  of  energy  travelling  at 
3  X  IQio  cm./sec,  it  is  about  600  cm.  long,  so  that  it  blows  for  about 
a  fifty- millionth  of  a  second  ;   and  one  electron  catches  the  lot. 

Evidently,  therefore,  it  is  a  thin  affair,  but  they  are  being  sent  out 
from  the  source  in  countless  milhons,  indifferently  in  all  directions, 
so  producing  the  effect  of  a  spherical  wave  system  ;  the  chance 
of  catching  a  number  varies  inversely  as  the  square  of  the  distance, 
giving  the  familiar  photometric  law;  and  they  are  long  enough 
to  enable  the  metre  to  be  measured  as  1,553,164  wave-lengths  of 
cadmium  red  light,  in  air  at  15°  C.  and  760  mm. 

Thus  the  Quantum  theory  does  no  violence  at  all  to  the  Wave 
theory. 

With  ordinary  light,  the  electron  will  get  600  X  10,000/0-55  = 
about  11  million  shakes  in  this  time  ;  with  ultra-violet  it  gets  twice 
as  many — each  single  shake  just  as  hard  because  the  total  energy  is 
proportional  to  the  frequency,  and  the  frequency  is  proportional  to 
the  total  number  in  the  600  cm.  :  every  uxive  in  every  quantum 
contains  an  identical  amount  of  energy,  one  three-trillionth  of  an  erg. 

§  984.  Thus  Ultra-violet  may  succeed  in  shaking  electrons  out 
of  zinc,  for  instance,  whereas  ordinary  light  has  no  such  effect  : 
like  a  single  equinoctial  gust  fetching  down  apples  in  the  orcliard 
that  have  survived  all  summer's  breezes.  This  is  the  Photoelectric 
Effect,  the  electron  has  been  dragged  clean  outside  the  atom,  which 
is  now  '  ionized,'  charged  +1.  It  begins  at  a  definite  wave-length, 
and  after  that  the  greater  energy  of  shorter-wave  photons  suffices 
not  only  to  release  the  electrons,  but  gives  them  a  surplus  of  energj', 
and  the '  photo-cell '  begins  to  exert  an  E.M.F.  and  drive  a  —  current. 

So  long  as  it  was  merely  known  that  ultra-violet  would  discharge 
a  —  charge  from  a  clean  plate  of  zinc,  the  effect  held  no  nractical 
interest,  but  presently  it  was  found  that  clean  films  of  the  more 
'  electro-positive '  alkali  metals,  melted  in  vacuo,  were  sensitive 
to  visible  light  also,  expelling  a  —  charge  on  to  a  grid  stretched  in 
front  of  them  across  the  bulb,  as  in  Fig.  413,  where  the  film  lines 
the  flat  back  of  the  two-inch  bulb.  The  currents  obtainable, 
however,  seldom  exceeded  a  micro-ampere,  and  it  remained  an  effect 
for  the  laboratory  until  amplifying  valves  became  commonplace, 
and  now  the  electric  eye  has  found  many  uses. 

The  curves  in  Fig.  414  show  the  currents  obtainable  from  a  Photo- 
cell lined  with  a  thin — sometimes  mono-molecular — layer  of  the 


804 


RADIATION 


[§984 


metals  named,  when  exposed  to  radiation  of  equal  strength  on  various 
wave-lengths  :  other  trade  mixtures  and  modifications  are  in  use, 
and  copper  oxide  is  also  an  extremely  effective  photo-electric  emitter. 
The  pecked  curve  is  that  of  visibility  to  the  human  eye,  cf .  Fig.  223, 
and  you  see  that  potassium-on-copper  still  has  plenty  of  sensitivity 
left,  well  into  the  infra-red. 

An  ordinary  H.T.  voltage  of  120  ensures  the  capture  of  the 
liberated  electrons,  and  for  many  common  purposes  the  cell  is 
filled  with  0*15  mm.  pressure  of  argon,  which  increases  the  current 
perhaps   20-fold   on   account   of    '  ionization   by   collision,'    §  886. 


Fig.  415. 


Fig.  414. 


Fig.  413. 


Amplifying  circuits  do  the  rest,  based  on  the  simple  one  of  Fig.  415, 
where  the  photo-electric  current  traverses  the  resistance,  and  the 
P.D.  between  its  ends,  modified  by  grid  bias,  is  applied  to  the  grid. 

You  already  know  many  uses  to  which  this  unfailing  Electric 
Eye  is  applied,  such  as  turning  on  or  off  street  lamps,  and  navigation 
lights  of  all  sizes,  measuring  mist  or  smoke,  §  302,  counting  bunches 
of  bananas,  or  visitors,  passing  between  it  and  a  lamp,  or,  with  the 
lamp  screened  to  practically  invisible  red,  giving  warning  of  un- 
welcome and  unwanted  prowlers  by  night. 

Without  any  complications,  a  cadmium  plate  connected  to  an 
electrometer  makes  a  self-contained  lonto-quantimeter  for  measuring 
Ultra- Violet  dosage,  and  a  similar  device  is  used  for  the  X-rays  of 
Deep  Therapy. 


§  985]  RADIATION  3^ 

Fig.  416  shows  how  sound  can  bo  recorded  on  moving  film  •  the 
0-2-mm.  thread  of  a  string  galvanometer,  §  762,  actuated  bv  the 
microphone  current,  swings  to  and  fro  across  a  0-1 -mm.  slit  set 
at  a  very  acute  angle,  so  that  it  increases  and  diminishetj  the  length 
of  sht  through  which  light  reaches  the  film,  and  this  is  recoixled  a« 
the  peaked  profile.  Cinema  film  so  marked  travels  over  the  re- 
producer slit,  altering  its  effective  length  correspondingly,  and  so 
modulates  the  light  passing  through  into  the  photo  cell,  and  the 
response  of  this,  which  does  not  lag  as  much  as  ten  miliiontha  of 
a  second,  is  amplified  up  into  the  loud  speakers. 

The  scanning  beam  of  a  Tele-picture-writer,  or  of  a  Television 
apparatus,  is  converted  into  current  in  the  photo-cell,  and  wirelen, 
synchronous  motors,  and  a  reproducer  such  as  the  oscillograph  of 
§  884,  do  the  rest.  ' 

In  ordinary  Photometry  the  Photo-cell  is  replacing  the  Eye; 
the  measurement  of  stellar  magnitudes  offers  an  infinite  field  for 
the  most  sensitive  forms  ;  and  the  modem  photographic  Exposure 
Meter  is  a  pocket  micro-ammeter  backed  by  an  oxidized  copper 
disc,  under  conducting  glass  :  one  opens 
it  to  the  light,  and  the  pointer  swings 
over  to  foot-candles,  or  lumens,  or  the  sto}) 
to  be  used,  or  the  strength  of  the  light 
measured  in  any  way  you  will. 

The  far  ultra-violet  of  the  sim  gets  used 
up  in  the  Stratosphere  (which  it  helps  to 
keep  warm)  in  photo-electric  removal  of 
electrons  from  the  gas  molecules  them- 
selves, these  electrons  being  speedily  cap- 
tured by  a  neighbour,  and  a  pair  of  ions 
thus  resulting.  To  this  is  owing  at  any 
rate  one  of  the  '  Heaviside  layers '  of  the  conductive  Ionosphere, 
which  shuts  in  our  radio  waves  from  wandering  off  into  space  : 
automatic  measurements,  on  the  fathometer  principle,  of  the  height 
of  this  reflecting  ceiling,  are  conducted  between  a  radio  trans- 
mitter just  west  of  the  City  and  a  receiver  2  miles  east. 

Ozone  is  a  product  of  this  ionization,  §  957,  which  at  one  swoop 
cuts  off  the  death-deaUng  ultra-violet,  attends  to  the  distribution 
of  atmospheric  electricity,  §  898,  and  saves  us  99%  on  our  long-dis- 
tance wireless. 

§  985.  To  get  an  X-ray  of  wave-length  X  we  saw  in  §  914  required 
a  minimum  voltage  1-2345/X,  to  drive  the  electron  into  the  target. 
and  now  we  can  see  why.  The  electron's  charge  e  is  15-9  x  lO"* 
coulomb,  making  its  energy  eV  =  15-9  X  10-»»  X  10»  joules  to  ergs 
X  1-23/X  =  2/(1012  X)  ergs,  and  you  recognize  that  as  the  energy 
of  the  X-ray  photon  its  impact  generates,  §  984. 

That  is  the  hardest  X-ray  it  can  :  it  generates  plenty  of  softer 
ones,  of  longer  X  and  feebler  energy— in  fivct  you  recollect  that  only 
one  electron  in  1,000  produces  an  X-ray  worth  having  at  all,  §  913. 


LIMITS  -^ 

1  wmc  ox«i« 

Of  WIR£S 
MOVtMlHT 

^ 

If 

Fj 

[o^Er«SUTO-l«« 
G.  416. 

806 


RADIATION 


[§985 


In  turn,  if  these  X-ray  photons  hit  a  suitable  target,  they  hand 
over  their  energy  complete,  and  superficial  electrons  get  shot  off 
as  Secondary  Cathode  Particles,  just  as  fast  as  the  originals,  though 
most  are  feebler  from  having  to  struggle  out  through  thickness  of 
metal.  And  these  again  can  generate  Secondary  X-rays  :  all  this 
is  a  great  nuisance  to  the  Radiographer,  who  finds  his  pictures 
fogged,  §  917.  In  abdo.  work,  a  bare  sixpence  used  to  be  placed 
on  the  umbilicus,  as  a  landmark,  but  the  practice  had  to  be 
given  up,  for  the  radiation  from  it  sometimes  caused  an  X-ray 
burn. 

Absorbed  in  gas  atoms,  they  cause  the  ejection  of  electrons,  which 
are  caught  by  neighbouring  atoms,  and  thus  pairs  of  ions  are  pro- 
duced, and  that  is  how  X-rays  make  any  gas  conductive. 

Received  in  silver  bromide,  they  dislodge  electrons ;  as  do  light 
photons  down  to  the  middle  of  the  visual  spectrum  only. 

§  986.  A  recipient   electron  is,   of   course,   entitled   to   unpack 

the  photon  and  use  its  contents 
for  his  own  purposes  ;  he  may  fly 
away  in  the  strength  of  it,  but  if 
he  wants  to  send  any  away  he 
must  repack  a  photon,  of  a  size 
he  carries  in  stock ;  any  loose 
tobacco  left  over  must  be  burnt, 
and  produces  common  compara- 
tively useless,  or  wholly  waste. 
Heat,  in  the  usual  way. 

What  ordinarily  happens  when 
a  photon  is  absorbed  in  an  atom 
can  be  followed  best  by  reference 
to  Fig.  417.  This  shows  the  differ- 
ent orbital  levels  possible  for 
electrons  in  a  specimen  atom ; 
the  scale  alongside  gives  an  idea 
of  the  quantity  of  energy  required 
to  drag  an  electron  out  of  each 
level,  against  the  attraction  of  the 
nucleus  down  below,  and  remove 
it  from  the  atom  altogether, 
leaving  that    '  ionized,'    charged 

+  1. 

The    photon    is    absorbed    in 

lifting  an  electron  from  some  one 

to  some  other  of  these  levels  (if 

it  lifts  it  right  out  free,  that  is  the 

Photo-electric  Effect). 

Marked  on  each  lift  is  a  wave-length  :  whichever  lifts  take  place, 

those  particular  photons  are  absorbed  from  the  incident  radiation, 

i.e.  those  lines  show  dark  in  its  Absorption  Spectrum.    For  instance 


-lOOO 

00 

$ 

a 

Zpoo                                         ^ 

o 
3ooo 

« 

£ 
5 

tv( 

-<• 

n 
•4 

y.ooo         ^ 

1 

^   2 

■» 

M 

V\ 

50 
00 

lOpoo        9 

! 

M 

I.R. 

^ 
^ 

9.0,000   ? 

I.R. 

VIS. 

U-V. 

lOO.OOO, 

UJ 

U-V. 

Fig.  417. 


988] 


RADIATION 


Wl 


0-656  is  red  and  0486  blue  ;   taking  these  colours  from  white  light, 
it  would  pass  through  yellowish -green. 

The  simplest  thing  that  can  happen  now  is  that  these  electrons 
immediately  drop  back  as  they  were,  along  the  arrows,  re-emitting 
the  missing  red  and  blue.  This  is  nothing  more  nor  less  than  white 
light  falling  on  almond  blossom,  or  sub-oxidized  copper,  and  their 
giving  you  back  a  mauve  colour. 

§  987.  But  already  there  is  another  possibility ;  the  0-486  may 
elect  to  drop  back  in  two  stages,  1-88  and  0-656.  The  first  is  quite 
invisible  infra-red,  the  second  is  the  original  red.  The  stuff  would 
be  yellowish-green  looked  through,  but  gives  you  back,  when  you 
look  at  it,  red.     See  Chlorophyll,  §  558. 

Next,  suppose  0-380  were  absorbed  :  look  at  the  return  possibili- 
ties now,  4  ultra-violets,  4  visibles,  8  infra-reds  {one  can't  do  them 
all,  of  course,  but  there  are  millions  of  atoms  and  plenty  of  time), 
a  whole  complicated  spectrum.  This  is  Fluorescence  ;  chlorophyll 
showed  the  simple  beginning  of  it :  you  see  therefore  that  all 
fluorescent  light  must  be  of  lower  photon  energy,  longer  wave-length, 
than  the  exciting  radiation,  and  also  what  immensely  better  chancea 
of  exciting  fluorescence  violet  and  ultra-violet  have,  cf .  §  959,  and 
a  fortiori  the  almost  omnipotent  X-photon,  §  913,  which  always 
lifts  the  electron  completely  out,  leaving  a  vacancy  for  any  new- 
comer to  rush  into. 

Of  course,  some  things  will  take  the  breakneck  leap  all  in  once, 
they  won't  fluoresce.  Others  will  not  only  take  it  in  easy  stages, 
but  will  take  their  time  about  it,  they  go  on  phosphorescing. 

§  988.  Parenthetically,  we  might  not  be  compelled  to  employ 
photons  to  lift  these  electrons.    We  have  just  been  using  electron 


® 


Sbo 

— h 


6oo 


^00 


800 


poc 


® 


® 


=1 


I^ET 


Thtit  are  hijht'line  sftctra, 


•30     IN    MiCKONt'tf 


.70  -tfa  So  -Uo     WAVC-CENCTM 

RED -OR.-  GR.  •    BLUevi.-»l  ULTRA-VIOLET 

Fig.  418. 

energy  of  motion  to  knock  out  X-photons,  and  X-photons  to  fire 
out  electrons,  and  so  on,  indifferently,  so  why  not  employ  flymg 
electrons  to  kick  up  0-656,  0-486  and  the  rest  ?         ^   ^  ^  .  ,, 

As  a  matter  of  fact,  this  is  the  Hydrogen  atom,  ana  *«»*  is  exactly 
what  we  do,  when  we  put  a  little  in  a  tube  and  hitch  on  a  few 
thousand  volts ;  it  glows  mauve-pink,  and  you  can  see  red,  blue, 
and  violet,  and  photograph  all  10  lines  of  this  Balmer  spectrum. 


808  RADIATION  X  §  988 

Fig,  418,  the  first  spectrum  series  ever  observed.  Beside  it  is  the 
principal  series  of  Sodium.    (These,  of  course,  are  bright  line  spectra.) 

Now  another  point.  You  see  sodium  vapour  glowing  yellow 
0*589,  and  you  have  made  sure  in  the  laboratory  to  be  shown  its 
light  absorbed  black  again  as  it  passes  out  from  an  arc  through 
surrounding  vapour,  §  557  ;  which  is  quite  as  it  should  be,  down  and 
up  one  jump  ;  and  you  know  the  black  D  line  in  the  sun. 

But  the  C,  F,  and  G  solar  lines  are  the  first  three  Balmer  lines 
right  enough,  yet  if  you  pass  white  light  through  any  length  of 
hydrogen,  at  any  temperature,  no  trace  of  them  is  to  be 
seen  ? 

The  electron  of  hydrogen  is  an  only  child,  and  there  is  no  opposi- 
tion to  him  living  as  near  home  as  possible,  and  in  the  natural  state 
of  hydrogen  he  is  in  the  lowest  orbit,  where  his  centrifugal  force 
keeps  him,  revolving  with  maximum  speed  and  energy  in  defence 
of  the  nucleus.  No  photon  of  visible  light  has  strength  enough  to 
lift  him  up  to  the  second  orbit,  you  see  it  takes  X  0-12,  which  is 
vacuum  ultra-violet,  §  954,  and  Lyman  had  to  carry  this  to  the 
extreme ;  and  then,  had  it  been  a  few  thousand  times  stronger, 
hydrogen  might  have  visibly  shone  its  usual  pink.  The  Sun  has 
got  this  ultra-violet,  and  keeps  plenty  of  hydrogen  ready  excited, 
so  has  no  difficulty  about  it. 

Don't  mistake  these  2,  3,  4  levels  for  termini ;  but  if  the  electron 
didn't '  stop  to  set  down  '  there  sometimes,  for  a  hundred-millionth 
of  a  second  or  so,  the  hydrogen  spectrum  would  be  solely  high  ultra- 
violet. 

§  989.  When  this  same  hydrogen  spectrum  was  also  obtained 
from  Helium  there  was  rather  a  to-do  about  it,  until  fine  measure- 
ment showed  a  sHght  displacement  in  the  lines.  Then  it  became 
clear  that  the  helium  was  in  an  ionized  condition,  bereft  of  its  second 
electron,  so  that  a  soUtary  electron  was  now  buzzing  round  the 
nucleus  of  mass  4  and  charge  +  2  :  naturally  it  gave  its  usual 
performance,  only  very  slightly  '  flat.' 

And  the  spectrum  of  Diplogen,  or  heavy  hydrogen,  is  necessarily 
the  same,  displaced  even  less,  1/3600  :  hiding  so  close,  and  being 
only  1/6000  as  strong,  it  was  never  seen  in  the  normal  hydrogen 
spectrum,  until  specially  hunted  for  in  1933,  after  the  isotope  had 
been  discovered. 

Argon  was  long  a  puzzle  because  an  increase  in  the  intensity 
of  the  exciting  electric  discharge  faded  out  a  spectrum  of  red  lines 
and  substituted  one  of  blues,  but  it  is  now  plain  that  it  had  been 
robbed  temporarily  of  one  of  its  18  electrons,  and  was  appearing 
as  chlorine,  17  ;  or  of  two,  as  sulphur,  16.  Similarly  sodium,  11, 
can  be  ionized  to  give  a  neon,  10,  spectrum  ;   and  so  on,  cf.  §  852. 

§  990.  Finally,  you  see  that  two  bodies  arrive  at  the  same  tempera- 
ture by  interchanging  photons  which  gradually  become  more  alike 
and  ultimately  equal.     A  recipient  cannot  repack  and  return  larger 


§  991]  RABIATION  ^f^ 

photons  unless,  of  his  own  power  of  evolving  energy,  he  has  been  able 
to  boost  electrons  further  out.  You  cannot  *  concentrate  *  the  heat 
of  a  kettle,  or  the  light  of  a  lamp,  into  an  image  hotter  or  brighter 
than  the  source  ;  the  solar  furnace  of  §  612  can  at  best  approximate 
to  the  temperature  of  the  solar  surface.  But  you  can  light  a  match 
by  holding  it  near  a  merely  red-hot  coal. 

§  991 .  See  how  a  Theory,  at  first  sight  unnecessary  and  comphcated 
and  forbidding,  has  disentangled,  and  then  joined  in  order,  things 
seemingly  very  diverse  ;  and  this  is  only  a  beginning.  In  the  course 
of  your  work  you  will  meet  with  many  Theories,  and  perhaps  make 
some  up  for  yourself  :  do  not  assess  them  by  their  new  long  words, 
but  make  them  work,  as  this  has  done  ;  else  they  are  of  little  worth! 


EXAM  QUESTIONS,   CHAPTER   LVI 


Radiation  used  to  be  treated  in  a  few  paragraphs  at  the  end  of  Heat 
the  Cosmos  consists  of  Matter  and  Radiation.  We  have  given  M«tier  •  good 
innings,  Radiation  must  have  fair  play — at  the  moment  I  am  son-bathing, 
and  it  has. 

In  this  long  and  complicated  chapter,  which  goes  as  far  aa  ia  accepted  at 
present,  you  can  scarcely  fail  to  find  some  points  of  interest :  your  »t  udy  of 
it  will  best  be  guided  by  the  Questions ;  for  the  rest,  it  ranges  from  the  amalleat 
of  created  things  to  recondite  problems  of  Creation ;  and  perhi^,  at  thin 
last,  you  will  gladly 


*  Leave  the  Wise  to  wrangle,  and  with  ino 
The  Quarrel  of  the  Universe  let  bo.* 


1.  '  Light  is  a  transverse  vibration  of  electromagnetic  character.' 
this.     Is  it  meant  to  apply  to  other  varieties  of  radiated  energy?     How  do 
they  differ  ? 

2.  Describe  a  method  of  measuring  the  speed  of  travel  of  Radiation. 

3.  By  what  experiments  would  you  show  that  Radiant  Heat  and  Light 
are  of  the  same  nature?  How  do  they  differ?  How  would  you  \-erify  the 
inverse-square  law  ?     (  X  3) 

4.  How  are  the  colours  of  the  spectrum  related  to  their  wave-lengtha.  and 
what  means  are  there  of  detecting  anything  outside  the  visible  spectrum  ? 

5.  How  would  you  compare  the  sensitivity  of  a  photographic  film  to  diflereai 
wave-lengths  in  the  spectrum  of  an  arc  ? 

6.  Describe  three  ways  of  detecting  Ultra-violet  radiation.  How  would 
you  establish  its  similarity  to  light  ?     (  X  2) 

7.  Describe  apparatus  for  producing  a  strong  beam  of  Ultra- Violet. 

8.  How  woTild  you  investigate  the  retention  of  an  electric  chaxge  by  a 
metal  in  ultra-violet  ? 


810  RADIATION 

9.  Describe  the  chief  characteristics  of  (o)  an  emission  spectrum,  (6)  an 
absorption  spectrum.     How  would  you  demonstrate  these  in  the  ultra-violet  ? 

10.  How  would  you  show  that  the  radiation  from  an  electric  arc  extends 
beyond  both  ends  of  the  visible  spectrum  ?  How  do  these  invisible  radiations 
differ  from  the  visible,  and  from  each  other  ?     (  X  4) 

11.  How  would  you  test  whether  a  source  of  light,  such  as  burning  mag- 
nesium, is  rich  in  ultra-violet  ?  Why  does  sunlight  vary  in  its  content  of 
ultra-violet  from  day  to  day  ?     What  effect  has  glass  ?     (  X  2) 

12.  Give  a  brief  account  of  the  spectrum  of  sunlight,  with  special  reference 
to  the  *  ultra-violet '  and  the  '  infra-red.'  State  concisely  what  you  know 
of  the  properties  of  these  radiations.     (  X  2) 

13.  Describe  experiments  which  illustrate  the  existence  of  radiant  heat. 
How  would  you  investigate  the  relative  transparency  of  materials  for  this 
type  of  radiation  ?     (  X  2) 

14.  How  would  you  show  that  a  large  amount  of  the  energy  radiated  by 
a  gas  flame  consists  of  non-luminous  heat  rays,  and  how  measure  the  percentage 
stopped  by  a  sheet  of  glass  ? 

15.  Describe  some  delicate  instrument  for  detecting  heat  radiation. 
Explain  how  it  can  be  used  (i)  to  determine  the  emissivity  of  a  surface, 

(ii)  to  compare  the  transparency  of  different  materials  to  heat  radiation. 

16.  What  experiments  would  determine  whether  glass  or  celluloid  absorbed 
less  radiation  from  a  body  below  red  heat  ? 

17.  Describe  a  method  for  comparing  the  intensities  of  heat  radiation  sent 
out  by  a  candle  and  a  black  kettle  filled  with  hot  water.  Explain  the  effect 
of  interposing  a  sheet  of  glass  in  front  of  each  source.     (  X  2) 

18.  What  is  meant  by  the  Diathermancy  of  a  substance  ?  How  would 
you  compare  those  of  two  plates  of  glass  for  radiation  from  a  lamp  ?  If  one 
plate  were  twice  as  thick  as  the  other,  how  would  you  reduce  your  results  ? 

19.  Compare  and  contrast  the  radiations  from  a  metal  filament  lamp,  an 
arc,  an  electric  discharge  tube,  and  sunlight. 

20.  Upon  what  does  the  rate  at  which  a  body  radiates  heat  depend  ?  Prove 
from  general  principles  that  the  radiating  power  of  a  body  is  equal  to  its 
absorbing  power  at  the  same  temperature,  and  describe  an  experiment  which 
illustrates  this  relation. 

21.  Describe  an  experiment  to  show  that  the  sums  of  the  emitting  and 
reflecting  powers  of  different  surfaces  for  heat  radiation  are  equal.  Dis- 
tinguish between  the  absorbing  power  of  a  surface  and  the  absorbing  power 
of  the  interior  of  a  solid. 

22.  How  can  it  be  shown  experimentally  that  the  heat  radiation  from  a 
hot  body  obeys  the  same  laws  of  reflection  and  refraction  as  light  ? 

23.  Mention  three  facts  bearing  upon  the  similarity  in  character  between 
light  and  radiant  heat.     Describe  generally  the  change  in  character  of  the 
radiation  from  a  body,  as  it  is  raised  from  the  ordinary  temperature  to  white  ' 
heat. 

24.  How  would  you  show  that  the  amount  of  heat  radiation  received 
from  a  surface  depends  (a)  on  the  nature  of  the  surface,  (6)  on  its  tempera- 
ture, (c)  on  the  distance  of  the  receiver  ? 

25.  Describe  some  form  of  radiation  Pyrometer,  and  how  to  test  its  accuracy. 


SOLUTIONS 

Th^ae  have  hem  checked  twice,  and  few  can  he  at  fault. 

CHAPTER   II,  p.  15. 
i.  m.p.h.  X  44-7. 

3.  37  X  6080/5280  =  37  x  1152,  i.e.  knots  to  m.p.h.  add  about  1  in  7 

4.  Knots  X  51-43. 

5.  20-4  sec,  30-6  sec,  and  40-8  sec  :  the  easiest  way  of  finding  your  «pe«<l 
in  a  train. 

6.  Assuming  fall  vertical,  then  horiz.  length  of  slanting  track/its  verticml 
height  =  speed  of  train/speed  of  fall  of  raindrop  (and  this  latter  has  a  maximum 
value  25  f t .  /sec . ) . 

7.  Draw  straight  cross-track,  draw  cross-line  parallel  to  it  1-5  upstream. 
From  A  draw  line  length  3  to  meet  this  where  it  can,  prolong  it  until  it  roachw 
bank  (actually  at  60°).  Time  =  (its  length /3  mi.)  hours  (actually  60/2v'S  -r 
3  =  5-75  min.) 

8.  Draw  AB  =  70  E.,  AC  =  50  N.E.,  wind  =  BC. 

9.  Easy  to  draw  and  measure,  or  =  \/[{0-3  +  0-2)*  +  0-2*]  =»  0-54  mile. 

10.  Draw  AB  =  16  to  W.,  BC  =  4  to  S.W.,  AC  is  her  course.  Draw 
AD  =  12  to  S.E.,  CD  is  her  smoke  trail,  AB  her  keel. 

13.  32-2. 

15.  9  sec 

16.  A  gift  to  a  formula  merchant. 

17.  10  cm./sec.2,  the  common  difference.  Average  of  10  and  20,  20  and  30, 
etc.  =  15,  25,  35,  45. 

18.  Average  speed  first  1000  ft.  =  1000/0-8  =  1250  ft. /sec;  second  « 
1162  ft. /sec  Loss  =  88  ft. /sec.  in  (0-4  -f  0-43)  sec,  from  mid  pt.  to  mid 
pt.  =  106  ft. /sec  in  one  second.     (Try  other  figures,  e.g.  0-85,  0-78.) 

19.  2000  =  i  X  981  X  t\t  =  2  sec;  30  m.  out. 

21.  First  ignore  soimd  speed,  s  =  i  X  32-2  X  2-7«  gives  117  ft.  Sound 
would  travel  this  in  117/1100  =  0106  8ec.,soactual  time  of  fall  =  2-7  -  010« 
=  2-6  sec  nearly  enough;    whence  s  =  ^  X  32-2  x  2-6"  =  109  ft. 

22.  20,000  =  i  X  981  X  t\  whence  total  time  =  6-4  +  200/330  «  7  sec. 


CHAPTER   III,  p.  31. 

3.  2200/0-006  ft./sec.2;   100  times  as  many  poundals,  over  500  tons. 

5.  2-24  X  108  dyne  =  0-228  ton  wt.,  33  ft. 

6.  Accelerative  force  =  0-5  X  0-5  X  mg  =  0-25  g  per  gra.  Average  sp««d 
for  t  sec  half  0-25  gt,  travelling  0-125  gt*  =  100  m.  .*.  <  =  9  sec.  /.  speed  - 
0-25  X  981  X  9  =  22-1  m./sec  =  79-5  km./hr. 

Stopping,  braking  all  four  wheels  halves  time,  4-5  see.  On  downgrBde  de- 
celerative  mfir/2  is  decreased  by  mgr/ 15,  time  increased  proportionately. 

7.  Average  speed  during  stop,  25  km./hr.  =  6-95  m./sec.  /.time  —  10/6-W 
=  144  sec  .-.  deceleration  =  1390/1-44  =  970,  force  =  800.000  xJtO  - 
775  million  dynes,  about  800  kg.,  which  is  at  least  twice  the  grip  of  good  tffr99 
on  a  dry  road.     Another  police-court  story. 

9.  500  X  1000  X  400  =  2  X  10«  dynes. 

811 


812  SOLUTIONS 

10.  5  X  60  (poundals). 

11.  Speed  to  lift  64  ft.  =  speed  at  end  of  64  ft.  fall.  Using  a  short  cut,  v* 
=  2  X  32  X  64  .-.  t;  =  64  ft. /sec.  /.  force  =  200  X  10  -^  60  X  64  =  2133 
poundals. 

12.  360  =  i  a  X  144.  /.  a  =  5.  2  tons  wt.  +  2  X  2240  X  6  poundals; 
2  tons  weight. 

14.  Average  speed  during  stopping  15,000,  taking  1/5000  sec.  to  destroy 
25  X  30,000  momentum,  or  at  the  rate  of  5000  X  25  X  30,000  per  sec.  = 
3750  megadynes,  about  3*8  tons. 

15.  Momentum  after  impact  200  X  30  +  50  X  60  =  momentum  before, 
200  X  45m./s. 

18.  1-5  +  0-5  kg.  is  accelerated  by  0-5  —  1-5  X  0-3  =  0-05  kg.  =  50  X  981 
dynes  =  2000  X  o,  .*.  a  =  2-45;  moves  about  1  ft. 

19.  1/100  -  1/200  =  1/200  of  gravity;   60  sf/200. 
21.  Pendulum. 

23.  s  =  7-7  =  i  fir  X  (32/256)^;    985-5  cm./sec.^ 

24.  30  ft./sec.2  instead  of  32-2. 

27.  a  =  981  X  (90  -  88)/(90  +  88)  =  11-02;    49-6  cm. 

28.  a  =  fif/ll,  v^  =  200  gr/ll;  v  =  4/3  m./sec.  Or,  thus,  energy  of  fall 
1  m.  =  (600  —  500)  X  100  g  ergs  =  kinetic  energy  0-5  X  1100  X  v^. 


CHAPTER  IV,  p.  39. 

3.  Steady  diminution,  straight  line  slanting  down.  Drawing  difficult, 
calculate  by  reversing  motion,  accelerating  from  rest  22-5  =  ^  at^  and  122-5  = 
^  a  {t  +  10)^  solve  for  o  and  t  {t  =  7-5  sees.,  a  =  —  0-8  yd. /sec),  etc. 

5.  From  food-reserves  of  body,  to  be  spent  in  acquiring  gravitational 
potential  energy,  and  in  friction.     Gets  hot  for  physiological  reasons. 

7.  Maintenance  against  friction;  stored  as  potential;  ditto  and  frictional 
losses ;   fluid  friction ;   ditto  and  lifted  higher. 

8.  Forces  away  atmospheric  pressure  outside  to  make  way  for  extruded 
air;  fluid  friction,  and  rise  to  higher  level;  friction  in  vessels,  §  333. 

9.  m  12-5,  V  40. 

10.  V  =  500,  mv  25,000. 

11.  500  X  50/10,000;   50/10,000. 

12.  V  =  (50  X  110  +  20  X  65)/(50  -f  20)  =  97;  330,000  ergs.  Work 
spent  in  crushing. 

14.  v^  only  a  quarter.     /.  only  quarter  rise;    3/4  lost  of  500  X  100  X  g  ergs. 

15.  mgs  =  imv2;  jqO  X  981  X  20  =  50  v^;  travels  to  height  20  =  40 
along  plane  expending  its  original  energy,  if  friction  negligible. 

16.  Energy  40  X  40  X  0-5  kgm.-metres,  1/4  lost  leaves  600,  which  is 
absorbed  in  25  m.  by  retarding  force  600/25  =  24  kg.    .*.  coeff.  24/40. 

17.  0-5  kgm.-metre,  (&)  add  ■v/3/2  pressure  on  plane  (§  79)  X  0-5. 

19.  i  X  5000  X  40,0002  ergs  ( =  400,000  joules) ;  then  4-  120  =  33,000 
megadynes. 

20.  140  X  40  =  5600ft.-lb.;  5600/33,000  h.p. ;  half  as  much. 

21.  Climber  helps  engine  to  drive  escalator;  other  wastes  it. 

22.  36  X  300  X  8  X  22/7  ft.-lb.  -^  33,000  =  8-2  h.p. 

24.  (15  X  112  X  44ft./sec.  X  1/20  +  15  X  44)  -^  33,000/60  =  8  h.p. 


SOLUTIONS 


813 


m^nteltW]''  ""  '^  ^^*"^  +  15  X  600  travelled  X  24  lb.)  -f  (20  x  33.000) 

48lb.  wt^  ""  ^^^  ""  ^^  ^*'^'^*''  X  1/22  +  R  X  44)  =  14  X  550  ft..lb./,ec.   R  - 

27.  Consider  average  velocities. 

28.  1440  X  7  =  10,080   min.   in   a   week.     h.p.  =  (2  X  20  y  1/ft  v  o/^w 
10,080  =  1/3000  h.p.  ^        I-  X  .ju  X  1/8  x  2/3)/ 

AEffiScy  0  24.  ^'^^  X  4  X  112  X  3/4  ft.)  =  62.450  ft.-Tb!/<liJ 


CHAPTER   V,  p.  50. 

1.  Fig.  12,  A,  H,  N. 

2.  Effectively  12-5  lb.  one  end  and  12-5  -  10  lb.  other,  hence  eg.  6  in.  from 

broken  end,  15  lb.  lift.  **  ^^ 


Fio.  419. 

3.  Fig.  419,  3.  Depends  on  how  held  deflected  :  if  in  fingrr  and  thumb, 
without  stress  on  cord,  mg  down  ;  if  by  horizontal  pull  mg  down  and  i  m§  ia 
cord,  draw  resultant  (which  is  equal  and  oppoaite  to  pull). 


814  SOLUTIONS 

4.  Fig.  419,  4.  Weight  W  vertical  through  centre  of  rod  meets  horizontal 
F,  therefore  also  reaction  R  in  same  point ;  complete  parallelogram. 

5.  See  Fig.  13,  S. 

6.  Only  solution  go  up  and  try. 

8.  Each  pier  carries  half  weight  of  bridge,  plus  its  share  of  load  inversely 
as  distance. 

9.  Lift  by  hook,  and  spring  balance,  at  any  two  distant  points;  inter- 
change exactly,  add  readings.     Prove  it  for  yourself. 

10.  Fig.  419,  10.     31b.;  2. 

11.  Fig.  419,  11.     12  lb.  X  5  =  40  at  centre  x  1-5.     Tangent  of  slope  1/3. 

12.  Fig.  419,  12.     2-5  ft. 

13.  Fig.  11,  §74. 

14.  Fig.  419,  14.  Taking  moments  about  iron  end  15-2  x  1  in.  -f  25-2  X 
3-5  +  10-4  X  7  =  50-8  total  wt.  X  3-46  in.  from  iron  end. 

15.  TKY  THIS,  and  puzzle  it  out. 

16.  Fig.  Taking  moments  about  hole  in  wall  (a)  15  x  1-5  =  T  X  2,  (6) 
15  X  1-5  =  T  X  V2. 

17.  Fig.  Taking  moments  about  elbow  1000  at  centre  X  20  +  1000  X 
30  =  T  X  5/^2  =  14,140  gm.  pull.  Draw  parallelogram  on  line  of  pull  and 
vertical  of  resultant  weight,  reaction  joins  elbow  to  their  common  point; 
scale  off. 

18.  Fig.     Cf.  Fig.  13,  P;    until  base /height  of  force  triangle  =  0-5. 

19.  Fig.  Same  2  :  1  triangle  occurs  at  45°  slant,  at  which  it  slips.  Half- 
way up,  diagram  same. 

20.  Fig.  On  ladder  AB  describe  circle,  draw  ZCN  vertical ;  this  is  weight, 
ZA,  ZB  are  supporting  reactions  equally  inclined  to  surfaces,  BN/ZN  =  coeff. 
=  1/(1  +  V^)-     Pretty,  but  too  clever  for  us. 

21.  Fig.  (a)  slant /height ;  (6)  base /height,  becomes  disadvantage  above 
45°. 

22.  Fig.  Draw  resultant  of  8  and  15,  anti-resultant  equal  and  opposite 
to  it,  construct  weight -reaction  parallelogram  on  it  as  diagonal. 

23.  Fig.     Boy  supplies  P  to  cart,  his  (spiked)  shoes  must  supply  2P. 

24.  Fig.     Simple  2/1  lever.     Must  lean  out,  then  climb. 

25.  Fig.  shows  forces  exerted  by  clip  on  stage,  forming  force  triangle. 
Half  thickness  of  stage/length  of  spring  must  not  exceed  coeff.  of  friction  in 
hole. 

26.  54/100  of  60  million  H-  981. 

27.  Efficiency  only  5/8  of  what  it  should  be,  .'.  8/5  of  150  kgm. -metres. 

28.  40/100  of  56  X  277  X  20  -^  1/4. 

29.  2  X  20  X  277  X  18/2  ^  1/2  (  x  efficiency). 


CHAPTER   VI,  p.  66. 

1.  Centrifugal  force,  felt  in  body,  or  shown  by  pendulum  swinging  out- 
wards, detects  curve  unless  banked  exactly  for  speed.  See  gyroscope,  §  92. 
No  detecting  uniform  straight-line  motion  (except  rail-end  taps). 

2.  mv^jr  =  mg;  v^  =  100  X  981;   313  cm. /sec.  overhead. 

3.  (2  X  22/7  X  6400  X  10^/86,400)^6400  X  10^  =  3-4  dynes  per  gm.  wt. 

4.  754  cm. /sec.  =  535  vertical  and  horizontal,  which  it  loses  in  535/981  = 
0-545  sec.     .*.  flies  1-09  sec.  to  same  level  as  let  go  =  5-87  m. 


SOLUTIONS  g|5 

SQnsT  ^U""^  1^^^  ^fl^^if  ft./sec  =  89  vertical  and  horizonUl;  lort  in 
89/32-2  =  2-75  sec.  . .  fly  horizontally  2  x  2-75  X  89  =  490  ft. 

6.  0-012  v^  =  2,000,000 ;    12,900  cm./sec.,  see  §  394. 

7  rnv^ir  =  ^. ;  ^  ggG  cm./sec  =  14  m.p.h.  Horizontally.  .mc«  only 
halt  the  pressure  has  to  prevent  skidding  down,  must  go  1-41  time*  fa«t«r 
20  m.p.h.  * 


CHAPTER   VII,  p.  86. 

2.  101  Ib./sq.  in.  =  total  force/area  of  valve. 

4.  179  gm./sq.  cm. 

5.  76  X  13-6  =  1033;    x  981  =  1,013,000. 

6.  13-6/000125  =  10,780  cm. 

7.  80  x  6  X  (62-3  X  6/2)  =  89,600  lb.;  44,8001b.;  80  X  40  X  6  X  62-S- 
1,196,000  lb. 

8.  10  m. 

9.  68  X  13-6  X  981  =  906,000  dynes;    (76  -  68)  x  13-6  «  100.000  cm.  x 
0-00109. 

10.  12  X  0-8  gm./cm.2  x  981  dynes/cm.« 

1 1 .  Inrush  of  air  bubbles  partly  obstructs  tail  end  of  tube. 

13.  200  X  10  X  50/33,000  =  3  h.p. 

14.  400  X  2  X  102  X  10  X  5/3  =  13,600  kgm.-metres  per  minute. 

15.  396,000  gals./min. 

16.  60  X  2  c.c.  raised  on  the  average  60/2  cm.,  each  13-6  gm.  =  49.000 
gm.-cm. 

17.  70  X  150/1-05    c.c.  X  24  X  13-6    gm.-cm./min.  x   981/60    ergs    eee. 
-=r  10'  =  6-33  watts. 

18.  Continuity  and  reduction  of  shock. 
21.  2000  X  000125  =  13-6  X  0186  cm. 

23.  Increases  total  contents  of  instrument;   when  zero  set,  none. 

24.  100  +  0-25  X  76  X  13-6  gm.  atmosphere  pressing  on  top  of  tube  aftd 
nothing  pressing  back  over  inner  area. 

25.  Speed  of  fall  from  height  h  ;   slowed  by  viscosity. 


CHAPTER   Vin,  p.  96. 

1.  1/1  +  1/0-89  =  2-25  c.c.  less  2-26  X  0-02  =  2-205  c.c.  weigh  2  gm.  » 
0-9075  s.g. 

12.  6/3  =  2;    and  1/1  +  1/2+1/3  c.c.  =  1-833  c.c.  weigh  3  gm.     /.  s^p 
1-64. 
4.  Measure  size  of  bacillus  with  micrometer  under  hi^h  power,  stir  theinas 
a  cloud  m  a  solution  of  a  neutral  substance,  and  alter  its  strength  until  Ihey 
refuse  to  centrifuge  out  :   measure  its  s.g. 
-    5.  (1290—90)    kg.  —  150.     Pressure   slightly  exceeds  stmosphero   «>   *^ 
whole,  thus  is  less  compressible  than  atmosphere,  has  compsrslively  lUed 
over-all  density,  and  floats  stably  in  air  of  that  density. 
6.  4,r/3  X  8  X  1-2  kg.  -  3  =  37  kg.     As  (14-4  -  2)  to  (14-4  -  I). 

7.  As  76  to  76  —  16-6,  neglectmg  wt.  of  balloon  fabric. 

8.  Of  each  c.c.  l/5th  is  glass  of  mass  0-5  gm.     .*.  floaU  hiOf  immersed,  and 
will  be  sunk  by  half  its  vol.  of  water  =  6/4  of  half  its  length. 


I 


»16  SOLUTIONS 

9.  Look  at  one. 

10.  Vol.  X  1-200  =  30  gm.  wt.  of  hydrometer  =  1-000  X  vol.  to  1-000 
which  is  .*.  l/5th  greater,  demanding  6  gm.  to  sink  in  1-2  liquid. 

11.  Vol.  X  1-3  =  (vol.  +  9  cm.)  X  1  =  (vol.  +  x  cm.)  X  1-1  =  (vol.  + 
y  cm.)  X  1-2.     Hence  x  2-5,  y  5-45;    draw  figure. 

12.  B  X  1-4  =  (B  +  5  in.)  X  1-2;    30  in. 

14.  h  X  13-5  =  20  X  1-1  -  10  X  0-73;  h  =  108  cm.;  1-08  X  13-6  = 
0-73  X  20. 

15.  Cf.  4.  If  air  and  gas  were  at  same  pressure  below,  as  in  a  balloon  with 
open  neck,  the  air,  forming  a  denser  column,  falls  off  faster  upwards  in  weight, 
i.e.  in  pressure. 

16.  14-86/(14-86  -  8-67)  =  2-40;    (14-86  -  9-85)/(14-86  -  8-67)  =  0-81. 

17.  80  gm.  transferred. 

18.  2  X  0-5  X  62-5  -  2/4  X  0-8  X  62-5  =  37-5  lb.  wt. 

19.  4x1/3x1/2  =  2/3  cu.  ft.  42  lb.  /.  1  cu.  ft.  63,  sinks  in  62-5,  floats 
in  62-5  X  1-03  by  1-3  lb.  per  cu.  ft.,  i.e.  by  about  l/50th  its  volume. 

20.  1  gm.  ice  =  1-0-917  =  1-090  c.c.  displaces  1/1-025  =  0*977  c.c.  of  sea 
water;  diff.  0-113  c.c./l-090  =  0-1037  its  volimie.  Often  more,  a/c  contained 
air  bubbles. 

21.  Nil  in  fresh,  3  to  4  lb.  in  sea  water. 

22.  (v  -  3)  X  5/6  =  V  X  3/4;    30  cu.  in. 

23.  42-5/5  =  s.g.  8-5.      /.  8-5/13-6  =  5/8ths. 

24.  15  -  15/0-6  +  57  -  57/11-4  =  42  gm. 

25.  1/2  oz.  or  5/6  oz. 

26.  5/6;    8  c.c.  X  5/6  +  1/8  c.c.  X  G  =  8-125  X  1;   G  =  11-66. 

27.  1-15  X  2t;/3  +  0-9  X  'y/3  are  the  masses  displaced  =  v  X  1-067. 

28.  Explore  the  fallacy. 

29.  Each  adds  its  volume  x  (its  density  less  that  of  petrol).  Floating 
blocks  would  add  nothing  at  all. 

30.  Nothing.  Sinks  1/2600  its  volume,  buoyancy  of  air  on  projecting  part 
being  removed.  The  air  presses  on  the  float  and  on  the  liquid  surface,  and 
heavier  on  the  lower,  by  the  weight  of  that  thickness  of  air. 


CHAPTER   IX,  p.   108. 

1.  (5000  X  981  dynes/ir  X  0-075^)  ^  2-5/3500  =  0-39  X  10^2  dynes/cm.^ 

2.  10  X  5  gm.-cm.  =  50  X  981  ergs. 

3.  i  X  20  X  14in.-lb.;    20  X  7/12  X  45  ft.-lb./33,000  =  0-016  h.p. 

4.  15  X  8  =  (15  +  4/12)  compressed  length  :   0-17  in.  below  top. 

5.  Let  mercury  enter  I  cm. ;  v  =  100  —  I;  p  =  75  +  50  —  Z;  pv  —  100  X 
75  originally.  Solving  the  quadratic  I  =  25  cm.,  vol.  being  75  and  pressure 
100. 

6.  1/6  cu.  ft. 

7.  Called  the  amphisboena  expt.,  after  that  fabled  two-headed  snake. 
Open  end  up,  volume  I  of  air  is  under  barometric  pressure  +  2  ft.  of  oil ;  open 
end  down  the  2  ft.  of  oil '  hangs  '  on  the  expanded  air.  24  in.  X  (H  +  2  ft.)  = 
26-5  X  (H  -  2).     .-.  H  =  40-4  ft. 

9.  1000  X  0-72  X  22/7  =  1540  gm.  No  barometric  height  being  given, 
assume  water  barometer  10  m.  high;  flask  half  fills. 

10.  76  X  3/2.     Depth  =  0-5  X  76  X  13-6/1-025  =  5  m. 


Solutions  gi7 

11.  For  graduation  for  this  tube  see  Fig.  37.  p  60/16  atmo«  3diMito«AW> 
Sounding  3  x  0-75  X  13-6/1.03  =  29-7  m.  /'°«^«n«»..  J  due  to  w*»(»r. 

12.  20  X  (H  +  9-5)  =  12  X  (H  +  65-5);   H  =  74-6  cm. 

13.  76/3  cm.,  air  making  up  the  2/3. 

14.  (25  +  0-5  in.  faU)  x  fall  =  0-1  x  30  in.;    1  in. 

15.  28-15  in. 

16.  Ship  is  flooded  to  raise  air  pressure  to  sea  pressure  (or  an  air-lock  worked 
by  compressed  air),  then  hatches  can  be  opened ;  speed  increaaee  by  expaiunon 
of  air  in  bag  with  reduction  of  depth  pressure. 

17.  See  Chap.  VIII,  No.  5,  balloon  more  resistant  to  compreMion  than 
surroundmg  air  :  thm  hull  of  submarine  more  compressible  with  depth  than 
sea-water,  crushed  if  too  deep. 

18.  (10/11),  (10/11)2,  (10/11)». 

20.  McLeod  gauge.     6  x  22/7  x  0-05«  c.c.  X2  =  30xp;p  =  0-026 
Hg. 


CHAPTER  X,  p.  121. 

2.  453  gm. ;    452/453. 

4.  Interchange. 

5.  Fig.  59 ;  100  gm.  X  distance  of  e.g.  to  left  of  fulcrum  =  10  X  0-05,  then 
distance  below  knife  edge  must  be  50  times  this  =  0-25  cm. 

7.  801-085. 

8.  M  =  25  +  25/800  X  2-5  -  25/7000  =  25-009. 

9.  Bulky  sphere  (a)  sinks,  (6)  rises. 

10.  Bulb  side,  for  it  is  now  buoyed  up  by  denser  air.  More  prcMcd  ck>wn 
on  top,  of  course,  but  even  more  pressed  up  from  heloxv  :  Everybody  forgot  iXis 
in  tvx)  exams.  Find  voliune  of  bulb  by  immersing  in  greuluated  jar  of  water, 
and  add  weight  of  this  volume  of  air  to  measured  weight. 

11.  A  micro -balance,  a  little  contrivance  of  fused  sUica,  slung  on  a  thread 
of  it,  having  a  1-c.c.  bulb  counterpoise,  usually  actuated  by  varying  the  air 
pressure  (and  thereby  density)  in  the  case  containing  it. 


CHAPTER   XI,  p.  133. 

8.  00000085  X  3  X  (40  -  4)  X  1000  =  0-92  c.c;   8000  X  100  X  0-00001 1 
X  3  X  60°  X  6/9  =  880  cu.  ft. 

9.  1000  X  0-000011  X  80"  X  5/9,  nearly  half  a  yard. 

10.  (a)  5/9  as  much,  (6)  not  at  aU.     10  X  (0000022  -  0-00001 1)  X  (I  - 
15°)  =  0-03,  t  =  288°. 

11.  Shrinkage   0000012  X  300°   per  cm.  =  0-0036   cm.     .-.    force  -  t  X 
10^=*  X  00036  dynes  X  10  sq.  cm.  =  about  72  tons.     Length  immat«rial. 

12.  Period  proportional  to  sq.  root  of  length,  11-5  sec. 

13.  (L  +  42)  X  0000011  =  L  X  0000028,  L  =  27-2  in. 

14.  Rod  expansion  =  that  of  lower  half  of  bob,   to  keep  e.g.  Hxed;    bob 
2/9ths  rod. 

15.  29-5  X  13°  (0-00018  -  000001)  =  0-065  in.  lew. 

16.  0-8/(24-25  X  40)  =  0000825. 

17.  760  X  15°  (000018  -  0-00002)  =  1-80  mm.  leM. 


818  SOLUTION^ 

19.  1-36/7500  =  0-000182. 

20.  It  is  —  the  coefficient  of  increase  of  density. 

21.  0-920  X  (0-000525  —  0-000025)  X  30°  heavier  =  0-934  as  near  as  cah 
be  read. 

22.  Not  asked  about  ice. 

23.  Glass  shifts  max.  density  temp,  to  6°,  where  water  and  glass  expand 
equally. 


CHAPTER   XII,  p.  148. 

6.  27/8  X  3/2  =  81/16. 

7.  B.  pt.  error  (2  —  1/2-7)  =  1-61°;   then  graph  it,  74-05°. 

8.  1000  X  0-000182  X  10  =  1-82  mm. 

9.  By  conversion  formula,  0-2°  F.  high;   other  0-1°  C.  low. 

10.  122,  -  40  and  -  459-4°  F. ;    -  17-8  and  37°  C. 

11.  22/7  X  0-00252  X  5  =  10  X  5/9  X  0-00015  B,  /.  B  =  0-118  c.c. ;  35 
to  40-6°  C. ;    37°  C. 

12.  1/300  X  9  X  10-5  =  o  X  1/4;    a  =  1-2  X  10-«  sq.  in. 
14,  15,  17.  §  199,  also  §§  778,  799. 

18.  §  147,  etc. 

19.  PV  =  RT,  34  X  1000  =  R  X  290,  54  x  V  =  R  X  280;   610  c.c. 

21.  29/38  =  (273  +  17)/(273  +  b.  pt.) ;    107°  C. 

22.  Normal  Temperature  and  Pressure,  0°  C.  and  760  mm.  Hg.  273  X  22/14 
=  429°  A. 

23.  (8-2  -  3-4)/(7-3  -  3-4)  X  100°  =  123°  C. 

25.  (61-1  -  32-4)/(13-7  -  32-4)  =  100° /«°;    -  65°  C. 

26.  Calculate  the  density,  then  fill  it  into  the  given  volume.  PV  =  P/D  = 
RT  gives  760/1-29  =  R  x  273and750/D  =  R  x  294orD  =  1-29  X  75/76  X 
273/294  =  1-182  gm. 

27.  0-00129  decreases  to  0-00129  X  273/546  X  1-05,  and  the  difference  X 
5000  is  pressure  reduction  in  gm./cm.^  =  3  '  cm.  of  water.' 

28.  As  26,  calculate  mass  under  both  conditions,  dj^  =  75/76  X  273/286  X 
1-293;  c?2  =  72/76  X  1-293,  difference  0-0077  X  10  X  8  X  3  X  1000  =  1850 
gm.  enter  the  room. 


CHAPTER   XIII,  p.  156. 

6.  Heat  given  up    50  X  (60  -  32)°  =  (50  +  c)  x  22°;     c  =  13-8   gm.   of 
water. 

7.  100  X  0-1  X  (100  -  22-4)  =  (105  X  0-1  +  300)  X  2-4  +  lost  21  cal. 

8.  19   oz.  X  (100  -  t)  =20  X  0-2  X  4/5  X  {t  -  15°);     t  =  88°. 

19   oz.  X  (100  -  0  =  15  X  0-06  silver  X  {t  -  15°);   t  =  96°. 

9.  82-8  X  s  X  (100  -  21-4)  =  (71-2  +  28  X  0-22)  X  (21-4  -  12-6);       s  = 
0-105. 

10.  200  X  0-032  X  79  =  132  X  5  X  6;   s  =  0-64. 

11.  80  X  0-032  X  («  -  21)  =  300  X  6;  t  =  725°. 

12.  2  X  (50  X  20  X  25)  X  1-3  kg.  mass  per  hour  X  0-24  X  10°  Cal.  per 
hour  X  24  hr.  -^  6000  Cal.  =  5/8  ton  of  coal. 


SOLUTIONS  gl9 

CHAPTER  XIV,  p.  161. 

10.  79-5. 

11.  80  X  a  X  10°  =  5  X  80cal. 

12.  940,000  tons. 

13.  4  X  0-5  X  («  +  20)  =  3  X  0-67  x  (17  -  «);   <  =  -  \'5\ 

14.  (0-2  X  0-918)  X  80  X  10,000  =  147,000  cals./hr. 

15.  280  X  0-095  X  (100  -  8)  =  (40  x  0-095  +  whole  120  gm.)  x  8  +  wt. 
of  ice  X  80;  wt.  of  ice  18-1  gm. 

16.  2000  X  (0-5  X  40  +  80  +  100)  =  400,000  cal.  required.  Alcohol 
yields  150  X  6000  =  900,000,  efficiency  4/9th8. 

17.  5  X  0095  X  100/80  =  0-595  gm.  ice  melting  contract  11-8  x  22/7  X 
00382  =  0-0535  c.c,  .'.  1  gm.  would  contract  0-090  c.c;  occupied  1-09  c.c. 
.-.  density  0-92. 

18.  80/95  kg. 

19.  6  X  1210  kg.  X  0-24  x  7°  -^  80  =  152  kg. 

20.  12  X  70  X  0-04  +  12  L  =  216  X  0-04  x  15;   L  =  8  cal./gm. 

21.  10  X  0-3  X  9°  given  out  by  warmth  of  salt  +  110-5  x  AV  by  water 
and  calorimeter  =  480  cal.  =  10  L. 

22.  500  X  2  X  10  X  (24  X  7  X  8)  X  1-29  X  10«  tons  of  air  X  0-24  x 
4°  ^  80  =  0-21  billion  tons. 

24.  Keep  away  from  water,  with  which  it  evolves  heat. 

25.  (3-38  X  (L  +  32-3))  =  107-2  x  (32-3  +  0-3  -  14-5)°;   L  =  543. 

26.  (15  X  12  X  10)  X  0-08  X  20  X  0-24  -^  590  =  1-18  lb. 

27.  2420  and  5350  tons. 

28.  100  X  80  -h  540,  and  this  H-  (80  melting  +  20  warming)  gm.  ic©. 

29.  1000  lb.  X  (212  -  60)  X  10/9  ^  100,000  tenpences  =  16-9  penc«,  add 
1000  X  (540  X  9/5)  X  10/9  -^  100,000,  a  further  9  shillings. 

30.  1000  X  «  X  85°  =  19  X  540;   s  =  0-121. 

31.  100  8  X  61-5°  -^  12  cal.  supplied  per  min.,  which  x  by  17-5  —  4400; 
8  =  0-515. 

32.  W  X  (50  X  0-3  +  540  +  20)  =  1  ton  x  (80  X  0-5  +  25);  W  =  01 13 
ton. 

33.  W  X  (540  +  100)  =  8  X  (10  X  0-5  +  80);   W  =  1-06  lb. 

34.  200  X  (80  +  20)  =  32-4  X  (L  +  100  -  20);   L  =  638. 

35.  10  X  (540  +  90)  =  150  X  10  +  W  X  80;   W  =  60  gm. 

36.  0-5  X  0-11  X  (14  +  196)  =  162  X  74/76  x  273/287  X  <H)OI2«Lj 
L  =  61-7,  the  nickel  having  nothing  to  do  with  warming  the  gM  altar 
vaporization. 


CHAPTER   XV,  p.  174. 

6.  As  100  to  55  X  0-8  (note,  sp.  ht.  given),  =  2-27. 

6.  200  gm.  turps  lost  196,  230  gm.  water  lost  435,  /.  sp.  hU.  -  196/43*  X 
230/200  =  0-54.  .  . 

7.  Good  vacuum,  bright  silvered  inside  to  still  further  r«duce  radtatioo, 
no  thick  part  to  crack  under  hot  or  cold  liquid. 

9.  d/(l  4-  at)  -dl(l  +  aT)  gm.-wt.     Viscosity. 
n.  vi8  =  34:53. 


820  SOLUTIONS 

13.  0-004  X  10,000  X  86,400  X  1°  -f  3,200  =  1080  cal. 

14.  0-00015  X  (6  X  25  X  10,000)  X  86,400  sec.  X  30°  -M5  =  38,800,000  cal. 

15.  388  cal. 

16.  0-416. 

17.  10  X  (<  -  16)  =  0-001  X  50  X  22/7  X  (100  -  average  {t  +  16)/2)  4- 
0-05;  t  =  38-9°. 

18.  H  =  0-005  X  5/4  per  sq.  cm.  per  sec.  Then  H/80  gm.  freeze,  and 
occupy  thickness  H/(80  X  0-918)  cm.  per  sec. ;    1  cm.  in  3-25  hr. 

19.  0-5  W  X  3000  =  0-003  X  200  X  10,000  X  86,400  x  5°  -f-  24;   69-4  kg. 

20.  6-3  million  cal.,  3  times  the  human  output.  The  fallacy  is,  that  we 
don't  wear  3  mm.  of  wool,  but  3  mm.  of  air  entangled  in  our  clothing;  the 
conductivity  of  air  brings  the  answer  about  right. 

21.  0-0025  X  3600  X  10  -f-  0-3  =  300  cal./cm.^ 

22.  The  emissivity  difficulty;  see  23. 
24.  25,  think  them  out. 

26.  W  X  5  =  0-9  X  6  X  60/20;   W  =  3-24  gm./sec. 

27.  H  =  0-9  X  10,000  X  60  X  3°/0-4  =  8000  L ;  L  =  507,  but  this  is  a 
hopeless  attempt  to  allow  for  the  small  emissivity ;  the  next  question  repre- 
sents actual  steam  practice. 

28.  500  W  =  0-00025  x  5000  sq.  cm.  X  3600  sec.  X  14075;  W  =  252  gm. 


CHAPTER   XVI,  p.  180. 

3.  1700  X  1,013,000  ergs /540  X  42,000,000  =  0-076  of  latent  heat. 

4.  Soil  heated  by  forced  displacement;    rope  and  hands;    in  muscles,  etc. 
(but  surprisingly  little) . 

6.  40°  F.;    40-1°  F.,  the  heat  is  developed  in  the  friction  of  the  turbulent 
eddies. 

6.  About  1/4°  C.     In  all  questions  be  careful  of  the  O's;     examination 
answers  to  this  varied  from  1/4200°  to  233,600°  C. 

7.  5830  cal. 

8.  6000  X  (100  X  2)  X  30  X  981  ergs  ^  42,000,000  cal./min.     Then  half 
this  heats  500  gm.  iron,  sp.  ht.  0-12;   by  5-83°  per  min. 

10.  30,000  X  981  ergs  per  gm.  H-  42  million,  cal.  -r  0-03  sp.  ht.  and  by 
2,  =  11-7°. 

11.  275  X  1680   X   981   =  J    X   (275   X   003  +  3-3)    X  0-85;      J  =  43-5 
million. 

12.  H  X  981  -f-  42  million  =  80  X  1/200;    172  metres. 

13.  i  m  X  16002  ergs/gm.  -r  42  milhon  =  80  m  x  1 /3000th. 

15.  imv2  =  ^  X  (285  x  0-03  +  5-4)  x  42  million;  342  m. /sec. 

16.  (10  X  746/4-2  X  3600)  -^-  2  =  (150,000  X  0-4)^;    t  =  53-2°. 

17.  7-5  km. /sec. ;  most  are  3  to  6  times  as  fast. 

18.  39-3°. 

19.  1/20°  C. 

20.  Perec.  (200,000  X  981  ergs  -=-  42  million)  -f  1-03  X  0-97  =  4-66°.    This 
is  the  '  homogenizer,'  used  to  hinder  the  cream  separating  out. 

21.  15  foot  tons. 

22.  0003,  contractor  quite  content,  0-000055. 

23.  0-00013  X  (1200  X  10,000)  X  1  sec.  x  22°  ^  15  calories  X  (4-2  X  3  4- 
46)h.p.  =  38-6  h.p. 


SOLUTIONS  821 

CHAPTER   XVII,  p.  191. 

3.  Height  of  flat  gives  f.pt.;   from  its  length  the  output  of  heat  deUvinc 
cooling  is  deducible.  ^^ 

4.  Successive  flats  on  cooling  curve.     Solid  sinks  or  swims. 

CHAPTER   XVIII,  p.  207. 

2.  Equal ;  less  inside  because  heated. 

3.  Conductivity;     volatility,    and    density    of    vapour;     vapour-oontont. 
expansibility. 

4.  The  36  cm.  increase  doubles  the  partial  pressure  on  the  air,  that  of  the 
ether  is  therefore  40. 

5.  Air  partial  pressure  (76  —  70)  —  1-5  is  doubled  by  extra  4-5  cm.,  which 
would  lower  mercury  to  65-5  cm. 

7.  In  Fig.  82  V  and  R  are  saturated,  their  mixture,  anywhere  along  the 
straight  line  VR,  is  in  the  supersaturation  space,  and  must  depodt  miat. 

8.  See  §  303. 

10.  Vapour  pressure  above  it  less,  by  weight  of  colunm  between  them. 

11.  50    c.c.   X   2/3    hydrogen  x  (74  -  1-44  -  10/13-6)/76  X  273/290  x 
1/11,160  =  00265  gm. 

12.  (Air  +  evaporating   ether  pressure)   exceeds   (air  prenore  +  stopper 
weight) ;  continues  until  about  half  air  expelled. 

15.  Steady  temperature,  needing  no  thermometer,  but  affected  by  baro- 
metric  change ;   only  a  few  temperatures  available,  however. 

16.  Fig.  83.     76  cm.  +  barometer  X  373/273. 

17.  Pressure  as  16;   mass  1-29  gm.  air  -}-  5/8  X  273/373  aa  much  vapour; 
less  0-1%  and  trifles. 

18.  Pressure  too  high  by  about  1600/2000  atmos. ;  now  all  acting  as  a  dry 
gas. 

19.  Never,  imtil  the  bulb  bursts. 


CHAPTER  XX,  p.  229. 

1.  Try  it. 

2.  Warm  damp,  following  cold. 
4.  In  open,  causes  pleasant  cooling  by  evaporation;   in  doeed  „___ 

but  saturates  space,  so  hindering  evaporation  of  your  own  perspiration, 
increasing  the  '  closeness.' 

6.  Fig.  82.     12/15-5  =  78%. 

7.  do.  doubled. 

8.  do.  5°;   2/6-5. 

9.  Upside  down,  with  your  hand  round  the  covered  bulb. 

10.  Mass  =  (sat.  press,  at  dew  pt./760)  X  5/8,  which  in  the  relative  dmadty 
of  water  vapour  and  air,  X  1-29  kg.  X  273 /absolute  t«mp.  of  air. 

11.  15-46/760  X  273/291  X  1000  X  18  mol.  wt.  of  H,0/22-8  -  15-4  gm. 

12.  9/760  X  273/291  X  60  cm.  X  1000  X  18/22-3  =  M8gm. 


822  SOLUTIONS 


CHAPTER   XXIII,  p.  269. 

3.  (75/0-025)  -f  981  =  3-05  cm. 

4.  Ignoring  angle,  3-1  cm.  :  an  actual  angle  of  contact  26°  would  reduce 
this  by  10%. 

5.  (2  X  75/0-05)  -f  981  =  3-05  and  235  cm.;  difidculty  in  entering,  depres- 
sion. 

6.  5/(2  X  0-8)  =  3-13. 

7.  Loop  pulls  out  into  ring  round  hole ;   collapse. 

8.  4T/r  =  100/2  =  50  dynes /sq.  mm. 

9.  DownpuU  77  X  (3  +  2-9)  X  30/981  =  hydrostatic  upthrust  (tt  X  2-95  X 
0-05)  X  1-05  X  depth  =  1-08  cm. 

10.  Weight  (77  X  2-95  x  0-05  X  15  x  77/4  X  2-952)  X  2-5  +  77  X  3  X  30/981  = 
77/4  X  32  X  1-05  X  h,  the  weight  of  liquid  displaced;  h  =  4-62  cm.  This  is 
closed  end  down;  inverted  liquid  enters  and  pulls  down  on  inside  surface 
also  0-28  gm.,  but  slight  compression  of  air  inside  as  tube  submerges  com- 
plicates calculation. 

12.  Drop  hangs  by  a  ring  wall  of  surface,  falls  off  when  weight  exceeds  lift 
of  this.     (W  —  w)  X  g  =  2a  X  T. 

13.  Completely  immersed  loses  0-6  gm.  .*,  vol.  0-6  c.c.  .'.  thickness  0-6  -f- 
77  X  22  =  0-048  cm.  also  density  =  2-5.  Half-immersed  displaces  0-3  c.c. 
/.  should  weigh  1-20  gm.  .'.  0-55  gm.  is  pull  of  s.t.  on  a  width  (2x4  +  2  x 
0-05)  =  8-1  cm.      /.  surface  tension  0-55  X  981/8-1  =  67  dynes /cm. 

14.  See  §  139.  Emergent  volimies  tt  X  1-25^  x  1-24  =  6-1  c.c,  and 
77  X  0-312  X  21-8  =  6-6  c.c,  difference  0-5  c.c  X  0-95=  0-475  gm.  X  981  = 
465  dynes,  which  is  the  difference  of  surface  tension  pull  on  circumference 
277  X  1-25  and  277  X  0-31.      .'.  T  =  79  dynes/cm. 

15.  A  X  T /radius  of  curvature. 

16.  Surface  477R2  becomes  1000  X  477(R/10)2,  an  increase  of  3677R2T  ergs 
of  surface  energy. 


CHAPTER   XXIV,  p.  287. 

3.  10/342  gm.-mol.  at  0°  and  760  mm.  occupies  22-3/34-2  =  0-653  litre, 
.'.  exerts  in  1  litre  a  pressure  0-653  atmo. 

4.  150/342  =  0-44  gm.-mol. /litre.     .*.  O.P.  =  0-44  X  22-3  =  9-8  atmos. 

5.  0-1  gm.-mol. /litre  =  2-23  X  303/273  =  2-47  atmos.  at  30°  C. 

8.  Vol.  of  solution  25/1-25  +  880/0-80  =  1120  c.c  contains  25/92  gm.-mol. 
.-.  O.P.  =  22-3  X  25/92  X  1000/1120  =  5-45  atmos. 

10.  A  5%  drop  of  vapour  pressure. 

11.  Fresh  water  dimLnishes  by  evaporation,  cooling  itself  and  air;  this 
then  deposits  a  little  moisture  in  salt  water,  slowly  increasing  its  bulk,  making 
it  a  little  warmer  than  fresh,  but  both  below  atmosphere.  The  other  way, 
both  diminish  by  evaporation,  there  is  a  similar  distribution  of  temperature. 

12.  (1-44  -  0-72)/76  X  4/22-3  X  18  x  273/290  =  2-88  gm. 

13.  2°  higher  b.pt.  means  about  l/14th  less  vapour  pressure,  or  1-34  mm. 
at  17°,  then  (1-34  x  0-72)/76  X  4/22-3  X  18  X  273/290  =  2-47  gm. 


CHAPTER   XXVII,  p.  335. 

Vq  usually  taken  as  330  m.  /s. 

10.  1090  X  V(293/273)  =  1090  X  (1  +  10/273)  =  1129. 

11.  1090  f./s.  X  (1  +  5/273):  pressure  no  effect ;   1110. 


SOLUTIONS  g23 

12.  340m./s.  atlS'';   14-70  sec. 

J!n  ifl'.Vr+%fl  Je  x^?3"?  arc'""  ^''"^'  "'  •»-  "— •  ^- 

14.  1300  m./s. 
57.5VpTf600,  5-57!^  =  «^0/600,then  1100/(1100  +  .)  =  p/600;  84-5  f./..  » 

17.  800  X  1150/1100=  836;    16/9;  how  you  tell  the  tmin  ia  commir 

18.  500  X  (1100  +  44)/(1100  -  44),  (a)  500  x  llOO/HIOO       aa\  i^'.u    >l. 
this  and  500  x  UOOIlU^A^ssun^ingii^SLZo^^^ 

CHAPTER   XXVIII,  p.  347. 

4.  292. 

5.  Pitch    ocy288/283  =1  +  25/283  =  1  +  00088    and    thi«   make,   a 
difference  of  either  1  or  7 ;   109  or  790.  »»"»*t7«   » 

QQn*   ^""^^ ^^ooT""^  1"^  '^•^■'  '■^'  ""^  ^^^quency,  =  3-3;    frequencies  333-3  and 
06O,  speed  333-3  m./sec. 

7.  40,  49,  etc. 

14.  V(21-8/l-3)  =  4-1. 

15.  n  =  V(</1-21)  -^  21  and  2n  =  V(T/90)  -^  2/;    1-86.  second  tenaor. 

16.  n  =  y/TId  ~  21  and  3n/2  =  V(T  +  10)/c/  H-  1-8  i;   12-2  lb. 

17.  Double  diameter. 

18.  480,  485. 

19.  572. 

20.  1-80  kg. 

21.  256. 

22.  Try  it. 

23.  P  =  3-93  X  106,  ^  0-0153,  n  80. 

24.  2  X  4000  cm. /sec.  =  ^(P/SO  gm./cm.);    1920  megadynee,  practicaUy 
2  tons. 

25.  9  tons. 

CHAPTER   XXIX,  p.  361. 

1.  Timbre  =  Quality. 

2.  Louder,  shorter.  (1100  -^  4)  -^  256  ft.  long,  stopped  end;  suite  770  fork. 

4.  ditto. 

5.  (336  m./s.  -^  4)  -^  448  =  18-75  cm.  and  any  odd  no.  of  multiples.     Move 
5/283rds  lower  down. 

6.  53  X  2  X  348  =  36,900  cm. /sec. 

7.  33,000  -^  (4  X  300)  =  27-5  —  1-2  and  every  55  cm.  beyond. 

8.  (a)  flattens  1/9000  per  °,  (6)  rises  0-5/273,  (c)  flattens  by  expanaion  of 
wire  more  than  wooden  frame,  less  effect  with  iron  frame. 

9.  34,000  -^  4  X  f  X  26  =  490. 

12.  33,000  -^  (4  X  (81  +  12))  =  100. 

14.  Semi-w.l.  =  1120—  (2  x  530)  =  1-06   ft.     Antinodes   in   middle  and 
006  ft.  beyond  ends,  nodes  0-03  ft.  outside  quarter-lengths. 

16.  B,  gets  impulse  every  so  many  periods. 

17.  Responds  to  780. 

19.  (a)  blown  pipes,  (6)  dust  tube. 
23.  256  X  (1  +  2-5/273)  =  279-5. 
30.  V  =  V(E/8-5);  w.l.  =  2  x  172,  and  344,000  -  V.     E  =  101  X  10»«. 


824  SOLUTIONS 


CHAPTER  XXXI,  p.  385. 

2.  50/52  ^  2.     Sine  30°  =  0-5.     .*.  at  this  angle  to  wall. 

3.  The  well-known  3,  4,  5  triangle,  500  -^  (182  +  242)  then  X  18/30;    1/3 
cu.  ft. 

4.  15/a;2  =  30/(5  -  x)^;   a;  =  12-1  (Rumford)  or  2-1  ft. 
6.  30/502  =  40/702  +  20/^2 ;   72-5  cm.  on  40  lamp  side. 

6.  Ald^=  0-8B/4d2;   b  =  5  A. 

7.  A/402  =  B/502  and  then  A/(40  -  x)^  =  0-81  B/502;    4-4  in. 

8.  C/302  =  L/1202  and  c/252  =  (L  +  card)/1252;    9/16ths. 

9.  C/302  ^  0-75  c/(30  +  10  +  10)2  ^  candle  +  virtual     image       §  403  = 
c/d2;   26-7  cm. 


CHAPTER   XXXII,  p.  401. 

7.  9-3/8-1  =  1-15  then  X  4/3  =  1-525. 

8.  13-3  cm.  above  sm'face. 

9.  0-5x5  =  2-5°  and  0-65  X  7  =  4-55°;    705  and  2-05. 

10.  Spectrometer  and  refractometer  methods  incomparably  most  accurate. 
13.  1-414. 

16.  6  cm. 

17.  §  487. 

18.  No  metal  to  tarnish,  but  some  glass  susceptible  to  moisture.     Become 
massive  and  costly. 

19.  Fig.  191,  195  and  198. 

20.  TRY  IT. 


CHAPTER   XXXIII,  p.  416. 

6.  Put  a  =  f  -{-  X  and  b  =  f  -\-  y,  and  you  get  the  product  of  the  distances 
outside  the  foci,  xy  =  /2,  a  hyperbola. 

8.  Use  7,  the  oculists'  quick  test  method ;  also  moving  to  and  fro,  when  + 
magnifies  and  —  diminishes,     try  these  with  any  spectacles  you  can  find. 

15.  1-5  ft.  focus.     As  14,  40 /a  +  40/(72  -  a)  =  40/12;    15  in.;   3-8. 

16.  Minimum  distance  between  object  and  image  is  4/,  equal  size,  remember 
THIS,     a  =  4&,  o  +  6  =  64.     .*.  b  =  64/5;  /  =  10-25;   a  -  b  ^  38-4. 

18.  0-4  +  1/a  =  8-33,  a  =  12-6,  (6  X  9)  X  500/12-6,  say  40  times.  Quicker 
exposiu-e. 

19.  Calculate  as  18;   7-5  and  16-5  mm.  forward. 

20.  A  non-axial  beam ;  draw  through  the  mid  point  of  convex  face  (optical 
centre,  §  501)  parallel  to  given  direction,  this  locates  image  on  focal  plane. 
Patch  11-5  X  0-5/57-3  cm.  diam.  but  elliptical,  and  blue  inner  end,  red  outer, 
Fig.  232. 

21.  Add  powers  straightaway,  as  of  course  you  naturally  do  with  double 
pocket-lenses,  etc. 

22.  3  in.  At  40/(—  5)  +  40/a  =  40/6  or  2-7  in.  beyond  second  lens,  using 
image  from  first  as  virtual  object  of  second.  On  centre  of  second  lens. 
Nowhere,  see  telescope. 


SOLUTIONS  gj5 

23.  1-67  4-  I  la  =  3-3.     Then  becomes  11-6  D 

26.  §  526,  Fig.  210. 

27.  8  D,  then  5-5  D.     (1/1-5)  metre. 

28.  Figs.  211,  IX,  262.     100/(-  6)  +1100/12  =  D  =  -  8-3. 

29.  36  in.  above;   second,  aim  at  apparent  bottom  only.  6  +  4-5  in  tiown. 
see  Figs.  138,  180,  192;   63  in.  ^        -r  .      wn, 

30.  6-6  in.  radii. 


CHAPTER   XXXIV,  p.  425. 

3.  Halfway  in  towards  centre,  virtual  and  erect,  real  inverted,  worth  your 
while  examining. 

4.  TBY  IT. 

5.  See  Chap.  XL. 

6.  1/0-8  +  1/d  =  2;   d  =  1-33  m. ;    x  2  x  r/57-3. 
8.  1/1-5  +  1/0-3  =  4D  or  1/4  m. 

10.  1/6  +  1/36  and  1/6  -  1/36  =1/3;   6  =  4  or  2  ft. 

11.  100/16  +  100/a  =  -  lOD;  virtual,  615  cm.  behind;    x  0-38. 

13.  1/0-40  +  1/a;  =  4  =  1/0-41  +  l/y;    2-6  cm. 

14.  Wider  view,  curved  surface  facing  more  places. 

15.  Fig.  210,  3. 

16.  2/44  =  a:/2000;    91  m. ;   none  on  size,  brightness  inoreaaed,  §  612. 

17.  Virtual  image  of  pin  must  be  at  centre  of  mirror,  so  wave  fit*  it  and 
comes  back  michanged  (rays  all  radial),  hence  6  =  15,  a  =  —  10;    —  3-3  D. 

18.  25  cm.,  easy. 

19.  The  mirror  fits  AEB,  Fig.  138,  the  wave  returns  to  centre  O  because  of 
the  refraction,  else  would  go  /x  times  as  far  to  centre  I  of  arc  AEB  (empty 
mirror). 

20.  Practically  Fig.  211,  HI,  only  piano  lens.  /  of  lens  =  r  of  mirror  — 
60,  by  §  508,  1/60  =  0-5/R;  R  =  30. 

21.  As  in  20,  /  =  40.  As  in  19,  /x  x  14  =  R.  Then  by  §  50«.  1/40 
=  ifi  -  1)/R;    1-54. 

22.  Real  image  6-7  cm.  from  convex,  real  object  20  cm. ;  gives  0-16  -f  1/a  — 
0-05  +  1/ci  =  -  0-08,  d  -  a  =  3-5. 

23.  The  keratometer;   6  =  2m,  magfn.  —  0-2. 

24.  Convex  lens  focussed  on  to  face  of  concave  mirror,  centre  of  curvature 
of  which  is  optical  centre  of  lens.     Make  diagram  :  the  beet  cycle  rear  roflector. 

25.  Fig.  211,  X. 

27.  Fig.  211,  V,  radius  is  8-1  cm.  less  than  a  in  100/30-5  -f  lOO/o  =  5  D; 
50  cm. 

28.  50  cm.  from  mirror  then  12-5  +  lOO/o  =  8  and  4  +  100/d  =»  8. 

29.  14-6  from  mirror;    13-3  from  lens. 

30.  Image  6-7  cm.  inside  lens,  then  1/23-3  +  l/o  =  2/20. 

31.  100/20  +  100/a  =  6-67,  a  =  60;  then  100/(-  30)  +  lOO/a  -  6-67. 
a  =  10,  total  separation  60  cm. 


826  SOLUTIONS 


CHAPTER   XXXIX,  p.  485. 

1.  Draw  radius  '  normal '  where  any  ray  meets  sphere,  construct  angles  i 
and  r;  it  emerges  with  exactly  same  angles  repeated,  and  crosses  at  F  its 
parallel  ray  which  you  have  drawn  straight  through  centre.  Ask  your 
seaside  meteorological  station. 

2.  In  Fig.  230,  O  the  goldfish  appears  at  0',  more  remote  and  larger. 
Instead  of  waves  you  can  use  construction  like  Q.  1,  only  incline  second  ray 
to  meet  first  inside  the  sphere,  this  is  object  point,  its  virtual  image  is  where 
original  rays  meet  when  produced. 

3.  Near  surface  almost  Fig.  180;  central,  radii  all  undeviated;  beyond, 
Q.  2. 

8.   100 /a  +  4  =  50;   0-18  cm. 

10.  -  6-25  D,  16  and  infinity. 

11.  —  4-25  D  nearest;    quarters  smallest  fractions  worked  to. 

12.  —  half  D  nearest. 

13.  1/20  —  10/20,  a  2-2-ft.  concave,  or  -  1-5  D. 

16.  Disregard  difference  between  15  ft.  and  infinity;    —  5  D,  13  in. 

17.  Puzzle  it  out. 

20.  +  3  D. 

21.  +  3  Dl 

22.  +  2-25  D. 

24.  +  3-25  D. 

25.  Optometer,  13-2  and  25,  deduct  13-7,  -  0-5  (hyper)  and  11-3  D;  11-8 
A.P.:   '  —.200  '  and  8-8  cm. 


CHAPTER  XL,  p.   532. 

2.  21-5  diams.  larger;   as  21*52  :  1. 

7.  Turn  over  in  its  bearings. 

8.  Fig.  259 ;   don't  attempt  to  show  object. 

9.  1-5  and  4-5  in. 

13.  1-1  in. 

14.  25/5  diams.;   6-25  cm. 

15.  40/6  +  40/(-  8)  =  20;    1-6  in. 

16.  Close;   at  3  in. ;   6  in.  or  more,  avoid  half  measures. 

18.  60,  because  your  unaided  eye  makes  EI  10  in. 

19.  Virtual  object   l/(-  6)  +  1/25  =  1//;     8  ft.;     25/6  X  the  36  of  the 
mirror. 

23.  6  and  60. 

24.  1/EI  +  l/(-  25)  -  1/5;     EI   4-16,    then    l/d  +  1/(20  -  4-16)  =  1/4; 
5-35  in  front. 

25.  EI  0-5  in.,  final  1  in.  from  E  (ridiculous,  draw  out  half  inch). 

26.  100/3  +  100 /a  =  50,    a  =  6    then    100/(GE  -  6)  +  100/(-  25)  =  10, 
lenses  apart  GE  13-14  cm.,  m.p.  2  X  25/7-14  =  7. 

27.  Only  if  lens  more  than  2  mm.  diameter,  §  629. 


SOLUTIONS 


827 


CHAPTER   XLIII,  p.  563. 

3.  MHsinA. 

4.  2  X  981  X  0-2  dyne-cms.  balance  0-44  M  which  =  0-44  (25  —  1  —  1)  p. 
M  890,  P  39. 

5.  §684. 

6.  P/8«  =  0-20;  12-8. 

7.  P/10,000  =  018,  800  P  =  M. 

8.  Hammer  E.  and  W. ;  along  dip  line;  N.  and  S.  horizontal,  and  vertical 
ratio  tan  dip.  ' 

13.  2M/253  =  0-2. 

14.  P/152  -  P/(15  +  20)2  =  018;    15  cm.  from  S.  pole;    P.  4Uo. 

21.  10%  longer. 

22.  (0-18  +  2M/1000)/018  =  (30/159)«'20/184)«;    =  3.  M  =  180. 
28.  0-36. 


CHAPTER  XLV,  p.  592. 

1.  Level  4  for  5  cm.,  then  hyperbola  falling  to  2  at  10  cm. 

2.  Two  pendulimas,  each  displaced  2-5  cm.  c'/5"  =  (2-5/60)  x  0-1  gin.  x 
981 ;   e  10-1  and  potential  e/r  =  20-2. 

3.  10/10  -  30/10  =  -  2;   30  X  10/20*  =  3/4  dyne. 

4.  Potentials  30/10  and  5/15,  flows  from  smaller  sphere.  Now  35  unita 
distributed  over  capacity  10  -\-  15  raises  all  to  35/25  =  1-4  potential.  Flow  =■ 
30  less  10  X  1-4  imits  left  to  it. 

5.  20/1;  0  and  double;   2/21  and  20  X  2/21. 

6.  Original  potentials  9/3  and  36/6,  energy  0-5  X  9  X  3  4-  0-5  X  36  X  6  -■ 
121-5  ergs.     Final  potential  45/9  =  5,  energy  0-5  X  45  X  6;  Iom  9  ergi. 

8.  The  P.D.  remains  constant. 

10.  Only  discharged  if  both  coatings  connected. 

11.  Q  10-«,  V  1000,  .-.  0001  mfd.     One-third  transfers,  V  667. 

12.  ParaUel  5/6;   series  1/(2  +  3)  =  1/5  mfd. 

13.  Make  each  in  tiun  share  charge  with  a  fixed  fourth,  and  find  which  falla 
most  in  potential.  Parallel  16.  In  series,  P.D.  in  first  Q/10,  in  second  Q.'i. 
in  third  Q/1,  total  1-3  Q  must  =  1.     .*.  Q  =  capacity  =  0-77  mfd. 

14.  100  7r/47rd=  10;    2-5  cm. 

15.  1/45  and  1/60;    0-5  X  10  X  1/45  -f  0-5  X  10  X  1/60,  about  0-2  erg. 

16.  Capacities  20,000/407r  and  5000/207r ;  energy  halved. 

17.  Halved. 

18.  Work  out  like  Q.  2,  assiuning  charges;  notice  17. 

19.  Reduced,  ebonite  weakens  lines;  more  reduced,  meUl  obliterates  linos; 
raised. 

20.  50K/47r  X  0-05  =  140;  1-75. 

21.  VC  becomes  (V/7)  x  (C  +  KC);   K  =  6. 

22.  0-5  X  (4007r/47r)  X  1/9  =  6-5  ergs.  Capacity  and  .*.  charge  doubled, 
energy  doubled ;   (6)  potential  halved,  energy  halved. 


828  SOLUTIONS 


CHAPTER  XLVI,  p.  610. 

2.  They  cling  tight  round  it,  in  circular  lapping. 

4.  i.e.  direction  and  value  of  resultant  field  of  earth  and  current. 

6.  (2/10)  ~2  X  1/10  =  0-01  dyne  per  cm. 

6.  Beware  of  iron  in  Fig.  288. 

10.  Coil's  axis  along  dip ;  vertically  down  ;  intermediately. 

12.  100  -  0-5  X  10. 

1-3.  Mowing  down  the  earth's  dipping  lines. 

14,   15,   16.   TRY. 

17.  Across  direction  of  motion  in  gap,  and  round  by  the  back  each  side. 

19.  Magnet  slowly  drags  after  disc,  and  stops  deflected,  dragging  on  disc' 
motion,  or  else  gets  carried  round  and  round. 

20.  (a)  no  current  induced,  no  resistance  loss. 

21.  Maximmn  when  cutting  lines  fastest,  zero  sliding  along  lines. 


CHAPTER  XLVII,  p.  621. 

6.  Same  no.  of  turns  would  give  field  8,  /.  5/8  of  same  no.  of  half-length 
coils;   6/16. 

7.  5  X  5/10  or  10  X  1/5  rds. 

8.  {2tt  X  3)/18  ±  (277  X  2)/27,  tans  1-5  and  0-58.     .*.  56-4°  and  30°. 

9.  (H  +  F)/(H  -  F)  =  225/81 ;   F  =  0-47  H. 

12.  230  milHamps. ;  use  2300/4  turns  fine  wire  2  cm.  radius. 

13.  Gets  only  1/3  current,  §  781 ;   H  0-16. 

14.  2-0/(200  +  galv.  resistance)  -f-  2-0/(500  +  g)  =  1/0-575.     .*.    g  =  206. 
Hence  2-0/406  amps,  gives  tan  =  1.     /.  k  is  1/203. 


CHAPTER  XLVIII,  p.    640. 

8.  240/0-6  =  400,  naturally,  well  over  5  times  the  absolute  temperature. 

10.  20/5  =  4  ohms  at  250°  becomes  40/6-7  =  6  at  420°,  a  change  of  50%  for 
170° ;   quote  as  such,  not  on  a  zero  basis. 

11.  100r/(7r  X  0-032)  =  1.16;   0-000033. 

12.  L  X  0-000050/77  X  0-0232  =  10;  332  cm. 

13.  91-5  X  0-0000016/(77  X  0-052);    io6  x  0-000095/0-01;    1/53. 

14.  The  reciprocal  of  (1/r  +  1/(10  -  r));   take  r  =  5,  4,  3,  2,  1. 

15.  R  was  0-5;    is  now   1/(1/0-6  +  1/0-2)  +  1(1/0-4  +  1/0-8),  two  pairs 
parallel  in  series.     0-15  +  0-27  =  0-42. 

16.  3,  6,  9;    1,  1-5,  4-5;   in  triangle  2. 

17.  1  =  1/(1/1-03  +  1/R);       33-3     ohm  =  L  X  45/(77  X  0-012652  X  10«); 
,3-7  m. 

18.  Takes  10  volts  to  drive  current  for  50  lamps  through  the  leads. 

19.  2/3,  2/5,  2/3  volt. 

20.  R  =  1(1  +0-5  +  0-33)  =  0-55;    2-2  metres. 


SOLUTIONS  829 

21.  Unshunted  ciarrent  1/(1  +  l  -f  8)  =  01  amp.;  try  ghunting  8  by  0-2 
ohm,  current  now  practically  1/2-2  =  0-45  amp.,  of  which  galv.  gets  a  fortioth 
Try  again. 

22.  Joint  R  =  1/(1/270  +  1/30)  =  27;  total  current  4-6/(3  +  27  4-  70)  - 
0045,  of  which  galv.  gets  00045  amp.  ' 

23.  V  =  51  X  0-04  =  2-04.  Joint  R  =  1/(1/10  +  1/60)  =  60/6;  current 
=  2-04/(50/6  +  1),  of  which  galv.  gets  5/6;   0-182.     Academic.  ^^ 

24.  AB  reduces  to  1/(1/2  +  1/20)  =  20/11,  then  CD/10  =  33/20;  16-5 
ohms. 


CHAPTER   XLIX,  p.  653. 

2.  (E  +  l-08)/(E  -  1-08)  =  44/13,  hence  31  E  =  57  x  1-08,  E  =  2. 
4.  0-1  millivolt  drive  1  micro-amp.  through  100  ohms;   add  900. 

7.  35/40  of  1-5  =  1-31. 

8.  Shunt  1/9  ohm  =  0-000034  L/tt  X  00612;   33  ^m. 

9.  9R;  no. 

10.  Shunt  by  l/20th  ohm,  (6)  put  490  ohms  in  series. 

11.  (a)  shunt  of  2-5  ohms,  (6)  150/R  =  0-005,  series  R  of  30,000. 

17.  (a)  small  (c)  larger  because  resistivity  larger,  ratio  unaltered  by 
reversal,  (6)  probably  much  larger  deflection,  changing  but  not  reversing  when 
cm-rent  reverses. 


CHAPTER   L,  p.  664. 
Recollect  VAT  is  the  general  expression,  and  use  it  in  preference  to  C"RT. 

2.  200-volt,  100-watt  lamp ;  4-2  shillings. 

3.  (a)  667,  (6)  0-3,  (c)  1000/60  =  16-7  hr. 

5.  20  X  50  X  100  hr.  -f-  1000  X  5d.  =  £2  Is.  8rf. 

6.  (0-55  +  0-05)  X  35  X  miles  =  50  volts  drop;   2-4  miles. 

8.  110;   2  X  110  X  3600  -i-  4-2  =  180,000  cal. 

9.  Wire  is  3  ohms /cm.,  0-52  x  3/4-2  =  0-179  cals./sec.,  cm.    Surface  area 
per  cm.  0017  sq.  cm.,  radiation  0-179/0017  =  10-6  cal./sec.  cm.« 

10.  cV,  mass,  sp.  ht. ;  c^r,  radiation,  §  968,  gas  cooling  if  any. 

11.  1-52  X  0-67/4-2  =  0-36  cal./sec.  =  0-02  X  {t  -  15);    33°. 

12.  (80  —  4-2  X  5)  joules  -^  50  =  1-18  amp. 

13.  (o)  2,400,000  a  penny,  (6)  100,000  X  30°  X  4-2/3,600,000  units  X  .1;2i/., 
fivepence  farthing  for  12,600,000  joules  or  4,150  foot  tons  per  bath. 

14.  Double  length  to  meet  voltage  gives  double  heating  surface,  so  halv© 
width. 

15.  Watts  X  630sec./4-2  =  500  X  15  cal.;    1-25  watts  per  candle. 

16.  (a)  Half  current,  same  voltage;   half  (6)  same  current,  double  voltage; 
double.     Actually  H,  2  H ;    4/9  H  and  2/9  H. 

17.  VA  100  X  100/300  and  100  X  100/500  X  2;   33-3  and  40. 

18.  c  X  10  +  c  X  4  =  V  of  battery;   then  0-^  X  JO  +  ^  X  4  -  V.  honco 
C/c  =  9/4;  current  in  shunt  .'.  6/4  c,  .'.  shunt  4/5  X  10  «  8  ohms.    Twistrr. 

19.  Same  voltage,  more  current,  think  of  lamps. 

20.  On  50  volts  loss  50^  X  0-02  =  50  watts,  on  260  low  I0«  X  CH»  -  2 
watts,  difference  per  1000  hours  48  kw.-h.  =  16/s. 


830  SOLUTIONS 


CHAPTER   LI,  p.  689. 

3.  '  Make'   current  lower  voltage,  longer  duration,  reversed. 

5.  Compass,   or  galvanometer,   or   electrolysis,  A.C.  no   effect;    noise  in 
telephone. 

6.  Slow  rise;   rapid  fall  and  flash. 

9.  Resistance  puts  a  brake  on,  and  if  large  enough  prevents  any  oscillation. 


CHAPTER   LII,  p.  715. 

2.  Petrol  insulator,  aq.  dest.  and  tea  nothing  perceptible,  milk  slightly 
conductive  from  Ca  salts  feebly  ionized,  sea  water  and  acid  electrolyse. 

3.  Try  it. 

4.  From  KI  saturated  pad,  as  kathode,  see  15. 

5.  1/00001045  X  1/3600  X  10  decamps,  to  amps.  X  2/18  of  HgO  =  3-23 
amp. 

6.  3  X  1800  X  63-6/2  x  e  -  1-72. 

7.  Area  240  sq.  cm.;  /.  mass  =  240  X  0005  X  10-5  =  12-6  gm.  =  1-5  X 
0-001118  X  7500  sees. 

8.  2/3  X  90  X  1/11,160  =  0-0054  gm.  hydrogen  X  273/290  X  (77  -  1-5)/ 
76  =  0-0050 gm.;  then  X  31-6  =  0-17  or  0158  Cu;    X  108  =  0-58  or  0-54  Ag. 

9.  0-23  amp.  X  1800  X  31-8/96,500  =  0-137  gm. 

11.  0-18. 

12.  100  X  273/288  X  (74  -  l-3)/76  X  1/11,160  X  96,500/300  =  2-63  amp. 

13.  96,500  X  5/(56  -^  2)  =  17,200;   5  X  200/28  -^  13-6  =  2-63  c.c. 

14.  112  X  453-6  gm.  -^  27/3  trivalent,  gm. -equivalents  X  96,500  coulombs 
-^  86,400  sec.  =  6300  amp. 

15.  Resistivity  =  1/0-002,  /.  R  =  500  X  7  ^  9  =  390  ohms;  0-0007  X 
1800  =  1-26  cm. 

16.  500  X  273/283  X  (76-  l-0)/76  X  1/11,160  X  96,500/3600  =  1-15,  error 
0-15  high. 

17.  (6  -  l-5)/3  X  3600  -^  96,.500  X  18/2  =  0-50  gm.;  4-5  X  (6  -  l-5)/3  -f- 
4-2  X  3600  =  5800  cals. 

18.  Iron  rusts,  mixed  gas  and  steam  pass  off.  More  water  and  new  iron 
rods. 

19.  Same  copper,  twice  heat  in  higher  R ;  half  copper  half  heat  in  ditto. 
21.  (a)  nil. 

24.  Liquid  R  =  1/0-7  X  0-8/144  =  0-008  add  002  then  2/0-028  =  71-5 
amp.,  about  50  times  overload. 

25.  Allowable  drop  10  v.  in  (0-3  +  0-1)  ohm  means  25  amp.  =  2500  watts; 
50  lamps. 

26.  (100  -  12  X  2-5)/3  =  23-3  ohm. 

31.  First,  total  resistance  about  1-4/0-02  =  70  ohms.  As  current  so  small, 
little  voltage  can  have  been  lost  in  the  cell  itself,  unless  it  is  of  high  resistance^ 
therefore  1-4  probably  nearly  its  full  E.M.F. 

Second,  this  1-4  drives  1-4  amp.,  .'.  (cell  +  ammeter)  only  about  1  ohm, 
cell  not  of  high  resistance.  Also  drives  only  trifling  current  through  volt- 
meter; this  must  have  high  resistance. 

Further,  only  0-02  volt  drove  1-4  amp.  through  ammeter,  which  is  therefore 
0-02/1-4  =  l/70th  ohm  only. 


SOLUTIONS  881 

.*.  cell  1  ohm,  and  voltmeter  70  ohms,  very  nearly ;  and  open  circuit  E.M.F. 
of  cell  l/70th  more,  1-44  volts. 

This  is  always  happening  in  Practical  Exains. 

Four  definite  values,  saved  from  four  readings  taken  blindly ,  by  a  littie  clear 
thinking.     Oo  thou  and  do  likewise. 

32.  l-5/(R  +  3)  =  0-4;  rest  of  circuit,  0-75  ohm. 

33.  High  external  resistance,  series;  low,  parallel. 

2  X  l-5/(intemal  26  +  10)  =  0-25,  .-.6  =  1. 
then  1-5/(10  +  1/2)  =  0-142  amp. 

34.  Voltmeters  can  only  measure  P.D.  between  lumps  of  metal,  cannot  go 
fishing  in  the  cell ;   E.M.F.  is  P.D.  of  brtiss  terminals  on  open  circuit. 

2  X  1-5/(26  +  10)  =  1-5/(6/2  +  10)  gives  6  =  10;  0-1  amp.,  half  and 
1  volt. 

35.  3  volts,  1-2  ohms;    1,0-13;   (1,  0-2)  +  (1,  0-4)  =  (2,  0-6),  no  other. 

36.  A,  1  volt  stays  inside  driving  cvu-rent  through  2  ohms,  i.e.  0-6  amp^ 

5  volts  drives  this  through  A,  =  10  ohms;  similarly  B  =  4. 

37.  First  ciurent  is  1-4/500  =  00028  amp.,  second  1-2/50  =  0-024  ainp.» 
0-2  volt  drives  their  difference  0-0212  amp.  through  6,  .*.  =  9-5  ohms. 

40.  Stay-at-home  volts  drive  (ext.  volts /ext.  resistance)  through  6,  0-18  ■■ 

6  X  (0-9/5);   6  =  1;    2-7;   4. 

41.  (3-3  -  2-5)/3  X  1-5  and  1-9/6;  inverting,  times  as  316  :  178. 

42.  Joint  R  6-7,  current  1-4/10-7  =  0-13,  P.D.  1-4  -  4  X  013  =  0-88, 
shares  0-43  and  0-87  amp. 


VALEDICTORY 


§  999.  And  now  are  you  come  to  the  day  of  your  exam. 

'  This,'  said  the  bo's'n  to  the  carpenter,  as  they  floated  side  by  side 
in  their  life-belts  in  a  wide,  wide  empty  sea, '  this  is  where  skill  ends 
and  luck  begins.' 

Maybe  you  have  '  looked  up  things  '  on  the  very  eve  of  the  trial. 
If  you  are  wiser,  you  have  taken  a  whole  day  off.  Particularly  is 
this  desirable  before  your  physics,  when  a  clear  head  for  argument 
will  stand  you  in  better  stead  than  a  stuffed  hassock. 

You  have  been  packing  the  gray  cortex  of  your  brain  with  electro-! 
chemical  messages  sent  along  the  fibres  which,  in  their  white  fatty  1 
insulation — imperfect  before  twenty-five — are  the  main  bulk  of  it, 
and  these  fibres  probably  contain  effete  chemical  products,  which 
must  be  given  time  to  diffuse  out  and  leave  them  clear  for  traffic 
the  other  way. 

Of  course  you  recollect  instances  where  at  the  last  minute  yoi 
dropped  on  some  odd  thing  which  came  in  immediately  useful,  an< 
of  course  you  do  forget  in  24  hours,  but  have  you  any  means  of  tell 
how  much  more  a  clear  head  would  have  set  down — forgotten 
before  the  fact,  forgotten  altogether  ?  Your  examiner  has ;  he 
could  look  through  and  spot  the  slips — if  he  wasn't  busy  with  the 
next  paper. 

When  that  wide  white  sheet  stretches  before  a  blank  mind,  what 
is  to  do  ?  Do  as  I  have  done  with  my  diagrams,  leave  it  to  your 
hands  :  many  of  my  diagrams  are  little  like  what  I  thought  they 
would  be,  my  hands  have  taken  charge,  and  theirs  is  the  credit. 

Only,  keep  a  watch  on  your  hands,  that  they  wander  not  too  farj 
repeating,  perhaps  even  contradicting.     A  book  like  this  has  to  tel 
you  things  this  way  and  that,  so  that  one  may  stick  :    if  all  ha"^ 
stuck  we  don't  want  to  hear  them ;    give  us  one  plain  intimatioi 
that  you  understand,  and  pass  on  :    two  more  pages  for  that  las 
mark  is  too  much  to  pay  for  perfection,  when  unhurried  thougl 
over  later  questions  may  rattle  down  an  extra  half-dozen. 

You  can  leave  gaps  to  fill,  only  run  an  arrow  along  them  to  tl 
next  answer.     Bulk  goes  for  nothing,  long  words  and  circumloci 
tion  fill  a  lot  of  it,  unthinking  repetition  and  pure  gibberish  puff 
out ;   don't  fear  brevity.     Cross  out  plainly  :    this  MS.  is  an  awfi 
blotch  in  places,  but  the  compositor  can't  read  what  is  blotted  out 

These  remarks  apply  to  all  exams. 

What  says  the  Tent-Maker,  whose  sea  was  of  thirsty  sand  ? 

'  Folks  of  a  surly  Tapster  tell. 
And  daub  his  Visage  with  the  Smoke  of  Hell, 
They  talk  of  some  strict  Testing  of  us — Tosh, 
He's  a  Good  Fellow,  and  'twill  all  be  well.' 
832 


§999] 


VALEDICTORY 


833 

I  hav«  never  worked  with  a  Medical  Physics  Examiner  who 
wasnt  a  good  fellow;  but  they  test  you,  and  the  being  well  is 
your  look  out.  It  is  deputed  to  us  to  hold  an  outer  gate  of  your 
proposed  profession,  and  we  cannot  ease  it  more  ajar  for  you  and 
squeeze  it  to  before  your  next  competitor,  as  mayhap  in  your 
worser  moments  you  wish  we  would.  Be  content  with  this  •  you 
get  more  marks  from  us  than  you  would  if  all  the  papers  were 
shuffled  round,  and  each  marked  by  another  fellow  with  the  aid  of 
the  book— more,  probably,  than  if  you  marked  them  that  way 
yourself.  ^ 

Marks  in  exams  of  all  sorts  run  as  in  Fig.  420,  where  you  see  that 
two  medical  physics  exams  that  I  happen  to  have  graphed,  years 


\  MARKS 

I 

• 

\ 

^. 

1 

^ 

h 

^^ 

''Jifc. 



"^ 

-^ 

L> 

~~ 

1              1 

N 

s\ 

„. 

DIOAl 

ES 

N 

Fig.  420. 


ago,  years  apart,  by  different  examiners,  adjusted  to  be  of  the  same 
horizontal  length  but  not  manipulated  in  any  such  way  as  lesser 
educational  authorities  love,  have  come  out  practically  identical. 
The  third,  a  this  year  written  paper  only,  rather  different  in  style, 
pulls  up  much  nearer  the  same  shape  when  the  practical  marks  are 
added. 

You  see  that,  taking  a  pass  standard  of  40%,  and  cutting  off 
a  tail  plainly  hopeless  (from  probably  a  diversity  of  contrary  causes), 
only  about  a  sixth  of  the  total  entry  is  left  in  honestly  hard  case. 

All  three  subjects  usually  drag  very  much  the  same  tail,  but  varia- 
tions  in  the  lumbar  region  account  for  the  final  pass  list  of  the 
triple  examination  being  nearer  50%  than  60. 

Pray  you,  never  let  yourself  think  of  *  forty  per  cent.* :  your 
dog  at  home  doesn't  carry  his  tail  at  *  forty  per  cent.*  elevation. 

In  your  Practical  Exam,  do  please  realize  that  the  official  regula- 
tion forbidding  you  to  communicate  with  your  exarauicrs  has  do 
force  when  you  meet  them  face  to  face. 

In  particular,  if,  after  a  tussle  with  it,  you  really  are  floored  by  the 

EE 


834 


VALEDICTORY 


[§  999 


test  experiment  that  has  become  yours  by  pure  chance,  consult 
your  examiner  straightaway  ;  for  he  wants  to  see  what  you  can  do, 
not  to  see  you  wasting  time  doing  nothing. 

On  the  other  hand,  it  is  just  as  well  not  to  try  to  delude  your 
examiner  ;  he  can  keep  a  straight  face,  he  has  heard  it  all  before. 

Some  of  you  can't  help  being  nervous  ;  so  far  as  we  are  concerned 
there  is  no  cause  for  it,  what  we  cannot  allay  we  try  to  allow  for  : 
the  awful  divinity  that  doth  hedge  a  visiting  School  examiner  we 
have  outgrown  :  we  be  your  elder  brethren,  glad  enough  to  welcome 
youngsters  into  the  family,  only  putting  back  those  who,  as  yet,  have 
over-estimated  the  natural  gifts  of  their  youth,  and  have  not  reached 
the  level  that,  a  few  months  later,  they  will  clear  with  ease. 


FRATER  VALE ! 

§  1000.  Well,  there  'tis ;  maybe  I  have  told  you  twice  too  much, 
but  I  would  not  have  you  as  the  beasts  that  perish.  This  is  a  book 
of  explanations,  and  the  first  few  words  printed  inside  that  far-distant 
front  cover  should  suffice  to  explain  why  I  have  set  no  stint  of  time 
nor  cost  in  turning  out  something  serviceable. 


Faithful  Reader,  here  I  leave  you. 


Fare  you  forth,  and  Fare  you  Well  I 


1) 


I 


INDEX 


{Note. — FigTires  refer  to  page-numbers.) 


Abbe  condenser,  524 
Aberration,  chromatic,  470 
Aberration,  spherical,  466 
Acceleration,  10 
Acceleration,  irregular,  1 1 
Accommodation,  visual,  478 
Accumulators,  acid,  703 

alkaUne,  706 

Accuracy  limits,  6 
Acetylene,  dissolved,  217 
Achromatic  lens,  472 
Acoustic  absorption,  333 
Activity,  23 

Adiabatic  condition,  209 
Adsorption,  268 
Advantage,  mechanical,  47 
Air  locks,  72 
Air  pumps,  72 
Alcarraza,  203 
Alcoholometer,  95 
a  jS  y,  766 

Alpha  particles,  757 
Alternating  Current,  A.C.,  666 
Altimeter,  81 
Altitude  and  pressure,  81 
Altitude  and  temperature,  234 
Aluminium  manufacture,  693 
Amber  disc,  493 
Ammeters,  613,  645 
Ammonia  machine,  219 
Ampere,  the,  602,  625 
Ampere  balance,  617 
Ampere's  rule,  597 
Amplitude,  289 
Anastigmats,  494 
Aneroid  barometer,  68,  79 
Angle,  113 
Angstrom  unit,  5 
Anode,  anions,  690 
Anticathode,  743 
Anticyclone,  246 
Anvil  cloud,  237,  245 
Aplanat,  515,  527 
Apochromatic  lenses,  473 
Approximations,  useful,  114 
Arcs,  729 

Arc  in  magnetic  field,  599 
Arcs,  ultra-violet,  779 
Archimedes,  89 
Archimedes  principle,  89,  91 


Architectural  acoustics,  330 

Areas,  113 

Argon,  monatomic,  323 

spectrum,  808 

Armature,  603 
Arms,  strong,  24 
Astatic  needles,  620 
Astigmatism,  383,  469,  481 
Atlantic  circulation,  241 
Atmosphere,  convective  equilibrium, 

234 
Atmospheric  electricity,  735 
Atmospheric  pressure,  normal,  79 
Atmospheric  refraction,  391 
Atmospherics,  738 
Atom,  history  of  ,760 
Atomic  diameter,  750 
Atomic  heat,  155 
Atomic  nucleus,  size  of,  750 
Atomic  Number,  751 
Atwood's  machine,  28 
Aurora,  460,  563 
Aiu^ra  polaris,  726 
Autoclave,  201 
Avignon,  26 

Avogadro's  number,  721,  762 
Avogadro's  principle,  145 
Azores,  246 


B 

B€tck  E.M.F.  of  motor,  609 

Bacon,  Roger,  488 

Balance,  117 

Balance  beam,  60 

Balance  stability  and  sensitiv 

119 
Balance  spring,  99 
Balance  wheel  and  spring,  127 
Ball,  billiard,  64 
Ball,  cricket,  65 
Ball,  golf,  27,  65 
Ball,  rolling.  64 
Barker's  mill,  86 
Barograph,  80 
Barometer,  aneroid,  68,  79 
Barometer  corrt'ct  ions.  78,  131 
Barometer,  heights  by,  81 
Barometer,  mercury,  77 
Barometer,  water,  etc.,  79 
Basilar  membrane,  370 
Bass,  bottle,  202 


835 


836 


INDEX 


Battery  arrangement,  713 
Battery,  Cadmium  Standard,  710 

Daniell,  etc.,  708 

E.M.F.,711 

internal  resistance,  711 

Leclanch6  and  Dry,  710 

Battery,  primary,  voltaic,  707 

Battery,  secondary  or  storage,  703 

Beats,  337 

Beaume  degrees,  95 

Bells,  366 

Beta  particles,  763 

Bicycle  wheel,  spinning,  27,  62 

Bifocals,  480 

Big  Ben,  40,  127 

Billion,  5 

Binocular  glasses,  508 

Binocular  vision,  477 

Biquartz,  451 

Bismuth,  polishing,  103 

Black  body,  789 

Blood  film,  corpuscles,  453 

Blood,  isotonic  solutions,  277 

Blood  spectrum,  446 

Blood  rain,  451 

Blood  worm,  281 

Boards,  loose,  24 

Boats,  rowing  and  sailing,  49 

Boiling,  197 

Boiling  with  bumping,  199,  266 

Boiling  point,  ultimate,  204 

Bolometer,  783 

Bora,  238 

Bottle,  to  dry,  133 

Bourdon  guage,  81 

Bowls,  63 

Boyle's  law,  105 

Brain,  floating,  69 

Brakes,  23 

Breezes,  land  and  sea,  238 

Bridge,  and  resonance,  386 

Brightness,  intrinsic,  491 

Brinell  test,  103 

Brittleness,  102 

Brown,  Robert,  515 

Brownian  motion,  274 

B.Th.Unit,  151 

B.T.Unit,  38,  656 

Bubbles,  264,  266 

Bunsen  burner,  84 

Burning  deck,  793 

Burnishing,  104 

Butterfly  wing,  455 


Cables,  aerial  electric,  714 
Cabot  quilts,  333 
Cadmium  cell,  642 
Calorex.  787 


Caloric,  151 
Calorie,  calorie,  151 
Calorifier,  169 
Calorimeter,  ice,  158 
Calorimeter,  steam,  1 60 
Calorimeter,  bomb,  gas,  etc.,  160 
Calorimetry,  151 

animal,  161 

Calorimetry,  latent  heat,  157 
Camera  lenses,  493 
Camera  lucida,  519 
Camera,  pinhole,  377 
Camphor,  melting,  193 

movements,  259 

Candle-foot,  379 
Candle-power,  379 
Candle  stool,  488 
Capacity,  electrical,  584 

thermal,  151 

Capillarity,  260 

Capillary  electrometer,  702 

Carbon  dioxide,  liquid,  219 

,  snow,  193 

Cassegrain  illuminator,  530 

Cassegrain  telescope,  497 

Cat,  64 

Cataphoresis,  699 

Cathode,  cations,  693 

Cathode,  hot,  719 

Cathode-ray  oscillograph,  722 

Cathode  stream,  719,  742 

Catoptric  lanterns,  492 

Caustics,  466,  467 

Caustic,  curve,  refraction,  391,  397 

Cautery,  cold,  683 

Cavendish  Laboratory,  720 

Celsius,  137 

Centre  of  gravity  or  mass,  eg,  43 

Centrifugal  force,  54  ' 

Centrifugal  pump,  84 

Centrifuge,  centrifugals,  57 

C.G.S.  system,  5 

Charles'  law,  133 

Chimney  ventilation,  175 

Chinook,  238 

Chladni's  figures,  365 

ChlorophyU,  446,  781,  807 

Choking  coils,  676 

Chromatic  aberration,  460 

dispersion,  470,  478 

Chromatic  scale,  374 
Chromosphere,  447 
Chronograph,  116,  338 
Cinema  projector,  495 
Circuits,  A.C.,  various,  679 
Circular  error,  56 
Cisterns,  flushing,  71 
Cleavage  planes,  104 
Clinical  thermometer,  142 
Clock  chimes,  367 


INDEX 


837 


Clocks,  A.C.,  116 

Clocks,  Shortt,  etc.,  114 

Clockspring,  energy  in,  714 

Clothing,  170 

Clouds,  235 

Coal  washer,  91 

Cochlea,  370 

Cod-liver  oil,  781 

Cold  front,  242 

Cold  storage,  218 

Collision,  in  Fig.,  770 

Colloids,  273 

Colour,  439 

Colour  by  artificial  light,  444 

Colours,  complementary,  446,  460 

Colour  printing,  462 

Colour  projection,  461 

Combustion,  heat  of,  161 

Comets'  tails,  790 

Commutator,  603 

Compass,  gyro,  63 

Compensation  contrivances,  125 

Compound  bar,  127 

Compound  harmonic  curves,  290 

Compound  microscope,  516 

Compound  pendulimi,  60 

Compressibilities,  107 

Concentration  cells,  701 

Concord  and  discord,  371 

Condenser,  electrolytic,  702 

Condensers,  electrical,  585 

;  coupled,  590 

,  energy  in,  588 

Condensers,  microscope,  534,  535 
Conductance,  624 
Conduction  of  heat,  170 
Conduction,  metallic,  630 
Conductivity,  thermal,  171,  174 
Conductor,  moving  in  field,  604 
Conductors  opaque,  688 
Conical  pendulum,  54 
Conservation  of  energy,  37 
Conservation  of  momentum,  27 
Conservative  system,  36 
Consonants,  368 
Constantan  or  eureka,  627 
Contrast,  destruction  of,  789 
Convection,  167 
Cooling  correction,  153,  164 
Cooling  curves,  183 
Cooling,  Newton's  law,  163 
Cooling,  processes  of,  165 
Copper  oxide,  rectifier,  651 

photo-electric,  383,  805 

Cornea,  475 

Corona,  electric,  733 

Coronae,  452 

Corrosion,  electrolytic,  713 

Corti,  organ  of,  370 

Cosmic  rays,  37,  771 


Coulometer,  697 

County  of  London  Electric  Co.,  660 

Cover-glass  thickne«8,  467,  525 

Cricket  ball,  65 

Critical  angle,  316,  395 

Critical  state,  t  and  p,  204 

Crookes'  radiant  matter,  719 

glasses,  780 

Cross  wires,  504 
Crystal  detector,  65 1 
Crystal,  polarizing,  536 
Crystals,  packing  distance,  750 
Crystalloids,  273 
Cryophorus,  204 
Curvature,  112,  409 
Curve,  motion  in,  53 
Current  crossing  field,  601 
Current  measuremen  t ,  6 1 2 
Current,  measuring  large,  648 
Cyclonic  depression,  242 
Cystoscope,  468,  512 


Dal  ton's  law,  197 
Damping,  mechanical,  295 
Daniell's  battery,  708 

hygrometer,  203 

Dark  ground  illumination,  529 
Dark  heat,  782 
Daylight,  384 
Deaf  aids,  329 
De-centred  lenses,  482 
Decibels,  328 
Dec,  magnetic,  560 
Decolorization  of  solutions,  266 
Deformation,   7 
Density,  69,  92 
Depression,  b€Ut)nietric,  242 
Depth,  true  and  apparent,  390 
Deviation  by  thin  prism,  316,  39H 
Deviation,  minimum,  394 
Dew,  233 
Dew  point,  223 
Dialysis,  273 
Diamagnetism,  551 
Diaphone,  340 
Diathermanous,  783,  786 
Diathermy,  683 
Diatom,  colour,  455,  523 
Diatonic  scale,  373 
Dichroism,  456 
Dictaphone,  339 
Dictionary,  13 
Dielectric  constant,  587 
Dielectric,  energy  in,  589 
Dielectrics  trttn8i)aront,  688 
Difference  tones,  371 
Diffraction,  307 
Diffraction  colour*.  452 


838 


INDEX 


Diffraction  grating,  308,  441 

,  finest,  455 

Diffusion,  271 

Dihydrol,  132 

Dilatometer,  129 

Dioptre,  408 

Dioptre,  prism,  482 

Dioptric  lanterns,  492 

Dioptric  strength,  408 

Diplopia,  482 

Diplogen,  754 

Diplon,  771 

Dip,  magnetic,  560 

Direct-vision  prisms,  440,  473 

Dispersion,  chromatic,  470 

Dissipation  of  energy,  37 

Diver,  high,  64 

Doctor,  239 

Doldrums,  239 

Doppler's  principle,  304,  448 

Double  refraction,  537 

Dough,  253 

Driving  mirror,  422 

Drop,  evaporation  of,  266,  268 

Droplets,  cloud  and  rain,  235 

Droplets,  mist,  232 

Dropping  electrode,  700 

Drops,  264 

Drops,  large,  266 

Dry  ice,  189,  193,  206 

Dry  tilth,  263 

Ducks  and  drakes,  64,  536 

Dulong  and  Petit,  155 

Dust  tube,  353 

Dyes,  fugitive,  451 

Dyes,  intense,  450 

Dynamo,  607 

Dynamometer,  38 

Dyne,  20 


Ear,  369 

Earth,  age  of,  768 

Earth,  black  body  temperature,  797 

Earth,  core  of,  303 

Earth,  eUiptic  orbit,  54 

Earth  inductor,  606 

Earth,  internal  heat  of,  766 

Earth,  magnetism  of,  560 

Earth,  mass  and  density,  30 

Earthquake  waves,  303,  322 

Echo,  325 

Echo,  musical,  309,  326 

Eclipses,  378,  447,  456 

Eddies,  84,  254 

Efficiency  of  engines,  214 

Efficiency  of  machine,  47 

Eikap,  Eikonometer,  526 

Einstein,  18,  31 


Ejectors,  83 

Elastic  data,  107 

Elastic  limit,  101 

Elasticity,  99 

Elasticitv,  adiabatic  and  isothermal, 
322 

Electric  bell  mechanism,  671 

Electric  discharge  tubes,  727 

Electric  lines,  571 

Electric  power  and  energy,  655 

Electric  power  measurement,  655 

transmission,  660 

Electric  shielding,  572 

Electric  shock,  740 

Electrical  machines,  576 

Electricity,  frictional,  566 

Electro-cardiograph,  613 

Electro -chemical  equivalent,  696 

Electrode,  690 

Electrode,  dropping,  700 

calomel  and  hydrogen,  701 

Electrolysis,  electrolyte,  691 

examples  of,  693 

laws  of,  696 

Electrolytic  corrosion,  753 

Electrolytic  gas  calculation,  197 

Electromagnetic  absolute  measure- 
ments, 625 

Electromagnetic  induction,  594 

Electromagnets,  598 

Electrometer,  capillary,  702 

Electrometers,  588 

Electromotive  force,  E.M.F.,  582,  642 

Electromotor,  D.C.,  602 

starter  and  back  E.M.F. ,  609 

Electrons,  cathode,  720 

Electrons,  energy  of  cathode,  744 

Electrons,  free,  759 

e/m,  721 

Electronic  charge,  721 

Electronic  mass,  722 

Electro-osmosis,  699 

Electrophorus,  574 

Electroplating,  695 

Electroscope,  gold-leaf,  573 

Electrostatic  voltmeters,  642 

Elements,  transmutation  of,  769 

Elinvar,  127 

E.M.F.  of  batteries,  comparing,  643 

ohmic,  etc.,  674 

E.M.F.,  thermo-electric,  649 

Emissivity,  thermal,  173 

for  radiation,  785,  789 

End  measure  bars.  111 

Energy,  34 

Energy  conservation,  etc.,  37 

Energy  in  a  fluid,  75 

Energy,  kinetic,  35 

Energy,  potential,  36 

Energy  of  vibration,  290 


INDEX 


830 


Engine,  steam,  75 

Engines,  heat,  212 

Engines,  steam,  oil,  etc.,  214 

Epidiascope,  494 

Equilibrium,  41 

Equilibrium,    statistical    of   vaj)our, 

212 
Equinoctiil  gales,  239 
Equipoter.tial  surfaces,  581 
Equivalents,  table  of,  5 
Erector,  509 
Erg,  35 

Ether  stopper  jumping,  197 
Eureka  or  constantan,  627 
Eustachian  tube,  369 
Evaporation,  192 

and  boiling,  197 

Evaporative  cooling,  203 

Evaporators,  multiple,  201 

Everest,  29,  81 

Evident,  2 

Exam  questions  discussed,  14 

Expansibilities,  tables  of,  125,  130 

Expansion,  thermal,  122 

,  true  and  apparent,  128 

,  of  gases,  133 

,  on  vaporizing,  193 

Exposure  meter,  780 

,  electric,  805 

Extraordinary  ray,  537 

Eye,  475 

Eye,  colour  sensibility,  443,  795,  804 

Eye,  electric,  803 

Eye,  seeing  colour,  460 

Eyepieces,  505 

Eyepieces,  erecting,  518 

Eyepieces,  fluorescent,  778,  782 

Eyepieces,  pancratic,  509 

Eye-ring,  500 


Fahrenheit,  137 

Fall,  free,  28 

Falling  plate  experiment,  28 

Fathometer,  322 

Fatty  acids,  255 

Feather  falling,  25 

Ferrous  iron  and  infra-red,  787 

Fibres,  drawn,  263 

Field,  electric,  580 

Field,  magnetic,  553,  594 

Field-glasses,  common,  508 

prismatic,  509 

Films  and  froth,  266 
Films,  soap,  267 
Filter  pump,  83 
Fire  alarms,  127 
Fireflies,  451,  795 
Fish  floating,  91 


Fish -eye  view,  397 
Flame,  manometric,  355 
Flame,  sensitive,  356 
Floating,  84 
Floating  ring,  605 
Floor,  elastic,  23 
Fluid,  speed  of  outflow,  82 
Fluids,  67 
Fluids,  energy,  75 
Fluids  in  motion,  82 
Fluorescence,  450 

and  photons,  807 

Fluorescence  in  ultra-violet.  781 

Fluorescent  eyepiece,  748,  778 

Fluorescent  screens,  720, 744, 748. 757 

F  numbers,  photographic.  494.  522 

Focal  distances,  406 

Focal  lines,  469 

Focus,  depth  of,  414 

Foetus,  69,  281 

Fog,  232 

Fohn,  238 

Foot,  poimd,  second,  21 

Foot-pound,  etc.,  35 

Force,  17 

Force,  centrifugal,  54 

Force,  Newtonian  philosophy  of.  24 

Force,  unit  of,  20 

Forced  oscillation,  293 

Forces,  parallelogram  of,  19 

Fraunhofer,  446 

Freezing  by  evaporation,  203 

Freezing  mixtures,  189 

Freon,  219 

Frequency,  290 

Frequency  meter  A.C.,  366 

Friction,  dry,  22 

Friction,  dry  and  fluid,  253 

Frigories,  219 

Frosty  fire,  132 

Fruit,  chilled,  218 

Fulcrum,  44 

Furnace,  electric,  659 

Furnace,  high-frequency,  682 

Furnace,  solar  vacuum,  493 

Fuses,  659 


Galileo,  56,  94,  489 

telescope,  507 

Galvani,  707 

Galvanizing,  714 

Galvanometers,  613 

Gamma  rays,  764 

Gas,  ionization  of,  731 

Gas  pressure,  kinetic  theon*.  145 

Gastroscope,  512 

Gauges,  pressure,  81 

Gauges,  screw,  1 1 1 


840 


INDEX 


Gauges,  vacuum,  73 
Gilbert,  William,  566 
Glacial  periods,  231,  798 
Glacier,  104,  188 
Glass,  cleaning,  510 
Glass,  optical,  470 
Glasses,  musical,  366 
Gliding  planes,  103,  188 
Glottis,  368 
Glow  worm,  451,  795 
Glucose,  201 

Golf  ball  and  club,  27,  65 
Gongs,  366 
Good  Samaritan,  277 
Graham's  law,  272 
Gramme  weight,  21,  29 
Gramophone,  339 
Gramophone  pick  up,  607 
Grating,  diffraction,  308,  441 
Gravitation,  28 
Gravity,  g.,  20,  29 
Gravity,  centre  of,  e.g.,  43 
Gray  bodies,  794 
Green  ray  or  flash,  459 
Greenhouse,  787 
Greenland  ice,  188,  322 
Greenwich,  56,  113,  115,  505 
Guinea,  25 

Gulf  stream,  241,  458 
Gun  recoil,  34 
Gyration,  radius  of,  59 
Gyro-compass,  63 
Gyroscope,  62 

H 

Haemoglobin,  281,  781 
Haidinger's  brushes,  538 
Hail,  236 
Hair  cells,  370 

Half-period,  radioactive,  759 
Halo,  454 

Haloes,  pleochroic,  768 
Hampton  Coiu-t,  228 
Hanse  merchants,  46 
Hardness,  103 
Hare's  apparatus,  93 
Harlequin  fly  larva,  281 
Harmonics,  346 
Harmonograph,  292 
Haze,  231 
Head-phones,  663 
Headwind,  37 
Heart,  72 

work  done  by,  74 

Heat,  atomic,  155 
Heat-engine  efficiency,  213 
Heat  engines,  212 
Heat,  quantity  of,  151 
Heating,  electric,  657 


Heating,  electro-domestic,  661 
Heaviside  layer,  805 
Heavy  water,  755 
Heights  by  barometer,  81 
Helium,  liquid,  217 

in  spectrum,  447,  808 

Henry,  670 

High  frequency,  681 

Hoar  frost,  234 

Hoe,  263 

Hooke's  law,  99 

Hooke's  microscope,  490 

Hoop,  revolving,  56 

Hope's  apparatus,  132 

Horse  latitudes,  235,  240 

Horse  power,  37 

Hot  iron,  colours  of,  136 

Hot-water  systems,  168,  241 

Hot-wire  ammeter,  658 

Hoy,  237 

Human  body,  heat  losses,  166 

Humidity,  222 

Hurricanes,  245 

Hydraulic  press,  69 

Hydro-extractors,  57 

Hydrogen,  heavy,  754 

Hydrogen,  liquid,  217 

Hydrogen-ion  concentration,  700 

Hydrometers,  95 

Hydrostatic  balance,  94 

Hygrometric  state,  222 

Hygroscopic  nuclei,  231 

Hygrostat,  224,  228 

Hypermetropia,  480 

Hypochlorite  and  chlorate,  694 

Hypsometer,  139 

Hypsometric  table,  81 

Hysteresis,  669 


Ice,  185 

Ice,  dry,  189,  193,  206 
Ice  factories,  220 
Illumination,  383,  658,  795 
Illumination,  micro,  523 

dark  ground,  529 

Images,  real  and  virtual,  407 
Immersion  object-glass,  468 
Impact,  26,  35 
Impedance,  676 
Impulse,  26 
Inclined  plane,  49 
Incubators,  148 
Indicator  diagrams,  75 
Inductance,  self  and  mutual,  670 
Induction  coil,  670 
Induction,  electric,  569 
Induction,  electromagnetic,  594 
Induction,  magnetic,  546 


INDEX 


Ml 


Inertia,  18 

Inertia,  moment  of,  59 

Infra-red,  782 

less  refracted,  788 

Infra-red  pictures,  782 
Injectors,  83 

Insulator  and  conductor,  688 
Interference  colours,  452 
Interference  of  waves,  305 
Interference  tube,  355 
Internal  heat  of  Earth,  766 
Introscope,  512 
Inverse-square  law,  light,  379 

,  electrical,  580 

,  magnetic,  554 

Ionic  speeds  in  solution,  692 
Ionic  theory,  286 
Ionization  by  collision,  724 
Ionosphere,  781,  805 
Ions,  690 

Ions,  speed  of  gaseous,  732 
lonto-quantimeter,  779,  804 
Iron,  melting,  188 
Isochronism  of  pendulum,  56 
Isothermal  condition,  209 
Isothermal  curves,  205 
Isotopes,  753 


Jellies,  274 

Jet-pump,  83 

Jet -reaction,  85 

Jew's  harp,  364 

J.  J.  Thomson,  720 

Joule,  35,  655 

Joule  mill,  38 

Joule's  equivalent,  178 

Joule's  law  of  electric  heating,  657 

Jiinctions,  hot  and  cold,  650 

K 

Katathermometer,  228 
Kelvin  double  bridge,  639 
Kelvin  scale,  146 
Kilowatt,  38 
Kilowatt-hour,  656 
Kinetic  energy,  35 
Kinetic  theory,  27,  209 

gas  pressure,  145 

of    liquid-vapour   change, 

211 
Knot,  9 
Kodacolor,  461 
Kimdt  dust-tube,  353 


Lactometer,  95 
Lag,  676 
Laminated  iron,  669 


Lkmp  and  scale,  387,  615 

Lamp,  wire,  658 

Lamps  in  parallel,  659 

Lamps,  arc,  pointolite,  etc.,  730 

Lamps,  recent,  795 

Lanolino,  256 

Lapse  rate,  234 

Larynx,  artificial,  385 

Latent  heat,  157 

Lawn  sprinkler,  86 

Laws  of  Motion,  Newtonian.  17 

Lead  oxides,  704 

Leaf,  growing,  281 

Leeuwenhoek,  491 

Length,  precise  measurement,  110 

Lens  combinations,  433 

Lens,  equivalent  thin,  434 

Lens  gauge,  112,  409 

Lens  in  liquid,  415 

Lens  making,  437 

Lens  measurement,  428 

Lenses,  403 

Lenses  in  contact,  415 

Lenses,  standard  construction,  412 

Lenses,  thin,  407 

Lenses,  thick,  432 

Lenz's  law,  605 

Levanter,  238 

Levers,  44 

Leyden  jars,  686 

Ley  den  jars  and  spark,  734 

Lift,  forces  in,  26 

Light,  cause  of  gtweous,  726 

Light,  nature  of,  377 

Light,  speed  of,  775 

Lighthouse,  492 

Lighting,  artificial,  385 

Lightning,  736 

Lightning  arrester,  729 

Lightning  conductors,  etc.,  739 

Lignum  vitse,  256 

Line  squall,  245 

Linnseus,  137 

Liquefaction  of  gases,  215 

Liquid  air,  216 

Liquid  crystals,  132 

Lissajou's  figures,  338 

Lister,  517 

Litre,  4 

Locomotive  blast  pipe,  23 

Lodestone,  543 

Log  line,  9 

Logarithms,  use  of.  14 

Long  conductor,  573 

Long  glass,  509 

Loud  speakers,  664 

Loudness,  327 

Low,  barometric,  242 

Lubrication,  254 

Lumens  per  watt,  659 


S42 


INDEX 


M 

Machine  efficiency,  47 

Make  and  break,  671 

Magnetic  fields,  comparison,  556,  558 

M  and  H, 559 

Magnetic  storms,  562,  726 

Magnetism,  magnetization,  543 

of  earth,  560 

Magnetite,  543 

Magneto,  608 

Magneto,  high-tension,  672 

Magnification,  414,  423 

Magnification  in  depth,  414 

Magnification  method  for  /,  434 

Magnifying  glass,  513 

Magnifying  power,  502,  513 

Manganin,  627 

Mangin  mirror,  493 

Manometers,  76,  79 

Marks,  distribution  of,  833 

Mass  and  weight,  21 

Mass  centre,  43 

Mass,  conversion  of,  775,  797 

Mass  of  Earth,  30 

Mass,  precise  measurement,  117 

Mass,  unit  of,  5 

Masses  and  force,  24 

Matter,  kinetic  theory,  27 

May,  Ice  men,  248 

Meat,  chilled  and  frozen,  218 

Mechanical  advantage,  47 

Mechanical  equivalent  of  heat,  177 

Megaphone,  367 

Megohm,  625 

Melde's  experiment,  341 

Melting  and  freezing,  182 

Melting  point  and  pressure,  184 

Membrane,  acoustic,  367 

Membrane,  basilar,  370 

Mercury  break,  671 

Mercury,  cleaning,  715 

Mercury-vapour  lamps,  727 

Metallic  colours,  449 

Meteorology,  221 

Meters,  current,  615 

Meters,  domestic  electric,  698 

Metre,  4,  5 

Metre  bridge,  636 

Micrometer  microscope,  504 

Micrometer  screw.  111 

Micrometer,  stage,  518 

Micron,  fi,  5 

Microphones,  662 

Microscope,  compound,  516 

MICROSCOPE  HINTS,  530 

Microscope,  simple,  513 

Microscope,  solar,  788 

Micro-slide  making,  400 

Micro -spectroscope,  440 


Milli-ammeters,  645 

Millibars,  mb,  79 

Millilitre,  4 

Million  volt,  735,  764 

Minus  colours,  463,  464 

Mirage,  392 

Mirror  making,  437 

Mirror  measurement,  428 

Mirror,  plane,  388 

Mirror,  tilting,  387 

Mirrors,  spherical,  420 

Mirrors,  standard  construction,  421 

Mist,  232 

Mistral,  238 

Mixed-gas  calculation,  197 

Modulus  of  elasticity,  100,  107 

Moisture,  film  on  glass,  259 

Molecular  speed,  etc.,  275 

Moment  of  inertia,  59 

Moment,  magnetic,  555 

Moments,  Principle  of,  42 

Momentum,  17 

Momentum,  conservation  of,  27 

Monochord,  343 

Monsoons,  239 

Mont  Blanc,  81,  139 

Month,  3 

Moon,  384 

Moon,  mass  and  density,  30 

Motion,  linear,  7 

Motion,  Newton's  Laws,  17 

Motion,  quantity  of,  17 

Motor  bus,  25,  27 

Motor,  electro-,  602 

Motor-generators,  668 

Motors,  A.C.  3-phase,  etc.,  677 

Mouth  correction,  352 

Moving-iron  instruments,  612 

jLi,  micron,  5 

fi,  refractive  index,  315,  389 

Mud  and  dust,  263 

Muscle,  24 

Musical  echo,  309,  326 

Musical  glasses,  366 

Musical  pitch,  374 

Musical  scale,  372 

Myopia,  479 


N 

Natural  law,  19 
Nautical  mile,  n.m.,  9 
Nebula,  growth  of,  791 
Neon  isotopes,  753 
Neon  tubes,  727,  766 
Neutron,  753,  770 
Newton,  19,  28,  29,  497 
Newtonian  constant,  30 
Newtonian  laws  of  motion,  17,  23 
Newton's  colours,  541 


INDEX 


843 


Newton's  law  of  cooling,  163 

Nicol  prism,  537 

Night  glasses,  502 

Nirvana,  35 

Nitrogen,  liquid,  217 

Nitrous  oxide,  sound  in,  324 

Nodes,  310 

Noise  and  note,  319 

Notes,  lowest  and  highest,  375 

Nuclei  of  bubble  formation,  199 

Nuclei,  hygroscopic,  231 

Numerical  aperture,  N.A.,  522,  525 


O 

Oar,  24 

Object,  virtual,  411 

Obvious,  2 

Ohm,  the,  625,  626 

Ohm's  law,  624 

Ohm-meter,  638,  657 

Oil,  lubricating,  255 

Oil  on  rough  water,  266 

Opacity,  400 

Open -fire  ventilation,  175 

Opera  glasses,  508 

Ophthalmoscope,  484 

Optical  centre  of  lens,  405 

Optical  instruments,  488 

Optometer,  478 

Orthochromatic  film,  451 

Oscillation  period,  56 

Oscillograph,  722 

Osmometers,  278 

Osmosis,  273,  281 

Osmosis,  electro-,  699 

Osmotic  pressure,  276 

Osmotic  pressure,  theory,  278 

Otoliths,  370 

Overheated  liquid,  199 

Overtones,  346 

Oxygen  of  atmosphere,  799 

Oxygen,  liquid,  217 

Ozone,  measurement  of,  780 

Ozone,  origin  and  action,  733 


Paget,  Sir  R.,  368 

Paint  box,  464 

Pancratic  eyepiece,  509 

Parabolic  mirrors,  467,  492 

Parachute,  25 

Paraffins,  256 

Parallelogram  laws,  19 

Particle,  7,  17 

Pascal,  68,  81 

Pendulum,  29 

Pendulum,  circular  error,  66 

Pendulum,  compensated,  125 


Pendulum,  compound,  60 

Pendulum,  conical,  54 

Pendulum,  isochroniam.  56 

Pendulum,  maator.  115 

Penflulum,  RocondA,  56 

Pendulum,  simplo,  55 

Penumbra,  378 

Perfect  gas,  PV  ^  RT.  146 

Perfect  gas  scale.  146 

Periodic  time,  289 

Periscopes,  512 

Periscopic  lenses,  482 

Permanent  gases,  205 

Permeability,  magnetic.  54K 

Perpetual  motion,  47,  (M)4 

Perrin,  274 

Personal  equation,  117 

Pfeiifor,  276 

pH,  700 

Phase,  289 

Phase  change  on  reflection.  31 1 

Phoneidoscope,  367 

Phonograph,  339 

Phosphorescence,  450,  782 

Photo-electricity,  803 

Photo-electric  cells,  383 

Photographic  lenses,  493 

Photometers,  381 

Photometry,  379 

Photometry,  electric,  805 

Photon,  801 

Piezo-electricity,  652 

Pinhole,  377 

Pinhole  camera,  377 

Pipes,  organ,  etc.,  356 

Pitch  of  screw,  1 1 1 

Planck's  constant,  802 

Plane,  inclined,  49 

Planetary  atmospheres,  448 

Planetary  perturbation,  30 

Planetary  temperatures,  784,  79H 

Planimeter,  1 1 3 

Plasmolysis,  276 

Plasticity,  101 

Plates,  vibration  of,  365 

Platinum  thermometer,  631 

Pleochroic  haloes.  768 

Pointolite  lamp.  730 

Polar  front.  242 

Polariraetr>',  540 

Polarization,  electpolytic.  691» 

Polarized  light,  535 

Polarizing  angle.  536 

Px)le-tinding  pai>or.  694 

Poles,  magnet  ic.  543 

Polish,  wax,  256 

Polygon  of  displaromontH.  7 

Porous  plug  experiment.  210 

Port  of  London,  214 

Positive  ions,  723 


844 


INDEX 


Positron,  771 

Post  office  box,  636 

Potential  difference,  P.D.,  642 

Potential,  electrical,  581 

Potentiometer,  645 

Poundal,  21 

Power,  37 

Power  from  sunshine,  800 

Power,  heritage  of,  799 

Precession,  62 

Presbyopia,  479 

Press,  hydraulic,  69 

Pressure,  67 

Pressure,  absolute,  70 

Pressure  gauges,  76,  79,  81 

Pressure,  normal  atmospheric,  79 

Prevost's  exchanges,  774,  809 

Primary  batteries,  707 

Primary  and  secondary  coils,  667 

Principal  planes,  433 

Principle  of  moments,  42 

Prismatic  binoculars,  509 

Prisms,  393 

Prisms,  thin,  316 

Prisms,  achromatic,  471 

Prisms,  direct -vision,  473 

Prism -dioptres,  482 

Proof  stress,  102 

Propeller  corrosion,  715 

Propeller  thrust,  50 

Proton,  753,  770 

Psychrometer,  sling,  225 

Pulleys,  49 

Pulse,  timing,  56 

Pumps,  air,  72 

Pumps,  centrifugal,  84 

Pumps,  diffusion,  etc.,  73 

Pumps,  reciprocating,  72 

Pumps,  work  done  by,  74 

Pyknometer,  92 

Pyro-electricity,  652 

Pyrometers,  optical,  793 

Pyrometers,  radiation,  790 

Q 

Quality,  or  timbre,  346 
Quanta,  775,  801 
Quantity  of  electricity,  580 
Quantity,  e-s.  and  e-m.,  687 
Quantity  of  heat,  151 
Quartz,  half- wave  plate,  541 
Quartz  oscillator,  116,  364 
Quartz,  piezo -electric,  652 
Quartz  plates,  541,  684 
Quartz  prisms  and  lenses,  537,  538 
Quenched  spark,  684 
Questions  and  solutions,  14 
Quicksand,  67,  91 
Quoits,  64 


R 

Radian,  5,  113 
Radiant  heat,  in  bulk,  784 
Radiant  heat  treatment,  787 
Radiation,  absorption  and  emission, 

784 
Radiation  emission  curves,  792 
Radiation,  great  spectrum,  776 
Radiation,    mechanism    of  emission, 

800 
Radiation  pressure,  790 
Radiation  pyrometers,  790 
Radiation,  rough  separation,  787 
Radiation,  speed  of,  775 
Radiation  and  temperature,  788 
Radiation,  temperature-quality,  785 
Radiation,  transmission,  786 
Radiator,  full,  788 
Radiators,  selective,  795 
Radio-active  equilibrium,  760 
Radio-active  measm-ements,  765 
Radio-active  series,  762 
Radio-activity,  induced,  770 
Radio-activity  of  rocks,  767 
Radiographic  tube,  746 
Radio -luminous  paint,  757 
Radio -transmitter,  684 
Radio-tube  or  valve,  722 

Radium,  757 

Radium  needles,  765 

Radon  (Ra  emanation),  762 

Rain,  236 

Rainband,  447 

Rainbows,  453 

Raindrops,  25,  236 

Range  finder,  512 

Rayleigh  disc,  328 

Rayleigh,  Lord,  767 

Rays  round  light,  477 

Razor  edge,  103 

Reactance,  676 

Reaction,  reactivity,  23 

Recoil  of  giua,  34 

Rectifier,  CuO,  651 

Rectifier,  mercury  vapour,  728 

Rectifiers,  668 

Rectifiers,  electrolytic,  702 

Red  glass,  456 

Red  and  white  heat,  136 

Red  seaweeds,  451 

Reduction  factor,  619 

Reeds,  356 

Reef  analogy,  399,  449 

Reflection,  314,  387 

Reflection,  total,  315,  395,  396 

Reflectors,  rear,  road,  etc.,  468 

Refraction,  314,  389 

Refraction,  atmospheric,  391 

Refraction,  double,  537 


INDEX 


84J 


Refraction  of  sound,  325 
Refraction  at  sphere,  475 
Refractions,  testing,  482 
Refractive  index,  315,  390 
Refractometry,  398 
Refrigeration,  218 
Regelation,  187 
Relativity,  31 
Resistance,  623 
Resistance,  body,  638 
Resistance  coils,  627 
Resistance  comparisons,  634 
Resistance  comparisons,  low,  648 
Resistance,  inductive,  638 
Resistance,  liquid,  636 
Resistance,  vanished,  673 
Resistances,  high  and  low,  639 
Resistivity,  628,  629 
Resistivity,  temperature  coeff.,  629 
Resolving  power,  502,  514,  519,  521, 

522 
Resonance,  294 
Resonance,  acoustic,  349 
Resonance,  spring  apparatus,  295 
Retardation,  10 
Retina,  476 
Retinoscopy,  483 
Rheostats,  628 
Rhone,  26 

Rider  apparatus,  46,  119 
Rigid  body,  7,  99 
Rigidity,  107 
Rim,  tension  in,  57 
River,  action  of,  57 
Rochelle  salt,  652 
Rock,  238,  722 
Rock  salt,  749,  783 
Room,  drying  a,  225 
Rotary  converters,  668 
Rotation,  7,  58 
Rotation,  optical,  540 
Rowing,  49 
Rumford,  177 


Sabouraud  pastille,  748 
SaccharLmeter,  541 
Sailing,  49 
Salinometer,  95 
Sand  blasting,  99 
Sand  dropper,  Kelvin,  576 
Saturated  steam  pressure,  196 
Saturated  vapour,  194 
Saturation  current,  732 
Saturation  fraction,  222 
Scale,  chromatic,  374 
Scale,  diatomic,  373 
Scattering  of  blue  light,  455 
Screw,  50 


Screw  gauges.  111 

Sea,  colour  of,  457 

Sea  miles,  speed  in,  9 

Searchlight,  492 

Second,  3,  56 

Secondary  X  and  cathode  i»y«,  80A 

Seismograph,  303 

Self -recorders,  olectrical,  616 

Sextant,  388 

Shadows,  378 

Shell,  rifle,  64 

Sherbet,  sound  of,  303 

Ship's  load  line,  90 

S.H.M.'s  combined,  291,  294 

Shock,  36 

Shock,  avoidance  of,  215 

Shock,  electric,  740 

Shocking  example,  334 

Shortt  clock,  115 

Shimts,  633 

Silent  areas,  327 

Silica  gel,  268 

Silver,  cleaning,  714 

Simple  harmonic  motion,  S.H.M.,  2SS 

Simple  pendulum,  55 

Sine,  sin,  114 

Sine  curve,  289 

Singing  of  kettle,  198 

Siphon,  71 

Skating,  23 

Skidding,  23 

Sky,  colour  of,  455 

Sky,  midnight  light  of,  459 

Slide  rule,  14 

Smoke,  232,  455 

Snow,  237 

Snow  blindness,  780 

Snowflakes,  25 

Soap  bubbles,  267 

Soil,  capillar}'  action  in,  263 

Solar  constant,  796 

Solar  11 -year  cycle,  663,  796 

Solenoids,  697 

Solute  and  solvent.  279 

Solution  boiling  poinU,  2S4 

Solution  freezing  points,  285 

Solution  pressure,  699 

Solution  vapour  preBsure,  282 

Solutions  to  Questions,  811 

Solvents,  associating,  etc.,  289 

Sonometer,  343 

Sound,  minimum  audible,  328 

Sound,  pitch,  337 

Sound,  prtxiuetion,  318 

Sound  ranging,  323 

Sound, speed,  319 

South-westerlies,  240 

Space,  3 

Spark,  etc.,  733 

Spark  coil,  670 


846 


INDEX 


Spark  potential,  734 

Spark,  shape  of,  737 

Specific  gravity,  92 

Specific  heat,  151 

Specific  inductive  capacity,  S.I.C.,  587 

Spectacles,  480 

Spectra,  442 

Spectra,  absorption,  445 

Spectra,  H,  He,  Hg,  Na,  807 

Spectra,  stellar,  448 

Spectrograph,  Mass,  753 

Spectrograph,  X-ray,  751' 

Spectrometer,  -graph,  441. 

Spectroscopes,  440 

Spectroscope  prisms,  direct-vision,  473 

Spectroscope,  ultra-violet,  778 

Spectrum,  diplogen  and  argon,  808 

Spectrum,  solar,  446 

Spectrum  theory,  806 

Speed,  8 

Sphere,  surface  and  volume,  113 

Spherical  aberration,  466 

Spheroidal  state,  192 

Spherometer,  112,  409 

Sphygmograph,  76 

Sphygmo -manometer,  77 

Spider's  web,  263 

Spinning  tops,  60 

Sprayers,  83 

Spring  weather,  248 

Squall  line,  242 

Squint,  482 

Staffordshire  miners  and  salt,  277 

Stage  micrometer,  518 

Stalloy,  669 

Star  glasses,  501 

Statical  equilibrium,  41 

Statics,  41 

Stationary  wave  motion,  310 

Steam  engine,  75,  214 

Steel,  constituents  of,  104 

Steelyard,  46 

Stefan's  law,  788 

Stereoscope,  482 

Stereoscopic  distance,  389 

Stereoscopic  vision,  477 

Sterilizers,  201 

Stethoscope,  318,  319 

Stick  in  water,  390 

Stratosphere,  235 

Stress  and  strain,  100 

Stress  diagram,  101 

Stress,  proof,  102 

Strings,  341 

Stroboscope,  339 

Sublimation,  193 

Submarine  bell,  321 

Submarine  spotting,  375 

Sugar  vacuum  pans,  201 

Sulphur  cloud,  455 


Summer,  St.  Martin's,  248 

Sun,  3 

Sun,  black  body  temperature,  796 

Sun,  how  it  radiates,  797 

Sun,  irregularity  of,  3 

Sun,  mass  and  density,  30 

Sun,  variable,  563,  796 

Sunburn,  780 

Sunset,  delayed,  392 

Sunset  tints,  456 

Sunshine,  temperature  in  direct,  800 

Sunspots,  511 

Super-conductivity,  674 

Supersaturated  solution,  183 

Supersaturated  vapour,  202,  724 

Supersonics,  375 

Surface  density,  electrical,  584 

Surface  energy,  267 

Surface  tension,  258 

Surface     tension,     and     hydrostatic 

pressure,  260 
Surface  tension,  and  vapour  pressure, 

265 
Sylvine,  783 
Syren,  340 


TABLES,    LISTS   AND    GRAPHS  : 

Altitude,  pressure,  and  B.Pt.,  81 
Atomic  diameters,  750 
Boiling  point  and  pressure,  196 
Change  of  state  data,  190 
Chromatic  dispersion,  471 
Colour  of  hot  iron,  136 
Earth's  Magnetism,  563 
Efficiency  of  engines,  214 
Elastic  constants,  107 
Electrical  conductors,  etc.,  629 
Electro -chemical  equivalents,  696 
Fall  of  drops,  235 
F.Pt.  depression,  285 
Gravity  and  latitude,  29 
Heats  of  combustion,  161 
Heavy  water  properties,  755 
Height  of  clouds,  236 
Horn  harmonics,  360 
Illumination,  383 
Intrinsic  brightness,  491 
Ionic  speeds  in  solution,  693 
Isotonic  solutions,  277 
J,  179 

Liquefaction  of  gases,  206 
Loudness  of  noises,  329 
Magnetic  permeability,  548 
Metric  and  English,  etc.,  5 
Moments  of  Inertia,  59 
Musical  pitch,  374 
Musical  scales,  373 
Optical  rotation,  540 
Para-  and  dia-magnetism,  551 


INDEX 


847 


TABLES,    LISTS    AND    GRAPHS  —  Con- 
tinued : — 
Radiation  and  temperature,  792 
Refractive  indices,  390 
Refrigerated  foodstuffs,  218 
Resistance  of  conductors,  629 
Sound  absorption,  333 
Spark  length  and  potential,  734 
Specific  heats,  155 
Specific  Inductive  Capacity,  587 
Spectra  visual,  443 
Spectra  photographic,  807 
Speed  of  sound,  321,  324 
Stellar  spectral  types,  448 
Surface  tensions,  262 
Temperatures,  very  high,  794 
Thermal  conductivity,  174 
Thermal  expansion,  liquids,  131 
Thermal  expansion,  solids,  125 
Thermometric  fixed  points,  144 
Vapour  pressure  of  water,  1 96 
Viscosities,  251 
Visual  accommodation,  479 
Water-waves,  300 
Wet  and  dry  bulb  hygrometers,  226 

Tangent,  114 

Tangent  galvanometer,  617 
Tanning,  ultra-violet,  779 
Tears  of  wine,  259 
Telephone,  662 
Telephoto  lenses,  511 
Telescopes,  496 
Telescopes,  erect  image,  507 
Telescopes,  focussing,  510 
Television,  723 
Temperament,  374 
Temperature  absolute,  138,  144 
Temperature  indicators,  135 
Temperature  of  hot  iron,  136 
Temperature  of  planets,  798 
Temperature,  radiation,  794 
Temperature  scales,  137 
Temperature,  true  scale,  145 
Tendencies,  13,  19 
Tension  in  hoop,  56 
Theory,  20 
Therm,  151 

Thermal  expansion,  122 
Thermal  unit,  British,  151 
Thermo-couple  and  pile,  650 
Thermo-electricity,  648 
Thermograph,  80,  127 
Thermometer,  Beckmann,  284 
Thermometer,  gas,  142 
Thermometer,  platinum,  631 
Thermometer,  stem  error,  140 
Thermometer  testing,  138 
Thermometers,  various,  141 
Thermometry,  135 


Thermometry,  fixed  points,  144 

Thermostats,  147 

Third  law  of  motion,  23 

Three-colour,  461 

Thundercloud,  236 

Thunderstorm,  736 

Time,  3 

Time  measivement,  1 14 

Tone,  musical,  346 

Tops,  spinning,  60 

Toric  or  toroidal  lenses,  482 

Tornado,  245 

Torpedo,  63,  91 

Torricellian  space,  78 

Torricellian  tube,  81 

Total  reflection,  315 

Tourmaline,  637,  652,  786 

Trade  winds,  240 

Train,  friction,  25 

Transformer,  A.C.,  666 

Translation,  7 

Transmutation  of  elements,  760 

Triangle  of  displacements,  7 

Trihydrol,  132,  185 

Tripod,  48 

Troposphere,  235 

Trough,  245 

Tube  length,  525 

Tug  and  tow,  26 

Tug  of  war,  25 

Tumbler,  ringing,  366 

Tuning-fork,  standard,  116 

Turbines,  86,  214 

Twaddell  degrees,  95 

Type,  383 

U 

Ultramicroscope,  529 
Ultra-violet,  in  atmosphere,  805 
Ultra-violet  fluorescence,  781 
Ultra-violet  glass,  etc.,  779,  781 
Ultra-violet  lamps,  728 
Ultra-violet  microscope,  528 
Ultra-violet  protective  glaaaes,  780" 
Ultra-violet  signalling,  782 
Ultra-violet,  sources  of,  778 
Ultra-violet  spectroscope,  778 
Ultra-violet,  vacuum,  778 
Umbra,  378 
Undercooling,  183 
Undertaker,  239 
Unilens,  508 
Units,  1 
Universe,  790 

Universe  expanding,  3,  448 
Universe,  fate  of,  37 
Unsaturated  vapour,  194 
Uranium  X,  756,  761 
Uranium-lead  ratio,  768 
Uviol,  781 


848 


INDEX 


Vaccine  dialysers,  273 

Vacuum  gauges,  73 

Vacuum  lamps,  73 

Vacuum  pans,  201 

Vacuum  pumps  and  getters,  73 

Van  der  Waals,  210 

Vaporization,  192,  211 

Vapour  pressure  and  temperature,  194 

Vapour,  satiu'ated  and  unsaturated, 

195 
Vapour,  superheated,  202 
Vapour,  supersaturated,  202,  724 
Vaseline,  256 
Vector,  7 

Vectors,  resolution  of,  9 
Velocities,  combining,  8 
Velocities,  parallelogram,  9 
Velocity  ratio,  47 
Ventilators,  84 
Venturi  meter,  83 
Vernier,  110 

Vibration  and  friction,  23 
Vibration  of  membranes,  367 
Vibration  of  pendulum,  56 
Vibration  of  plates,  365 
Vibration  of  rods  and  bars,  364 
Vibration  microscope,  338 
Virtual  work,  47 
Viscosity,  250 

Visibility  and  invisibility,  399 
Visual  accommodation,  478 
Visual  purple,  477 
Vitaglass,  779 
Vitamin  D,  781 
Voice,  367 
Volt,  the,  606,  624 
Volta,  707 

Voltage  measurement,  642 
Voltameters,  697 
Voltmeters,  644 
Voltmeters,  electrostatic,  642 
Volumes,  113 

W 

Waller,  dry  ice  baton,  193,  365 

Warm  front,  242 

Water  equivalent,  153,  154 

Water,  heavy,  132,  775 

Water,  maximum  density,  132 

Water  vapour  in  air,  231 

Waterwheels,  85 

Watt,  38,  656 

Watts  per  c.p.,  658 

Wattmeter,  656 

Wave  charts,  312 


Wave  diffraction,  307 

Wave  energy,  303 

Wave  interference,  305 

Wave  motion,  297 

Wave  motion,  stationary,  309 

Wave  reflection  and  refraction,  309 

Wave,  straight,  306 

Waves,  electromagnetic,  685 

Waves,  e.-m.,  speed  and  length,  687 

Waves,  push  and  shake,  302 

Waves,  water,  298 

Wax  polishing,  256 

Weather,  231 

Weather-glass,  cottage,  79  ' 

Weather  wisdom,  227 

Wedge,  50 

Weighing,  correction  for  air,  120 

Weighing,  double,  etc.,  118 

Weight  and  mass,  21 

Weight  dilatometer,  129 

Wheatstone  bridge,  635 

Whispering  galleries,  325 

Whistles,  West's,  354 

Wien's  laws,  791 

Wilson  expansion  chatnber,  724 

Wimshurst  machine,  577 

Wind,  238,  248 

Wind  and  sound,  326 

Wind  instruments,  360 

Work,  34 

Work  of  elastic  stretching,  107 

Work  of  heart,  74 

Work,  virtual,  47 

Wound,  healing,  12 


X  radiation,  742 

X-ray  dosage  and  erythema,  748 

X-ray  gas  tubes,  743 

X-ray  modern  apparatus,  745 

X-ray  spectrograph,  751 

X-ray  wave-length,  744,  750 

X-rays,  diffraction  by  crystal,    749, 

751 
X-rays,  origin  and  effects,  743  , 


Year  3 

Yellow,  460,  464 
Yield  point,  67,  101 
Young's  modulus,  100,  107 


Zero,  absolute,  217 
Zinc  blende,  757,  769 
Zircon,  769 


Printed  in  Great  Britain  by  Richard  Clay  db  Sons,  Limited,  Bungay,  Suffoli. 


^1 


PLEASE  DO  NOT  REMOVE 
CARDS  OR  SUPS  PROM  THIS  POCKET 


UNIVERSITY  OF  TORONTO  LIBRARY